function abram0 ( xvalue ) !*****************************************************************************80 ! !! ABRAM0 evaluates the Abramowitz function of order 0. ! ! Discussion: ! ! The function is defined by: ! ! ABRAM0(x) = Integral ( 0 <= t < infinity ) exp ( -t^2 - x / t ) dt ! ! The code uses Chebyshev expansions with the coefficients ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) ABRAM0, the value of the function. ! implicit none real ( kind = 8 ), dimension(0:8) :: ab0f = (/ & -0.68121927093549469816d0, & -0.78867919816149252495d0, & 0.5121581776818819543d-1, & -0.71092352894541296d-3, & 0.368681808504287d-5, & -0.917832337237d-8, & 0.1270202563d-10, & -0.1076888d-13, & 0.599d-17 /) real ( kind = 8 ), dimension(0:8) :: ab0g = (/ & -0.60506039430868273190d0, & -0.41950398163201779803d0, & 0.1703265125190370333d-1, & -0.16938917842491397d-3, & 0.67638089519710d-6, & -0.135723636255d-8, & 0.156297065d-11, & -0.112887d-14, & 0.55d-18 /) real ( kind = 8 ), dimension(0:8) :: ab0h = (/ & 1.38202655230574989705d0, & -0.30097929073974904355d0, & 0.794288809364887241d-2, & -0.6431910276847563d-4, & 0.22549830684374d-6, & -0.41220966195d-9, & 0.44185282d-12, & -0.30123d-15, & 0.14d-18 /) real ( kind = 8 ), dimension(0:27) :: ab0as = (/ & 1.97755499723693067407d+0, & -0.1046024792004819485d-1, & 0.69680790253625366d-3, & -0.5898298299996599d-4, & 0.577164455305320d-5, & -0.61523013365756d-6, & 0.6785396884767d-7, & -0.723062537907d-8, & 0.63306627365d-9, & -0.989453793d-11, & -0.1681980530d-10, & 0.673799551d-11, & -0.200997939d-11, & 0.54055903d-12, & -0.13816679d-12, & 0.3422205d-13, & -0.826686d-14, & 0.194566d-14, & -0.44268d-15, & 0.9562d-16, & -0.1883d-16, & 0.301d-17, & -0.19d-18, & -0.14d-18, & 0.11d-18, & -0.4d-19, & 0.2d-19, & -0.1d-19 /) real ( kind = 8 ) abram0 real ( kind = 8 ) asln real ( kind = 8 ) asval real ( kind = 8 ) cheval real ( kind = 8 ) fval real ( kind = 8 ) gval real ( kind = 8 ), parameter :: gval0 = 0.13417650264770070909D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 real ( kind = 8 ) hval real ( kind = 8 ), parameter :: lnxmin = -708.3964D+00 integer ( kind = 4 ), parameter :: nterma = 22 integer ( kind = 4 ), parameter :: ntermf = 8 integer ( kind = 4 ), parameter :: ntermg = 8 integer ( kind = 4 ), parameter :: ntermh = 8 real ( kind = 8 ), parameter :: onerpi = 0.56418958354775628695D+00 real ( kind = 8 ), parameter :: rt3bpi = 0.97720502380583984317D+00 real ( kind = 8 ), parameter :: rtpib2 = 0.88622692545275801365D+00 real ( kind = 8 ), parameter :: six = 6.0D+00 real ( kind = 8 ) t real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) v real ( kind = 8 ) x real ( kind = 8 ), parameter :: xlow1 = 1.490116D-08 real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'ABRAM0 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' abram0 = zero else if ( x == zero ) then abram0 = rtpib2 else if ( x < xlow1 ) then abram0 = rtpib2 + x * ( log ( x ) - gval0 ) else if ( x <= two ) then t = ( x * x / two - half ) - half fval = cheval ( ntermf, ab0f, t ) gval = cheval ( ntermg, ab0g, t ) hval = cheval ( ntermh, ab0h, t ) abram0 = fval / onerpi + x * ( log ( x ) * hval - gval ) else v = three * ( ( x / two ) ** ( two / three ) ) t = ( six / v - half ) - half asval = cheval ( nterma, ab0as, t ) asln = log ( asval / rt3bpi ) - v if ( asln < lnxmin ) then abram0 = zero else abram0 = exp ( asln ) end if end if return end subroutine abram0_values ( n_data, x, fx ) !*****************************************************************************80 ! !! ABRAM0_VALUES returns some values of the Abramowitz0 function. ! ! Discussion: ! ! The function is defined by: ! ! ABRAM0(x) = Integral ( 0 <= t < infinity ) exp ( -t^2 - x / t ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 21 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.87377726306985360531D+00, & 0.84721859650456925922D+00, & 0.77288934483988301615D+00, & 0.59684345853450151603D+00, & 0.29871735283675888392D+00, & 0.15004596450516388138D+00, & 0.11114662419157955096D+00, & 0.83909567153151897766D-01, & 0.56552321717943417515D-01, & 0.49876496603033790206D-01, & 0.44100889219762791328D-01, & 0.19738535180254062496D-01, & 0.86193088287161479900D-02, & 0.40224788162540127227D-02, & 0.19718658458164884826D-02, & 0.10045868340133538505D-02, & 0.15726917263304498649D-03, & 0.10352666912350263437D-04, & 0.91229759190956745069D-06, & 0.25628287737952698742D-09 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0078125000D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.2500000000D+00, & 1.5000000000D+00, & 1.8750000000D+00, & 2.0000000000D+00, & 2.1250000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 5.0000000000D+00, & 6.0000000000D+00, & 7.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 40.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function abram1 ( xvalue ) !*****************************************************************************80 ! !! ABRAM1 evaluates the Abramowitz function of order 1. ! ! Discussion: ! ! The function is defined by: ! ! ABRAM1(x) = Integral ( 0 <= t < infinity ) t * exp ( -t^2 - x / t ) dt ! ! The code uses Chebyshev expansions with the coefficients ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) ABRAM1, the value of the function. ! implicit none real ( kind = 8 ) ab1as(0:27) real ( kind = 8 ) ab1f(0:9) real ( kind = 8 ) ab1g(0:8) real ( kind = 8 ) ab1h(0:8) real ( kind = 8 ) abram1 real ( kind = 8 ) asln real ( kind = 8 ) asval real ( kind = 8 ) cheval real ( kind = 8 ) fval real ( kind = 8 ) gval real ( kind = 8 ), parameter :: half = 0.5D+00 real ( kind = 8 ) hval real ( kind = 8 ) lnxmin integer ( kind = 4 ), parameter :: nterma = 23 integer ( kind = 4 ), parameter :: ntermf = 9 integer ( kind = 4 ), parameter :: ntermg = 8 integer ( kind = 4 ), parameter :: ntermh = 8 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) onerpi real ( kind = 8 ), parameter :: rt3bpi = 0.97720502380583984317D+00 real ( kind = 8 ), parameter :: six = 6.0D+00 real ( kind = 8 ) t real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) v real ( kind = 8 ) x real ( kind = 8 ) xlow real ( kind = 8 ) xlow1 real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 data ab1f/1.47285192577978807369d0, & 0.10903497570168956257d0, & -0.12430675360056569753d0, & 0.306197946853493315d-2, & -0.2218410323076511d-4, & 0.6989978834451d-7, & -0.11597076444d-9, & 0.11389776d-12, & -0.7173d-16, & 0.3d-19/ data ab1g/0.39791277949054503528d0, & -0.29045285226454720849d0, & 0.1048784695465363504d-1, & -0.10249869522691336d-3, & 0.41150279399110d-6, & -0.83652638940d-9, & 0.97862595d-12, & -0.71868d-15, & 0.35d-18/ data ab1h/0.84150292152274947030d0, & -0.7790050698774143395d-1, & 0.133992455878390993d-2, & -0.808503907152788d-5, & 0.2261858281728d-7, & -0.3441395838d-10, & 0.3159858d-13, & -0.1884d-16, & 0.1d-19/ data ab1as(0)/ 2.13013643429065549448d0/ data ab1as(1)/ 0.6371526795218539933d-1/ data ab1as(2)/ -0.129334917477510647d-2/ data ab1as(3)/ 0.5678328753228265d-4/ data ab1as(4)/ -0.279434939177646d-5/ data ab1as(5)/ 0.5600214736787d-7/ data ab1as(6)/ 0.2392009242798d-7/ data ab1as(7)/ -0.750984865009d-8/ data ab1as(8)/ 0.173015330776d-8/ data ab1as(9)/ -0.36648877955d-9/ data ab1as(10)/ 0.7520758307d-10/ data ab1as(11)/-0.1517990208d-10/ data ab1as(12)/ 0.301713710d-11/ data ab1as(13)/-0.58596718d-12/ data ab1as(14)/ 0.10914455d-12/ data ab1as(15)/-0.1870536d-13/ data ab1as(16)/ 0.262542d-14/ data ab1as(17)/-0.14627d-15/ data ab1as(18)/-0.9500d-16/ data ab1as(19)/ 0.5873d-16/ data ab1as(20)/-0.2420d-16/ data ab1as(21)/ 0.868d-17/ data ab1as(22)/-0.290d-17/ data ab1as(23)/ 0.93d-18/ data ab1as(24)/-0.29d-18/ data ab1as(25)/ 0.9d-19/ data ab1as(26)/-0.3d-19/ data ab1as(27)/ 0.1d-19/ data onerpi/ 0.56418958354775628695d0/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow,xlow1,lnxmin/1.11023d-16,1.490116d-8,-708.3964d0/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'ABRAM1 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' abram1 = zero else if ( x == zero ) then abram1 = half else if ( x < xlow ) then abram1 = half else if ( x < xlow1 ) then abram1 = ( one - x / onerpi - x * x * log ( x ) ) * half else if ( x <= two ) then t = ( x * x / two - half ) - half fval = cheval ( ntermf, ab1f, t ) gval = cheval ( ntermg, ab1g, t ) hval = cheval ( ntermh, ab1h, t ) abram1 = fval - x * ( gval / onerpi + x * log ( x ) * hval ) else v = three * ( ( x / two ) ** ( two / three ) ) t = ( six / v - half ) - half asval = cheval ( nterma, ab1as, t ) asln = log ( asval * sqrt ( v / three ) / rt3bpi ) - v if ( asln < lnxmin ) then abram1 = zero else abram1 = exp ( asln ) end if end if return end subroutine abram1_values ( n_data, x, fx ) !*****************************************************************************80 ! !! ABRAM1_VALUES returns some values of the Abramowitz1 function. ! ! Discussion: ! ! The function is defined by: ! ! ABRAM1(x) = Integral ( 0 <= t < infinity ) t * exp ( -t^2 - x / t ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 21 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.49828219848799921792D+00, & 0.49324391773047288556D+00, & 0.47431612784691234649D+00, & 0.41095983258760410149D+00, & 0.25317617388227035867D+00, & 0.14656338138597777543D+00, & 0.11421547056018366587D+00, & 0.90026307383483764795D-01, & 0.64088214170742303375D-01, & 0.57446614314166191085D-01, & 0.51581624564800730959D-01, & 0.25263719555776416016D-01, & 0.11930803330196594536D-01, & 0.59270542280915272465D-02, & 0.30609215358017829567D-02, & 0.16307382136979552833D-02, & 0.28371851916959455295D-03, & 0.21122150121323238154D-04, & 0.20344578892601627337D-05, & 0.71116517236209642290D-09 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0078125000D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.2500000000D+00, & 1.5000000000D+00, & 1.8750000000D+00, & 2.0000000000D+00, & 2.1250000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 5.0000000000D+00, & 6.0000000000D+00, & 7.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 40.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function abram2 ( xvalue ) !*****************************************************************************80 ! !! ABRAM2 evaluates the Abramowitz function of order 2. ! ! Discussion: ! ! The function is defined by: ! ! ABRAM2(x) = Integral ( 0 <= t < infinity ) t^2 * exp ( -t^2 - x / t ) dt ! ! The code uses Chebyshev expansions with the coefficients ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) ABRAM2, the value of the function. ! implicit none real ( kind = 8 ) abram2 real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ), parameter :: nterma = 23 integer ( kind = 4 ), parameter :: ntermf = 9 integer ( kind = 4 ), parameter :: ntermg = 8 integer ( kind = 4 ), parameter :: ntermh = 7 real ( kind = 8 ), parameter :: six = 6.0D+00 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) ab2f(0:9),ab2g(0:8),ab2h(0:7),ab2as(0:26), & asln,asval,fval,gval,hval,lnxmin, & onerpi,rtpib4,rt3bpi,t, & v,xlow,xlow1 data ab2f/1.03612162804243713846d0, & 0.19371246626794570012d0, & -0.7258758839233007378d-1, & 0.174790590864327399d-2, & -0.1281223233756549d-4, & 0.4115018153651d-7, & -0.6971047256d-10, & 0.6990183d-13, & -0.4492d-16, & 0.2d-19/ data ab2g/1.46290157198630741150d0, & 0.20189466883154014317d0, & -0.2908292087997129022d-1, & 0.47061049035270050d-3, & -0.257922080359333d-5, & 0.656133712946d-8, & -0.914110203d-11, & 0.774276d-14, & -0.429d-17/ data ab2h/0.30117225010910488881d0, & -0.1588667818317623783d-1, & 0.19295936935584526d-3, & -0.90199587849300d-6, & 0.206105041837d-8, & -0.265111806d-11, & 0.210864d-14, & -0.111d-17/ data ab2as(0)/ 2.46492325304334856893d0/ data ab2as(1)/ 0.23142797422248905432d0/ data ab2as(2)/ -0.94068173010085773d-3/ data ab2as(3)/ 0.8290270038089733d-4/ data ab2as(4)/ -0.883894704245866d-5/ data ab2as(5)/ 0.106638543567985d-5/ data ab2as(6)/ -0.13991128538529d-6/ data ab2as(7)/ 0.1939793208445d-7/ data ab2as(8)/ -0.277049938375d-8/ data ab2as(9)/ 0.39590687186d-9/ data ab2as(10)/-0.5408354342d-10/ data ab2as(11)/ 0.635546076d-11/ data ab2as(12)/-0.38461613d-12/ data ab2as(13)/-0.11696067d-12/ data ab2as(14)/ 0.6896671d-13/ data ab2as(15)/-0.2503113d-13/ data ab2as(16)/ 0.785586d-14/ data ab2as(17)/-0.230334d-14/ data ab2as(18)/ 0.64914d-15/ data ab2as(19)/-0.17797d-15/ data ab2as(20)/ 0.4766d-16/ data ab2as(21)/-0.1246d-16/ data ab2as(22)/ 0.316d-17/ data ab2as(23)/-0.77d-18/ data ab2as(24)/ 0.18d-18/ data ab2as(25)/-0.4d-19/ data ab2as(26)/ 0.1d-19/ data rt3bpi/ 0.97720502380583984317d0/ data rtpib4/ 0.44311346272637900682d0/ data onerpi/ 0.56418958354775628695d0/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow,xlow1,lnxmin/2.22045d-16,1.490116d-8,-708.3964d0/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'ABRAM2 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' abram2 = zero else if ( x == zero ) then abram2 = rtpib4 else if ( x < xlow ) then abram2 = rtpib4 else if ( x < xlow1 ) then abram2 = rtpib4 - half * x + x * x * x * log ( x ) / six else if ( x <= 2.0D+00 ) then t = ( x * x / two - half ) - half fval = cheval ( ntermf, ab2f, t ) gval = cheval ( ntermg, ab2g, t ) hval = cheval ( ntermh, ab2h, t ) abram2 = fval / onerpi + x * ( x * x * log ( x ) * hval - gval ) else v = three * ( ( x / two ) ** ( two / three ) ) t = ( six / v - half ) - half asval = cheval ( nterma, ab2as, t ) asln = log ( asval / rt3bpi ) + log ( v / three ) - v if ( asln < lnxmin ) then abram2 = zero else abram2 = exp ( asln ) end if end if return end subroutine abram2_values ( n_data, x, fx ) !*****************************************************************************80 ! !! ABRAM2_VALUES returns some values of the Abramowitz2 function. ! ! Discussion: ! ! The function is defined by: ! ! ABRAM2(x) = Integral ( 0 <= t < infinity ) t^2 * exp ( -t^2 - x / t ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 22 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.44213858162107913430D+00, & 0.43923379545684026308D+00, & 0.42789857297092602234D+00, & 0.38652825661854504406D+00, & 0.26538204413231368110D+00, & 0.16848734838334595000D+00, & 0.13609200032513227112D+00, & 0.11070330027727917352D+00, & 0.82126019995530382267D-01, & 0.74538781999594581763D-01, & 0.67732034377612811390D-01, & 0.35641808698811851022D-01, & 0.17956589956618269083D-01, & 0.94058737143575370625D-02, & 0.50809356204299213556D-02, & 0.28149565414209719359D-02, & 0.53808696422559303431D-03, & 0.44821756380146327259D-04, & 0.46890678427324100410D-05, & 0.20161544850996420504D-08 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0078125000D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.2500000000D+00, & 1.5000000000D+00, & 1.8750000000D+00, & 2.0000000000D+00, & 2.1250000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 5.0000000000D+00, & 6.0000000000D+00, & 7.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 40.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function airy_ai_int ( xvalue ) !*****************************************************************************80 ! !! AIRY_AI_INT calculates the integral of the Airy function Ai. ! ! Discussion: ! ! The function is defined by: ! ! AIRY_AI_INT(x) = Integral ( 0 <= t <= x ) Ai(t) dt ! ! The program uses Chebyshev expansions, the coefficients of which ! are given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) AIRY_AI_INT, the value of the function. ! implicit none real ( kind = 8 ) aaint1(0:25) real ( kind = 8 ) aaint2(0:21) real ( kind = 8 ) aaint3(0:40) real ( kind = 8 ) aaint4(0:17) real ( kind = 8 ) aaint5(0:17) real ( kind = 8 ) airy_ai_int real ( kind = 8 ) airzer real ( kind = 8 ) arg real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ) forty1 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ) fr996 real ( kind = 8 ) gval real ( kind = 8 ) hval real ( kind = 8 ) nine real ( kind = 8 ) ninhun integer ( kind = 4 ), parameter :: nterm1 = 22 integer ( kind = 4 ), parameter :: nterm2 = 17 integer ( kind = 4 ), parameter :: nterm3 = 37 integer ( kind = 4 ) nterm4 integer ( kind = 4 ) nterm5 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) piby4 real ( kind = 8 ) pitim6 real ( kind = 8 ) rt2b3p real ( kind = 8 ) t real ( kind = 8 ) temp real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xhigh1 real ( kind = 8 ) xlow1 real ( kind = 8 ) xneg1 real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) z data aaint1(0)/ 0.37713517694683695526d0/ data aaint1(1)/ -0.13318868432407947431d0/ data aaint1(2)/ 0.3152497374782884809d-1/ data aaint1(3)/ -0.318543076436574077d-2/ data aaint1(4)/ -0.87398764698621915d-3/ data aaint1(5)/ 0.46699497655396971d-3/ data aaint1(6)/ -0.9544936738983692d-4/ data aaint1(7)/ 0.542705687156716d-5/ data aaint1(8)/ 0.239496406252188d-5/ data aaint1(9)/ -0.75690270205649d-6/ data aaint1(10)/ 0.9050138584518d-7/ data aaint1(11)/ 0.320529456043d-8/ data aaint1(12)/-0.303825536444d-8/ data aaint1(13)/ 0.48900118596d-9/ data aaint1(14)/-0.1839820572d-10/ data aaint1(15)/-0.711247519d-11/ data aaint1(16)/ 0.151774419d-11/ data aaint1(17)/-0.10801922d-12/ data aaint1(18)/-0.963542d-14/ data aaint1(19)/ 0.313425d-14/ data aaint1(20)/-0.29446d-15/ data aaint1(21)/-0.477d-17/ data aaint1(22)/ 0.461d-17/ data aaint1(23)/-0.53d-18/ data aaint1(24)/ 0.1d-19/ data aaint1(25)/ 0.1d-19/ data aaint2(0)/ 1.92002524081984009769d0/ data aaint2(1)/ -0.4220049417256287021d-1/ data aaint2(2)/ -0.239457722965939223d-2/ data aaint2(3)/ -0.19564070483352971d-3/ data aaint2(4)/ -0.1547252891056112d-4/ data aaint2(5)/ -0.140490186137889d-5/ data aaint2(6)/ -0.12128014271367d-6/ data aaint2(7)/ -0.1179186050192d-7/ data aaint2(8)/ -0.104315578788d-8/ data aaint2(9)/ -0.10908209293d-9/ data aaint2(10)/-0.929633045d-11/ data aaint2(11)/-0.110946520d-11/ data aaint2(12)/-0.7816483d-13/ data aaint2(13)/-0.1319661d-13/ data aaint2(14)/-0.36823d-15/ data aaint2(15)/-0.21505d-15/ data aaint2(16)/ 0.1238d-16/ data aaint2(17)/-0.557d-17/ data aaint2(18)/ 0.84d-18/ data aaint2(19)/-0.21d-18/ data aaint2(20)/ 0.4d-19/ data aaint2(21)/-0.1d-19/ data aaint3(0)/ 0.47985893264791052053d0/ data aaint3(1)/ -0.19272375126169608863d0/ data aaint3(2)/ 0.2051154129525428189d-1/ data aaint3(3)/ 0.6332000070732488786d-1/ data aaint3(4)/ -0.5093322261845754082d-1/ data aaint3(5)/ 0.1284424078661663016d-1/ data aaint3(6)/ 0.2760137088989479413d-1/ data aaint3(7)/ -0.1547066673866649507d-1/ data aaint3(8)/ -0.1496864655389316026d-1/ data aaint3(9)/ 0.336617614173574541d-2/ data aaint3(10)/ 0.530851163518892985d-2/ data aaint3(11)/ 0.41371226458555081d-3/ data aaint3(12)/-0.102490579926726266d-2/ data aaint3(13)/-0.32508221672025853d-3/ data aaint3(14)/ 0.8608660957169213d-4/ data aaint3(15)/ 0.6671367298120775d-4/ data aaint3(16)/ 0.449205999318095d-5/ data aaint3(17)/-0.670427230958249d-5/ data aaint3(18)/-0.196636570085009d-5/ data aaint3(19)/ 0.22229677407226d-6/ data aaint3(20)/ 0.22332222949137d-6/ data aaint3(21)/ 0.2803313766457d-7/ data aaint3(22)/-0.1155651663619d-7/ data aaint3(23)/-0.433069821736d-8/ data aaint3(24)/-0.6227777938d-10/ data aaint3(25)/ 0.26432664903d-9/ data aaint3(26)/ 0.5333881114d-10/ data aaint3(27)/-0.522957269d-11/ data aaint3(28)/-0.382229283d-11/ data aaint3(29)/-0.40958233d-12/ data aaint3(30)/ 0.11515622d-12/ data aaint3(31)/ 0.3875766d-13/ data aaint3(32)/ 0.140283d-14/ data aaint3(33)/-0.141526d-14/ data aaint3(34)/-0.28746d-15/ data aaint3(35)/ 0.923d-17/ data aaint3(36)/ 0.1224d-16/ data aaint3(37)/ 0.157d-17/ data aaint3(38)/-0.19d-18/ data aaint3(39)/-0.8d-19/ data aaint3(40)/-0.1d-19/ data aaint4/1.99653305828522730048d0, & -0.187541177605417759d-2, & -0.15377536280305750d-3, & -0.1283112967682349d-4, & -0.108128481964162d-5, & -0.9182131174057d-7, & -0.784160590960d-8, & -0.67292453878d-9, & -0.5796325198d-10, & -0.501040991d-11, & -0.43420222d-12, & -0.3774305d-13, & -0.328473d-14, & -0.28700d-15, & -0.2502d-16, & -0.220d-17, & -0.19d-18, & -0.2d-19/ data aaint5/1.13024602034465716133d0, & -0.464718064639872334d-2, & -0.35137413382693203d-3, & -0.2768117872545185d-4, & -0.222057452558107d-5, & -0.18089142365974d-6, & -0.1487613383373d-7, & -0.123515388168d-8, & -0.10310104257d-9, & -0.867493013d-11, & -0.73080054d-12, & -0.6223561d-13, & -0.525128d-14, & -0.45677d-15, & -0.3748d-16, & -0.356d-17, & -0.23d-18, & -0.4d-19/ data nine,forty1/ 9.0d0, 41.0d0/ data ninhun,fr996/ 900.0d0, 4996.0d0 / data piby4/0.78539816339744830962d0/ data pitim6/18.84955592153875943078d0/ data rt2b3p/0.46065886596178063902d0/ data airzer/0.35502805388781723926d0/ ! ! Machine-dependant constants (suitable for IEEE machines) ! data nterm4,nterm5/15,15/ data xlow1,xhigh1,xneg1/2.22045d-16,14.480884d0,-2.727134d10/ x = xvalue if ( x < xneg1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'AIRY_AI_INT - Fatal error!' write ( *, '(a)' ) ' X too negative for accurate computation.' airy_ai_int = -two / three return else if ( x < -eight ) then z = - ( x + x ) * sqrt ( -x ) / three arg = z + piby4 temp = nine * z * z t = ( fr996 - temp ) / ( ninhun + temp ) gval = cheval ( nterm4, aaint4, t ) hval = cheval ( nterm5, aaint5, t ) temp = gval * cos ( arg ) + hval * sin ( arg ) / z airy_ai_int = rt2b3p * temp / sqrt ( z ) - two / three else if ( x <= -xlow1 )then t = -x / four - one airy_ai_int = x * cheval ( nterm3, aaint3, t ) else if ( x < xlow1 ) then airy_ai_int = airzer * x else if ( x <= four ) then t = x / two - one airy_ai_int = cheval ( nterm1, aaint1, t ) * x else if ( x <= xhigh1 ) then z = ( x + x ) * sqrt ( x ) / three temp = three * z t = ( forty1 - temp ) / ( nine + temp ) temp = exp ( -z ) * cheval ( nterm2, aaint2, t ) / sqrt ( pitim6 * z ) airy_ai_int = one / three - temp else airy_ai_int = one / three end if return end subroutine airy_ai_int_values ( n_data, x, fx ) !*****************************************************************************80 ! !! AIRY_AI_INT_VALUES returns some values of the integral of the Airy function. ! ! Discussion: ! ! The function is defined by: ! ! AIRY_AI_INT(x) = Integral ( 0 <= t <= x ) Ai(t) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 22 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & -0.75228838916610124300D+00, & -0.57348350185854889466D+00, & -0.76569840313421291743D+00, & -0.65181015505382467421D+00, & -0.55881974894471876922D+00, & -0.56902352870716815309D+00, & -0.47800749642926168100D+00, & -0.46567398346706861416D+00, & -0.96783140945618013679D-01, & -0.34683049857035607494D-03, & 0.34658366917927930790D-03, & 0.27657581846051227124D-02, & 0.14595330491185717833D+00, & 0.23631734191710977960D+00, & 0.33289264538612212697D+00, & 0.33318759129779422976D+00, & 0.33332945170523851439D+00, & 0.33333331724248357420D+00, & 0.33333333329916901594D+00, & 0.33333333333329380187D+00 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & -12.0000000000D+00, & -11.0000000000D+00, & -10.0000000000D+00, & -9.5000000000D+00, & -9.0000000000D+00, & -6.5000000000D+00, & -4.0000000000D+00, & -1.0000000000D+00, & -0.2500000000D+00, & -0.0009765625D+00, & 0.0009765625D+00, & 0.0078125000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 4.0000000000D+00, & 4.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 12.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function airy_bi_int ( xvalue ) !*****************************************************************************80 ! !! AIRY_BI_INT calculates the integral of the Airy function Bi. ! ! Discussion: ! ! The function is defined by: ! ! AIRY_BI_INT(x) = Integral ( 0 <= t <= x ) Bi(t) dt ! ! The program uses Chebyshev expansions, the coefficients of which ! are given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) AIRY_BI_INT, the value of the function. ! implicit none real ( kind = 8 ) abint1(0:36) real ( kind = 8 ) abint2(0:37) real ( kind = 8 ) abint3(0:37) real ( kind = 8 ) abint4(0:20) real ( kind = 8 ) abint5(0:20) real ( kind = 8 ) airy_bi_int real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 integer ( kind = 4 ), parameter :: nterm1 = 33 integer ( kind = 4 ), parameter :: nterm2 = 30 integer ( kind = 4 ), parameter :: nterm3 = 34 integer ( kind = 4 ) nterm4 integer ( kind = 4 ) nterm5 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) arg,birzer,f1,f2,nine,ninhun, & onept5,piby4,rt2b3p,sixten,seven,t,temp, & thr644,xlow1,xhigh1,xmax,xneg1, & z data abint1(0)/ 0.38683352445038543350d0/ data abint1(1)/ -0.8823213550888908821d-1/ data abint1(2)/ 0.21463937440355429239d0/ data abint1(3)/ -0.4205347375891315126d-1/ data abint1(4)/ 0.5932422547496086771d-1/ data abint1(5)/ -0.840787081124270210d-2/ data abint1(6)/ 0.871824772778487955d-2/ data abint1(7)/ -0.12191600199613455d-3/ data abint1(8)/ 0.44024821786023234d-3/ data abint1(9)/ 0.27894686666386678d-3/ data abint1(10)/-0.7052804689785537d-4/ data abint1(11)/ 0.5901080066770100d-4/ data abint1(12)/-0.1370862587982142d-4/ data abint1(13)/ 0.505962573749073d-5/ data abint1(14)/-0.51598837766735d-6/ data abint1(15)/ 0.397511312349d-8/ data abint1(16)/ 0.9524985978055d-7/ data abint1(17)/-0.3681435887321d-7/ data abint1(18)/ 0.1248391688136d-7/ data abint1(19)/-0.249097619137d-8/ data abint1(20)/ 0.31775245551d-9/ data abint1(21)/ 0.5434365270d-10/ data abint1(22)/-0.4024566915d-10/ data abint1(23)/ 0.1393855527d-10/ data abint1(24)/-0.303817509d-11/ data abint1(25)/ 0.40809511d-12/ data abint1(26)/ 0.1634116d-13/ data abint1(27)/-0.2683809d-13/ data abint1(28)/ 0.896641d-14/ data abint1(29)/-0.183089d-14/ data abint1(30)/ 0.21333d-15/ data abint1(31)/ 0.1108d-16/ data abint1(32)/-0.1276d-16/ data abint1(33)/ 0.363d-17/ data abint1(34)/-0.62d-18/ data abint1(35)/ 0.5d-19/ data abint1(36)/ 0.1d-19/ data abint2(0)/ 2.04122078602516135181d0/ data abint2(1)/ 0.2124133918621221230d-1/ data abint2(2)/ 0.66617599766706276d-3/ data abint2(3)/ 0.3842047982808254d-4/ data abint2(4)/ 0.362310366020439d-5/ data abint2(5)/ 0.50351990115074d-6/ data abint2(6)/ 0.7961648702253d-7/ data abint2(7)/ 0.717808442336d-8/ data abint2(8)/ -0.267770159104d-8/ data abint2(9)/ -0.168489514699d-8/ data abint2(10)/-0.36811757255d-9/ data abint2(11)/ 0.4757128727d-10/ data abint2(12)/ 0.5263621945d-10/ data abint2(13)/ 0.778973500d-11/ data abint2(14)/-0.460546143d-11/ data abint2(15)/-0.183433736d-11/ data abint2(16)/ 0.32191249d-12/ data abint2(17)/ 0.29352060d-12/ data abint2(18)/-0.1657935d-13/ data abint2(19)/-0.4483808d-13/ data abint2(20)/ 0.27907d-15/ data abint2(21)/ 0.711921d-14/ data abint2(22)/-0.1042d-16/ data abint2(23)/-0.119591d-14/ data abint2(24)/ 0.4606d-16/ data abint2(25)/ 0.20884d-15/ data abint2(26)/-0.2416d-16/ data abint2(27)/-0.3638d-16/ data abint2(28)/ 0.863d-17/ data abint2(29)/ 0.591d-17/ data abint2(30)/-0.256d-17/ data abint2(31)/-0.77d-18/ data abint2(32)/ 0.66d-18/ data abint2(33)/ 0.3d-19/ data abint2(34)/-0.15d-18/ data abint2(35)/ 0.2d-19/ data abint2(36)/ 0.3d-19/ data abint2(37)/-0.1d-19/ data abint3(0)/ 0.31076961598640349251d0/ data abint3(1)/ -0.27528845887452542718d0/ data abint3(2)/ 0.17355965706136543928d0/ data abint3(3)/ -0.5544017909492843130d-1/ data abint3(4)/ -0.2251265478295950941d-1/ data abint3(5)/ 0.4107347447812521894d-1/ data abint3(6)/ 0.984761275464262480d-2/ data abint3(7)/ -0.1555618141666041932d-1/ data abint3(8)/ -0.560871870730279234d-2/ data abint3(9)/ 0.246017783322230475d-2/ data abint3(10)/ 0.165740392292336978d-2/ data abint3(11)/-0.3277587501435402d-4/ data abint3(12)/-0.24434680860514925d-3/ data abint3(13)/-0.5035305196152321d-4/ data abint3(14)/ 0.1630264722247854d-4/ data abint3(15)/ 0.851914057780934d-5/ data abint3(16)/ 0.29790363004664d-6/ data abint3(17)/-0.64389707896401d-6/ data abint3(18)/-0.15046988145803d-6/ data abint3(19)/ 0.1587013535823d-7/ data abint3(20)/ 0.1276766299622d-7/ data abint3(21)/ 0.140578534199d-8/ data abint3(22)/-0.46564739741d-9/ data abint3(23)/-0.15682748791d-9/ data abint3(24)/-0.403893560d-11/ data abint3(25)/ 0.666708192d-11/ data abint3(26)/ 0.128869380d-11/ data abint3(27)/-0.6968663d-13/ data abint3(28)/-0.6254319d-13/ data abint3(29)/-0.718392d-14/ data abint3(30)/ 0.115296d-14/ data abint3(31)/ 0.42276d-15/ data abint3(32)/ 0.2493d-16/ data abint3(33)/-0.971d-17/ data abint3(34)/-0.216d-17/ data abint3(35)/-0.2d-19/ data abint3(36)/ 0.6d-19/ data abint3(37)/ 0.1d-19/ data abint4(0)/ 1.99507959313352047614d0/ data abint4(1)/ -0.273736375970692738d-2/ data abint4(2)/ -0.30897113081285850d-3/ data abint4(3)/ -0.3550101982798577d-4/ data abint4(4)/ -0.412179271520133d-5/ data abint4(5)/ -0.48235892316833d-6/ data abint4(6)/ -0.5678730727927d-7/ data abint4(7)/ -0.671874810365d-8/ data abint4(8)/ -0.79811649857d-9/ data abint4(9)/ -0.9514271478d-10/ data abint4(10)/-0.1137468966d-10/ data abint4(11)/-0.136359969d-11/ data abint4(12)/-0.16381418d-12/ data abint4(13)/-0.1972575d-13/ data abint4(14)/-0.237844d-14/ data abint4(15)/-0.28752d-15/ data abint4(16)/-0.3475d-16/ data abint4(17)/-0.422d-17/ data abint4(18)/-0.51d-18/ data abint4(19)/-0.6d-19/ data abint4(20)/-0.1d-19/ data abint5(0)/ 1.12672081961782566017d0/ data abint5(1)/ -0.671405567525561198d-2/ data abint5(2)/ -0.69812918017832969d-3/ data abint5(3)/ -0.7561689886425276d-4/ data abint5(4)/ -0.834985574510207d-5/ data abint5(5)/ -0.93630298232480d-6/ data abint5(6)/ -0.10608556296250d-6/ data abint5(7)/ -0.1213128916741d-7/ data abint5(8)/ -0.139631129765d-8/ data abint5(9)/ -0.16178918054d-9/ data abint5(10)/-0.1882307907d-10/ data abint5(11)/-0.220272985d-11/ data abint5(12)/-0.25816189d-12/ data abint5(13)/-0.3047964d-13/ data abint5(14)/-0.358370d-14/ data abint5(15)/-0.42831d-15/ data abint5(16)/-0.4993d-16/ data abint5(17)/-0.617d-17/ data abint5(18)/-0.68d-18/ data abint5(19)/-0.10d-18/ data abint5(20)/-0.1d-19/ data onept5/ 1.5d0 / data seven/ 7.0d0 / data nine,sixten/ 9.0d0 , 16.0d0 / data ninhun,thr644/900.0d0 , 3644.0d0 / data piby4/0.78539816339744830962d0/ data rt2b3p/0.46065886596178063902d0/ data birzer/0.61492662744600073515d0/ ! ! Machine-dependent parameters (suitable for IEEE machines) ! data nterm4,nterm5/17,17/ data xlow1,xhigh1/2.22044604d-16,104.587632d0/ data xneg1,xmax/-2.727134d10,1.79d308/ x = xvalue if ( x < xneg1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'AIRY_BI_INT - Warning!' write ( *, '(a)' ) ' Argument is too negative for accurate computation.' airy_bi_int = zero else if ( x < -seven ) then z = - ( x + x ) * sqrt ( -x ) / three arg = z + piby4 temp = nine * z * z t = ( thr644 - temp ) / ( ninhun + temp ) f1 = cheval ( nterm4, abint4, t ) * sin ( arg ) f2 = cheval ( nterm5, abint5, t ) * cos ( arg ) / z airy_bi_int = ( f2 - f1 ) * rt2b3p / sqrt ( z ) else if ( x <= -xlow1 ) then t = - ( x + x ) / seven - one airy_bi_int = x * cheval ( nterm3, abint3, t ) else if ( x < xlow1 ) then airy_bi_int = birzer * x else if ( x <= eight ) then t = x / four - one airy_bi_int = x * exp ( onept5 * x ) * cheval ( nterm1, abint1, t ) else if ( x <= xhigh1 ) then t = sixten * sqrt ( eight / x ) / x - one z = ( x + x ) * sqrt ( x ) / three temp = rt2b3p * cheval ( nterm2, abint2, t ) / sqrt ( z ) temp = z + log ( temp ) airy_bi_int = exp ( temp ) else write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'AIRY_BI_INT - Warning!' write ( *, '(a)' ) ' Argument is too large for accurate computation.' airy_bi_int = xmax end if return end subroutine airy_bi_int_values ( n_data, x, fx ) !*****************************************************************************80 ! !! AIRY_BI_INT_VALUES returns some values of the integral of the Airy function. ! ! Discussion: ! ! The function is defined by: ! ! AIRY_BI_INT(x) = Integral ( 0 <= t <= x ) Bi(t) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 23 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.17660819031554631869D-01, & -0.15040424806140020451D-01, & 0.14756446293227661920D-01, & -0.11847304264848446271D+00, & -0.64916741266165856037D-01, & 0.97260832464381044540D-01, & 0.50760058495287539119D-01, & -0.37300500963429492179D+00, & -0.13962988442666578531D+00, & -0.12001735266723296160D-02, & 0.12018836117890354598D-02, & 0.36533846550952011043D+00, & 0.87276911673800812196D+00, & 0.48219475263803429675D+02, & 0.44006525804904178439D+06, & 0.17608153976228301458D+07, & 0.73779211705220007228D+07, & 0.14780980310740671617D+09, & 0.97037614223613433849D+11, & 0.11632737638809878460D+15 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & -12.0000000000D+00, & -10.0000000000D+00, & -8.0000000000D+00, & -7.5000000000D+00, & -7.0000000000D+00, & -6.5000000000D+00, & -4.0000000000D+00, & -1.0000000000D+00, & -0.2500000000D+00, & -0.0019531250D+00, & 0.0019531250D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 4.0000000000D+00, & 8.0000000000D+00, & 8.5000000000D+00, & 9.0000000000D+00, & 10.0000000000D+00, & 12.0000000000D+00, & 14.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function airy_gi ( xvalue ) !*****************************************************************************80 ! !! AIRY_GI computes the modified Airy function Gi(x). ! ! Discussion: ! ! The function is defined by: ! ! AIRY_GI(x) = Integral ( 0 <= t < infinity ) sin ( x*t+t^3/3) dt / pi ! ! The approximation uses Chebyshev expansions with the coefficients ! given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) AIRY_GI, the value of the function. ! implicit none real ( kind = 8 ) airy_gi real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: four = 4.0D+00 integer ( kind = 4 ), parameter :: nterm1 = 28 integer ( kind = 4 ), parameter :: nterm2 = 23 integer ( kind = 4 ), parameter :: nterm3 = 39 integer ( kind = 4 ) nterm4 integer ( kind = 4 ) nterm5 integer ( kind = 4 ) nterm6 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) argip1(0:30),argip2(0:29),argin1(0:42), & arbin1(0:10),arbin2(0:11),arhin1(0:15), & bi,cheb1,cheb2,cosz,five,five14, & gizero,minate,nine,onebpi,one76,one024,piby4, & rtpiin,seven,seven2,sinz,t,temp,twelhu,twent8, & xcube,xhigh1,xhigh2,xhigh3,xlow1,xminus, & zeta data argip1(0)/ 0.26585770795022745082d0/ data argip1(1)/ -0.10500333097501922907d0/ data argip1(2)/ 0.841347475328454492d-2/ data argip1(3)/ 0.2021067387813439541d-1/ data argip1(4)/ -0.1559576113863552234d-1/ data argip1(5)/ 0.564342939043256481d-2/ data argip1(6)/ -0.59776844826655809d-3/ data argip1(7)/ -0.42833850264867728d-3/ data argip1(8)/ 0.22605662380909027d-3/ data argip1(9)/ -0.3608332945592260d-4/ data argip1(10)/-0.785518988788901d-5/ data argip1(11)/ 0.473252480746370d-5/ data argip1(12)/-0.59743513977694d-6/ data argip1(13)/-0.15917609165602d-6/ data argip1(14)/ 0.6336129065570d-7/ data argip1(15)/-0.276090232648d-8/ data argip1(16)/-0.256064154085d-8/ data argip1(17)/ 0.47798676856d-9/ data argip1(18)/ 0.4488131863d-10/ data argip1(19)/-0.2346508882d-10/ data argip1(20)/ 0.76839085d-12/ data argip1(21)/ 0.73227985d-12/ data argip1(22)/-0.8513687d-13/ data argip1(23)/-0.1630201d-13/ data argip1(24)/ 0.356769d-14/ data argip1(25)/ 0.25001d-15/ data argip1(26)/-0.10859d-15/ data argip1(27)/-0.158d-17/ data argip1(28)/ 0.275d-17/ data argip1(29)/-0.5d-19/ data argip1(30)/-0.6d-19/ data argip2(0)/ 2.00473712275801486391d0/ data argip2(1)/ 0.294184139364406724d-2/ data argip2(2)/ 0.71369249006340167d-3/ data argip2(3)/ 0.17526563430502267d-3/ data argip2(4)/ 0.4359182094029882d-4/ data argip2(5)/ 0.1092626947604307d-4/ data argip2(6)/ 0.272382418399029d-5/ data argip2(7)/ 0.66230900947687d-6/ data argip2(8)/ 0.15425323370315d-6/ data argip2(9)/ 0.3418465242306d-7/ data argip2(10)/ 0.728157724894d-8/ data argip2(11)/ 0.151588525452d-8/ data argip2(12)/ 0.30940048039d-9/ data argip2(13)/ 0.6149672614d-10/ data argip2(14)/ 0.1202877045d-10/ data argip2(15)/ 0.233690586d-11/ data argip2(16)/ 0.43778068d-12/ data argip2(17)/ 0.7996447d-13/ data argip2(18)/ 0.1494075d-13/ data argip2(19)/ 0.246790d-14/ data argip2(20)/ 0.37672d-15/ data argip2(21)/ 0.7701d-16/ data argip2(22)/ 0.354d-17/ data argip2(23)/-0.49d-18/ data argip2(24)/ 0.62d-18/ data argip2(25)/-0.40d-18/ data argip2(26)/-0.1d-19/ data argip2(27)/ 0.2d-19/ data argip2(28)/-0.3d-19/ data argip2(29)/ 0.1d-19/ data argin1(0)/ -0.20118965056732089130d0/ data argin1(1)/ -0.7244175303324530499d-1/ data argin1(2)/ 0.4505018923894780120d-1/ data argin1(3)/ -0.24221371122078791099d0/ data argin1(4)/ 0.2717884964361678294d-1/ data argin1(5)/ -0.5729321004818179697d-1/ data argin1(6)/ -0.18382107860337763587d0/ data argin1(7)/ 0.7751546082149475511d-1/ data argin1(8)/ 0.18386564733927560387d0/ data argin1(9)/ 0.2921504250185567173d-1/ data argin1(10)/-0.6142294846788018811d-1/ data argin1(11)/-0.2999312505794616238d-1/ data argin1(12)/ 0.585937118327706636d-2/ data argin1(13)/ 0.822221658497402529d-2/ data argin1(14)/ 0.132579817166846893d-2/ data argin1(15)/-0.96248310766565126d-3/ data argin1(16)/-0.45065515998211807d-3/ data argin1(17)/ 0.772423474325474d-5/ data argin1(18)/ 0.5481874134758052d-4/ data argin1(19)/ 0.1245898039742876d-4/ data argin1(20)/-0.246196891092083d-5/ data argin1(21)/-0.169154183545285d-5/ data argin1(22)/-0.16769153169442d-6/ data argin1(23)/ 0.9636509337672d-7/ data argin1(24)/ 0.3253314928030d-7/ data argin1(25)/ 0.5091804231d-10/ data argin1(26)/-0.209180453553d-8/ data argin1(27)/-0.41237387870d-9/ data argin1(28)/ 0.4163338253d-10/ data argin1(29)/ 0.3032532117d-10/ data argin1(30)/ 0.340580529d-11/ data argin1(31)/-0.88444592d-12/ data argin1(32)/-0.31639612d-12/ data argin1(33)/-0.1505076d-13/ data argin1(34)/ 0.1104148d-13/ data argin1(35)/ 0.246508d-14/ data argin1(36)/-0.3107d-16/ data argin1(37)/-0.9851d-16/ data argin1(38)/-0.1453d-16/ data argin1(39)/ 0.118d-17/ data argin1(40)/ 0.67d-18/ data argin1(41)/ 0.6d-19/ data argin1(42)/-0.1d-19/ data arbin1/1.99983763583586155980d0, & -0.8104660923669418d-4, & 0.13475665984689d-6, & -0.70855847143d-9, & 0.748184187d-11, & -0.12902774d-12, & 0.322504d-14, & -0.10809d-15, & 0.460d-17, & -0.24d-18, & 0.1d-19/ data arbin2/0.13872356453879120276d0, & -0.8239286225558228d-4, & 0.26720919509866d-6, & -0.207423685368d-8, & 0.2873392593d-10, & -0.60873521d-12, & 0.1792489d-13, & -0.68760d-15, & 0.3280d-16, & -0.188d-17, & 0.13d-18, & -0.1d-19/ data arhin1/1.99647720399779650525d0, & -0.187563779407173213d-2, & -0.12186470897787339d-3, & -0.814021609659287d-5, & -0.55050925953537d-6, & -0.3763008043303d-7, & -0.258858362365d-8, & -0.17931829265d-9, & -0.1245916873d-10, & -0.87171247d-12, & -0.6084943d-13, & -0.431178d-14, & -0.29787d-15, & -0.2210d-16, & -0.136d-17, & -0.14d-18/ data five,seven,minate/ 5.0d0, 7.0d0 , -8.0d0 / data nine,twent8,seven2/ 9.0d0, 28.0d0 , 72.0d0 / data one76,five14/ 176.0d0 , 514.0d0 / data one024,twelhu/ 1024.0d0, 1200.0d0 / data gizero/0.20497554248200024505d0/ data onebpi/0.31830988618379067154d0/ data piby4/0.78539816339744830962d0/ data rtpiin/0.56418958354775628695d0/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data nterm4,nterm5,nterm6/9,10,14/ data xlow1,xhigh1/2.22045d-16,208063.8307d0/ data xhigh2,xhigh3/0.14274d308,-2097152.0d0/ x = xvalue if ( x < -xhigh1 * xhigh1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'AIRY_GI - Fatal error!' write ( *, '(a)' ) ' Argument too negative for accurate computation.' airy_gi = zero else if ( x <= xhigh3 ) then xminus = -x t = xminus * sqrt ( xminus ) zeta = ( t + t ) / three temp = rtpiin / sqrt ( sqrt ( xminus ) ) cosz = cos ( zeta + piby4 ) sinz = sin ( zeta + piby4 ) / zeta xcube = x * x * x bi = ( cosz + sinz * five / seven2 ) * temp t = ( xcube + twelhu ) / ( one76 - xcube ) airy_gi = bi + cheval ( nterm6, arhin1, t ) * onebpi / x else if ( x < minate ) then xminus = -x t = xminus * sqrt ( xminus ) zeta = ( t + t ) / three temp = rtpiin / sqrt ( sqrt ( xminus ) ) cosz = cos ( zeta + piby4 ) sinz = sin ( zeta + piby4 ) / zeta xcube = x * x * x t = - ( one024 / ( xcube ) + one ) cheb1 = cheval ( nterm4, arbin1, t ) cheb2 = cheval ( nterm5, arbin2, t ) bi = ( cosz * cheb1 + sinz * cheb2 ) * temp t = ( xcube + twelhu ) / ( one76 - xcube ) airy_gi = bi + cheval ( nterm6, arhin1, t ) * onebpi / x else if ( x <= -xlow1 ) then t = -( x + four ) / four airy_gi = cheval ( nterm3, argin1, t ) else if ( x < xlow1 ) then airy_gi = gizero else if ( x <= seven ) then t = ( nine * x - twent8 ) / ( x + twent8 ) airy_gi = cheval ( nterm1, argip1, t ) else if ( x <= xhigh1 ) then xcube = x * x * x t = ( twelhu - xcube ) / ( five14 + xcube ) airy_gi = onebpi * cheval ( nterm2, argip2, t ) / x else if ( x <= xhigh2 ) then airy_gi = onebpi / x else airy_gi = zero end if return end subroutine airy_gi_values ( n_data, x, fx ) !*****************************************************************************80 ! !! AIRY_GI_VALUES returns some values of the Airy Gi function. ! ! Discussion: ! ! The function is defined by: ! ! AIRY_GI(x) = Integral ( 0 <= t < infinity ) sin ( x*t+t^3/3) dt / pi ! ! The data was reported by McLeod. ! ! Modified: ! ! 24 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.20468308070040542435D+00, & 0.18374662832557904078D+00, & -0.11667221729601528265D+00, & 0.31466934902729557596D+00, & -0.37089040722426257729D+00, & -0.25293059772424019694D+00, & 0.28967410658692701936D+00, & -0.34644836492634090590D+00, & 0.28076035913873049496D+00, & 0.21814994508094865815D+00, & 0.20526679000810503329D+00, & 0.22123695363784773258D+00, & 0.23521843981043793760D+00, & 0.82834303363768729338D-01, & 0.45757385490989281893D-01, & 0.44150012014605159922D-01, & 0.39951133719508907541D-01, & 0.35467706833949671483D-01, & 0.31896005100679587981D-01, & 0.26556892713512410405D-01 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & -0.0019531250D+00, & -0.1250000000D+00, & -1.0000000000D+00, & -4.0000000000D+00, & -8.0000000000D+00, & -8.2500000000D+00, & -9.0000000000D+00, & -10.0000000000D+00, & -11.0000000000D+00, & -13.0000000000D+00, & 0.0019531250D+00, & 0.1250000000D+00, & 1.0000000000D+00, & 4.0000000000D+00, & 7.0000000000D+00, & 7.2500000000D+00, & 8.0000000000D+00, & 9.0000000000D+00, & 10.0000000000D+00, & 12.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function airy_hi ( xvalue ) !*****************************************************************************80 ! !! AIRY_HI computes the modified Airy function Hi(x). ! ! Discussion: ! ! The function is defined by: ! ! AIRY_HI(x) = Integral ( 0 <= t < infinity ) exp(x*t-t^3/3) dt / pi ! ! The approximation uses Chebyshev expansions with the coefficients ! given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) AIRY_HI, the value of the function. ! implicit none real ( kind = 8 ) airy_hi real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: four = 4.0D+00 integer ( kind = 4 ), parameter :: nterm1 = 29 integer ( kind = 4 ), parameter :: nterm2 = 17 integer ( kind = 4 ), parameter :: nterm3 = 22 integer ( kind = 4 ) nterm4 integer ( kind = 4 ) nterm5 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) arhip(0:31),arbip(0:23),argip1(0:29), & arhin1(0:21),arhin2(0:15), & bi,five14,gi,hizero,lnrtpi, & minate,onebpi,one76,seven,t,temp, & thre43,twelhu,twelve,xcube, & xhigh1,xlow1,xmax,xneg1,xneg2, & zeta data arhip(0)/ 1.24013562561762831114d0/ data arhip(1)/ 0.64856341973926535804d0/ data arhip(2)/ 0.55236252592114903246d0/ data arhip(3)/ 0.20975122073857566794d0/ data arhip(4)/ 0.12025669118052373568d0/ data arhip(5)/ 0.3768224931095393785d-1/ data arhip(6)/ 0.1651088671548071651d-1/ data arhip(7)/ 0.455922755211570993d-2/ data arhip(8)/ 0.161828480477635013d-2/ data arhip(9)/ 0.40841282508126663d-3/ data arhip(10)/0.12196479721394051d-3/ data arhip(11)/0.2865064098657610d-4/ data arhip(12)/0.742221556424344d-5/ data arhip(13)/0.163536231932831d-5/ data arhip(14)/0.37713908188749d-6/ data arhip(15)/0.7815800336008d-7/ data arhip(16)/0.1638447121370d-7/ data arhip(17)/0.319857665992d-8/ data arhip(18)/0.61933905307d-9/ data arhip(19)/0.11411161191d-9/ data arhip(20)/0.2064923454d-10/ data arhip(21)/0.360018664d-11/ data arhip(22)/0.61401849d-12/ data arhip(23)/0.10162125d-12/ data arhip(24)/0.1643701d-13/ data arhip(25)/0.259084d-14/ data arhip(26)/0.39931d-15/ data arhip(27)/0.6014d-16/ data arhip(28)/0.886d-17/ data arhip(29)/0.128d-17/ data arhip(30)/0.18d-18/ data arhip(31)/0.3d-19/ data arbip(0)/ 2.00582138209759064905d0/ data arbip(1)/ 0.294478449170441549d-2/ data arbip(2)/ 0.3489754514775355d-4/ data arbip(3)/ 0.83389733374343d-6/ data arbip(4)/ 0.3136215471813d-7/ data arbip(5)/ 0.167865306015d-8/ data arbip(6)/ 0.12217934059d-9/ data arbip(7)/ 0.1191584139d-10/ data arbip(8)/ 0.154142553d-11/ data arbip(9)/ 0.24844455d-12/ data arbip(10)/ 0.4213012d-13/ data arbip(11)/ 0.505293d-14/ data arbip(12)/-0.60032d-15/ data arbip(13)/-0.65474d-15/ data arbip(14)/-0.22364d-15/ data arbip(15)/-0.3015d-16/ data arbip(16)/ 0.959d-17/ data arbip(17)/ 0.616d-17/ data arbip(18)/ 0.97d-18/ data arbip(19)/-0.37d-18/ data arbip(20)/-0.21d-18/ data arbip(21)/-0.1d-19/ data arbip(22)/ 0.2d-19/ data arbip(23)/ 0.1d-19/ data argip1(0)/ 2.00473712275801486391d0/ data argip1(1)/ 0.294184139364406724d-2/ data argip1(2)/ 0.71369249006340167d-3/ data argip1(3)/ 0.17526563430502267d-3/ data argip1(4)/ 0.4359182094029882d-4/ data argip1(5)/ 0.1092626947604307d-4/ data argip1(6)/ 0.272382418399029d-5/ data argip1(7)/ 0.66230900947687d-6/ data argip1(8)/ 0.15425323370315d-6/ data argip1(9)/ 0.3418465242306d-7/ data argip1(10)/ 0.728157724894d-8/ data argip1(11)/ 0.151588525452d-8/ data argip1(12)/ 0.30940048039d-9/ data argip1(13)/ 0.6149672614d-10/ data argip1(14)/ 0.1202877045d-10/ data argip1(15)/ 0.233690586d-11/ data argip1(16)/ 0.43778068d-12/ data argip1(17)/ 0.7996447d-13/ data argip1(18)/ 0.1494075d-13/ data argip1(19)/ 0.246790d-14/ data argip1(20)/ 0.37672d-15/ data argip1(21)/ 0.7701d-16/ data argip1(22)/ 0.354d-17/ data argip1(23)/-0.49d-18/ data argip1(24)/ 0.62d-18/ data argip1(25)/-0.40d-18/ data argip1(26)/-0.1d-19/ data argip1(27)/ 0.2d-19/ data argip1(28)/-0.3d-19/ data argip1(29)/ 0.1d-19/ data arhin1(0)/ 0.31481017206423404116d0/ data arhin1(1)/ -0.16414499216588964341d0/ data arhin1(2)/ 0.6176651597730913071d-1/ data arhin1(3)/ -0.1971881185935933028d-1/ data arhin1(4)/ 0.536902830023331343d-2/ data arhin1(5)/ -0.124977068439663038d-2/ data arhin1(6)/ 0.24835515596994933d-3/ data arhin1(7)/ -0.4187024096746630d-4/ data arhin1(8)/ 0.590945437979124d-5/ data arhin1(9)/ -0.68063541184345d-6/ data arhin1(10)/ 0.6072897629164d-7/ data arhin1(11)/-0.367130349242d-8/ data arhin1(12)/ 0.7078017552d-10/ data arhin1(13)/ 0.1187894334d-10/ data arhin1(14)/-0.120898723d-11/ data arhin1(15)/ 0.1189656d-13/ data arhin1(16)/ 0.594128d-14/ data arhin1(17)/-0.32257d-15/ data arhin1(18)/-0.2290d-16/ data arhin1(19)/ 0.253d-17/ data arhin1(20)/ 0.9d-19/ data arhin1(21)/-0.2d-19/ data arhin2/1.99647720399779650525d0, & -0.187563779407173213d-2, & -0.12186470897787339d-3, & -0.814021609659287d-5, & -0.55050925953537d-6, & -0.3763008043303d-7, & -0.258858362365d-8, & -0.17931829265d-9, & -0.1245916873d-10, & -0.87171247d-12, & -0.6084943d-13, & -0.431178d-14, & -0.29787d-15, & -0.2210d-16, & -0.136d-17, & -0.14d-18/ data seven/ 7.0d0 / data minate,twelve,one76/ -8.0d0 , 12.0d0 , 176.0d0 / data thre43,five14,twelhu/ 343.0d0, 514.0d0, 1200.0d0 / data hizero/0.40995108496400049010d0/ data lnrtpi/0.57236494292470008707d0/ data onebpi/0.31830988618379067154d0/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data nterm4,nterm5/19,14/ data xlow1,xhigh1/2.220446d-16,104.4175d0/ data xneg1,xneg2,xmax/-0.14274d308,-208063.831d0,1.79d308/ x = xvalue if ( xhigh1 < x ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'AIRY_HI - Fatal error!' write ( *, '(a)' ) ' Argument too large.' airy_hi = xmax return end if ! ! Code for x < 0.0 ! if ( x < zero ) then if ( x < minate ) then if ( x < xneg1 ) then airy_hi = zero else if ( x < xneg2 ) then temp = one airy_hi = - temp * onebpi / x else xcube = x * x * x t = ( xcube + twelhu ) / ( one76 - xcube ) temp = cheval ( nterm5, arhin2, t ) airy_hi = - temp * onebpi / x end if end if else if ( -xlow1 < x ) then airy_hi = hizero else t = ( four * x + twelve ) / ( x - twelve ) airy_hi = cheval ( nterm4, arhin1, t ) end if end if ! ! Code for x >= 0.0 ! else if ( x <= seven ) then if ( x < xlow1 ) then airy_hi = hizero else t = ( x + x ) / seven - one temp = ( x + x + x ) / two airy_hi = exp ( temp ) * cheval ( nterm1, arhip, t ) end if else xcube = x * x * x temp = sqrt ( xcube ) zeta = ( temp + temp ) / three t = two * ( sqrt ( thre43 / xcube ) ) - one temp = cheval ( nterm2, arbip, t ) temp = zeta + log ( temp ) - log ( x ) / four - lnrtpi bi = exp ( temp ) t = ( twelhu - xcube ) / ( xcube + five14 ) gi = cheval ( nterm3, argip1, t ) * onebpi / x airy_hi = bi - gi end if end if return end subroutine airy_hi_values ( n_data, x, fx ) !*****************************************************************************80 ! !! AIRY_HI_VALUES returns some values of the Airy Hi function. ! ! Discussion: ! ! The function is defined by: ! ! AIRY_HI(x) = ! Integral ( 0 <= t < infinity ) exp ( x * t - t^3 / 3 ) dt / pi ! ! The data was reported by McLeod. ! ! Modified: ! ! 24 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.40936798278458884024D+00, & 0.37495291608048868619D+00, & 0.22066960679295989454D+00, & 0.77565356679703713590D-01, & 0.39638826473124717315D-01, & 0.38450072575004151871D-01, & 0.35273216868317898556D-01, & 0.31768535282502272742D-01, & 0.28894408288051391369D-01, & 0.24463284011678541180D-01, & 0.41053540139998941517D+00, & 0.44993502381204990817D+00, & 0.97220515514243332184D+00, & 0.83764237105104371193D+02, & 0.80327744952044756016D+05, & 0.15514138847749108298D+06, & 0.11995859641733262114D+07, & 0.21472868855967642259D+08, & 0.45564115351632913590D+09, & 0.32980722582904761929D+12 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & -0.0019531250D+00, & -0.1250000000D+00, & -1.0000000000D+00, & -4.0000000000D+00, & -8.0000000000D+00, & -8.2500000000D+00, & -9.0000000000D+00, & -10.0000000000D+00, & -11.0000000000D+00, & -13.0000000000D+00, & 0.0019531250D+00, & 0.1250000000D+00, & 1.0000000000D+00, & 4.0000000000D+00, & 7.0000000000D+00, & 7.2500000000D+00, & 8.0000000000D+00, & 9.0000000000D+00, & 10.0000000000D+00, & 12.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function arctan_int ( xvalue ) !*****************************************************************************80 ! !! ARCTAN_INT calculates the inverse tangent integral. ! ! Discussion: ! ! The function is defined by: ! ! ARCTAN_INT(x) = Integral ( 0 <= t <= x ) arctan ( t ) / t dt ! ! The approximation uses Chebyshev series with the coefficients ! given to an accuracy of 20D. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 24 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) ARCTAN_INT, the value of the function. ! implicit none real ( kind = 8 ), dimension ( 0:22 ) :: atnina = (/ & 1.91040361296235937512d0, & -0.4176351437656746940d-1, & 0.275392550786367434d-2, & -0.25051809526248881d-3, & 0.2666981285121171d-4, & -0.311890514107001d-5, & 0.38833853132249d-6, & -0.5057274584964d-7, & 0.681225282949d-8, & -0.94212561654d-9, & 0.13307878816d-9, & -0.1912678075d-10, & 0.278912620d-11, & -0.41174820d-12, & 0.6142987d-13, & -0.924929d-14, & 0.140387d-14, & -0.21460d-15, & 0.3301d-16, & -0.511d-17, & 0.79d-18, & -0.12d-18, & 0.2d-19 /) real ( kind = 8 ) arctan_int real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) ind integer ( kind = 4 ), parameter :: nterms = 19 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) t real ( kind = 8 ), parameter :: twobpi = 0.63661977236758134308D+00 real ( kind = 8 ) x real ( kind = 8 ), parameter :: xlow = 7.4505806D-09 real ( kind = 8 ), parameter :: xupper = 4.5036D+15 real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 ind = 1 x = xvalue if ( x < zero ) then x = -x ind = -1 end if if ( x < xlow ) then arctan_int = x else if ( x <= one ) then t = x * x t = ( t - half ) + ( t - half ) arctan_int = x * cheval ( nterms, atnina, t ) else if ( x <= xupper ) then t = one / ( x * x ) t = ( t - half ) + ( t - half ) arctan_int = log ( x ) / twobpi + cheval ( nterms, atnina, t ) / x else arctan_int = log ( x ) / twobpi end if if ( ind < 0 ) then arctan_int = -arctan_int end if return end subroutine arctan_int_values ( n_data, x, fx ) !*****************************************************************************80 ! !! ARCTAN_INT_VALUES returns some values of the inverse tangent integral. ! ! Discussion: ! ! The function is defined by: ! ! ARCTAN_INT(x) = Integral ( 0 <= t <= x ) arctan ( t ) / t dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 25 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.19531241721588483191D-02, & -0.39062433772980711281D-02, & 0.78124470192576499535D-02, & 0.15624576181996527280D-01, & -0.31246610349485401551D-01, & 0.62472911335014397321D-01, & 0.12478419717389654039D+00, & -0.24830175098230686908D+00, & 0.48722235829452235711D+00, & 0.91596559417721901505D+00, & 0.12749694484943800618D+01, & -0.15760154034463234224D+01, & 0.24258878412859089996D+01, & 0.33911633326292997361D+01, & 0.44176450919422186583D+01, & -0.47556713749547247774D+01, & 0.50961912150934111303D+01, & 0.53759175735714876256D+01, & -0.61649904785027487422D+01, & 0.72437843013083534973D+01 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & -0.0039062500D+00, & 0.0078125000D+00, & 0.0156250000D+00, & -0.0312500000D+00, & 0.0625000000D+00, & 0.1250000000D+00, & -0.2500000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & -2.0000000000D+00, & 4.0000000000D+00, & 8.0000000000D+00, & 16.0000000000D+00, & -20.0000000000D+00, & 25.0000000000D+00, & 30.0000000000D+00, & -50.0000000000D+00, & 100.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function bessel_i0_int ( xvalue ) !*****************************************************************************80 ! !! BESSEL_I0_INT computes the integral of the modified Bessel function I0(X). ! ! Discussion: ! ! The function is defined by: ! ! I0_INT(x) = Integral ( 0 <= t <= x ) I0(t) dt ! ! The program uses Chebyshev expansions, the coefficients of ! which are given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 29 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) BESSEL_I0_INT, the value of the function. ! implicit none real ( kind = 8 ) ari01(0:28) real ( kind = 8 ) ari0a(0:33) real ( kind = 8 ), parameter :: ateen = 18.0D+00 real ( kind = 8 ) bessel_i0_int real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) ind real ( kind = 8 ), parameter :: lnr2pi = 0.91893853320467274178D+00 integer ( kind = 4 ), parameter :: nterm1 = 25 integer ( kind = 4 ), parameter :: nterm2 = 27 real ( kind = 8 ) t real ( kind = 8 ) temp real ( kind = 8 ), parameter :: thirt6 = 36.0D+00 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ) x real ( kind = 8 ), parameter :: xhigh = 713.758339D+00 real ( kind = 8 ), parameter :: xlow = 0.5161914D-07 real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 data ari01(0)/ 0.41227906926781516801d0/ data ari01(1)/ -0.34336345150081519562d0/ data ari01(2)/ 0.22667588715751242585d0/ data ari01(3)/ -0.12608164718742260032d0/ data ari01(4)/ 0.6012484628777990271d-1/ data ari01(5)/ -0.2480120462913358248d-1/ data ari01(6)/ 0.892773389565563897d-2/ data ari01(7)/ -0.283253729936696605d-2/ data ari01(8)/ 0.79891339041712994d-3/ data ari01(9)/ -0.20053933660964890d-3/ data ari01(10)/ 0.4416816783014313d-4/ data ari01(11)/-0.822377042246068d-5/ data ari01(12)/ 0.120059794219015d-5/ data ari01(13)/-0.11350865004889d-6/ data ari01(14)/ 0.69606014466d-9/ data ari01(15)/ 0.180622772836d-8/ data ari01(16)/-0.26039481370d-9/ data ari01(17)/-0.166188103d-11/ data ari01(18)/ 0.510500232d-11/ data ari01(19)/-0.41515879d-12/ data ari01(20)/-0.7368138d-13/ data ari01(21)/ 0.1279323d-13/ data ari01(22)/ 0.103247d-14/ data ari01(23)/-0.30379d-15/ data ari01(24)/-0.1789d-16/ data ari01(25)/ 0.673d-17/ data ari01(26)/ 0.44d-18/ data ari01(27)/-0.14d-18/ data ari01(28)/-0.1d-19/ data ari0a(0)/ 2.03739654571143287070d0/ data ari0a(1)/ 0.1917631647503310248d-1/ data ari0a(2)/ 0.49923334519288147d-3/ data ari0a(3)/ 0.2263187103659815d-4/ data ari0a(4)/ 0.158682108285561d-5/ data ari0a(5)/ 0.16507855636318d-6/ data ari0a(6)/ 0.2385058373640d-7/ data ari0a(7)/ 0.392985182304d-8/ data ari0a(8)/ 0.46042714199d-9/ data ari0a(9)/ -0.7072558172d-10/ data ari0a(10)/-0.6747183961d-10/ data ari0a(11)/-0.2026962001d-10/ data ari0a(12)/-0.87320338d-12/ data ari0a(13)/ 0.175520014d-11/ data ari0a(14)/ 0.60383944d-12/ data ari0a(15)/-0.3977983d-13/ data ari0a(16)/-0.8049048d-13/ data ari0a(17)/-0.1158955d-13/ data ari0a(18)/ 0.827318d-14/ data ari0a(19)/ 0.282290d-14/ data ari0a(20)/-0.77667d-15/ data ari0a(21)/-0.48731d-15/ data ari0a(22)/ 0.7279d-16/ data ari0a(23)/ 0.7873d-16/ data ari0a(24)/-0.785d-17/ data ari0a(25)/-0.1281d-16/ data ari0a(26)/ 0.121d-17/ data ari0a(27)/ 0.214d-17/ data ari0a(28)/-0.27d-18/ data ari0a(29)/-0.36d-18/ data ari0a(30)/ 0.7d-19/ data ari0a(31)/ 0.6d-19/ data ari0a(32)/-0.2d-19/ data ari0a(33)/-0.1d-19/ ind = 1 x = xvalue if ( xvalue < zero ) then ind = -1 x = -x end if if ( x < xlow ) then bessel_i0_int = x else if ( x <= ateen ) then t = ( three * x - ateen ) / ( x + ateen ) bessel_i0_int = x * exp ( x ) * cheval ( nterm1, ari01, t ) else if ( x <= xhigh ) then t = ( thirt6 / x - half ) - half temp = x - half * log ( x ) - lnr2pi + log ( cheval ( nterm2, ari0a, t )) bessel_i0_int = exp ( temp ) else write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'BESSEL_I0_INT - Fatal error!' write ( *, '(a)' ) ' Argument magnitude too large.' bessel_i0_int = exp ( xhigh - lnr2pi - half * log ( xhigh ) ) end if if ( ind == -1 ) then bessel_i0_int = -bessel_i0_int end if return end subroutine bessel_i0_int_values ( n_data, x, fx ) !*****************************************************************************80 ! !! BESSEL_I0_INT_VALUES returns some values of the Bessel I0 integral. ! ! Discussion: ! ! The function is defined by: ! ! I0_INT(x) = Integral ( 0 <= t <= x ) I0(t) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 29 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.19531256208818052282D-02, & -0.39062549670565734544D-02, & 0.62520348032546565850D-01, & 0.12516285581366971819D+00, & -0.51051480879740303760D+00, & 0.10865210970235898158D+01, & 0.27750019054282535299D+01, & -0.13775208868039716639D+02, & 0.46424372058106108576D+03, & 0.64111867658021584522D+07, & -0.10414860803175857953D+08, & 0.44758598913855743089D+08, & -0.11852985311558287888D+09, & 0.31430078220715992752D+09, & -0.83440212900794309620D+09, & 0.22175367579074298261D+10, & 0.58991731842803636487D+10, & -0.41857073244691522147D+11, & 0.79553885818472357663D+12, & 0.15089715082719201025D+17 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & -0.0039062500D+00, & 0.0625000000D+00, & 0.1250000000D+00, & -0.5000000000D+00, & 1.0000000000D+00, & 2.0000000000D+00, & -4.0000000000D+00, & 8.0000000000D+00, & 18.0000000000D+00, & -18.5000000000D+00, & 20.0000000000D+00, & -21.0000000000D+00, & 22.0000000000D+00, & -23.0000000000D+00, & 24.0000000000D+00, & 25.0000000000D+00, & -27.0000000000D+00, & 30.0000000000D+00, & 40.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function bessel_j0_int ( xvalue ) !*****************************************************************************80 ! !! BESSEL_J0_INT calculates the integral of the Bessel function J0. ! ! Discussion: ! ! The function is defined by: ! ! J0_INT(x) = Integral ( 0 <= t <= x ) J0(t) dt ! ! The code uses Chebyshev expansions whose coefficients are ! given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) BESSEL_J0_INT, the value of the function. ! implicit none real ( kind = 8 ) bessel_j0_int real ( kind = 8 ) cheval integer ( kind = 4 ) ind integer ( kind = 4 ), parameter :: nterm1 = 22 integer ( kind = 4 ), parameter :: nterm2 = 18 integer ( kind = 4 ), parameter :: nterm3 = 16 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) arj01(0:23),arj0a1(0:21),arj0a2(0:18), & five12,one28,pib41,pib411,pib412, & pib42,rt2bpi,sixten,t,temp,xhigh,xlow, & xmpi4 data sixten/ 16.0d0 / data one28,five12/ 128.0d0 , 512d0 / data rt2bpi/0.79788456080286535588d0/ data pib411,pib412/ 201.0d0 , 256.0d0/ data pib42/0.24191339744830961566d-3/ data arj01(0)/ 0.38179279321690173518d0/ data arj01(1)/ -0.21275636350505321870d0/ data arj01(2)/ 0.16754213407215794187d0/ data arj01(3)/ -0.12853209772196398954d0/ data arj01(4)/ 0.10114405455778847013d0/ data arj01(5)/ -0.9100795343201568859d-1/ data arj01(6)/ 0.6401345264656873103d-1/ data arj01(7)/ -0.3066963029926754312d-1/ data arj01(8)/ 0.1030836525325064201d-1/ data arj01(9)/ -0.255670650399956918d-2/ data arj01(10)/ 0.48832755805798304d-3/ data arj01(11)/-0.7424935126036077d-4/ data arj01(12)/ 0.922260563730861d-5/ data arj01(13)/-0.95522828307083d-6/ data arj01(14)/ 0.8388355845986d-7/ data arj01(15)/-0.633184488858d-8/ data arj01(16)/ 0.41560504221d-9/ data arj01(17)/-0.2395529307d-10/ data arj01(18)/ 0.122286885d-11/ data arj01(19)/-0.5569711d-13/ data arj01(20)/ 0.227820d-14/ data arj01(21)/-0.8417d-16/ data arj01(22)/ 0.282d-17/ data arj01(23)/-0.9d-19/ data arj0a1(0)/ 1.24030133037518970827d0/ data arj0a1(1)/ -0.478125353632280693d-2/ data arj0a1(2)/ 0.6613148891706678d-4/ data arj0a1(3)/ -0.186042740486349d-5/ data arj0a1(4)/ 0.8362735565080d-7/ data arj0a1(5)/ -0.525857036731d-8/ data arj0a1(6)/ 0.42606363251d-9/ data arj0a1(7)/ -0.4211761024d-10/ data arj0a1(8)/ 0.488946426d-11/ data arj0a1(9)/ -0.64834929d-12/ data arj0a1(10)/ 0.9617234d-13/ data arj0a1(11)/-0.1570367d-13/ data arj0a1(12)/ 0.278712d-14/ data arj0a1(13)/-0.53222d-15/ data arj0a1(14)/ 0.10844d-15/ data arj0a1(15)/-0.2342d-16/ data arj0a1(16)/ 0.533d-17/ data arj0a1(17)/-0.127d-17/ data arj0a1(18)/ 0.32d-18/ data arj0a1(19)/-0.8d-19/ data arj0a1(20)/ 0.2d-19/ data arj0a1(21)/-0.1d-19/ data arj0a2(0)/ 1.99616096301341675339d0/ data arj0a2(1)/ -0.190379819246668161d-2/ data arj0a2(2)/ 0.1539710927044226d-4/ data arj0a2(3)/ -0.31145088328103d-6/ data arj0a2(4)/ 0.1110850971321d-7/ data arj0a2(5)/ -0.58666787123d-9/ data arj0a2(6)/ 0.4139926949d-10/ data arj0a2(7)/ -0.365398763d-11/ data arj0a2(8)/ 0.38557568d-12/ data arj0a2(9)/ -0.4709800d-13/ data arj0a2(10)/ 0.650220d-14/ data arj0a2(11)/-0.99624d-15/ data arj0a2(12)/ 0.16700d-15/ data arj0a2(13)/-0.3028d-16/ data arj0a2(14)/ 0.589d-17/ data arj0a2(15)/-0.122d-17/ data arj0a2(16)/ 0.27d-18/ data arj0a2(17)/-0.6d-19/ data arj0a2(18)/ 0.1d-19/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow,xhigh/3.650024d-8,9.0072d15/ x = xvalue ind = 1 if ( x < zero ) then x = -x ind = -1 end if if ( x < xlow ) then bessel_j0_int = x else if ( x <= sixten ) then t = x * x / one28 - one bessel_j0_int = x * cheval ( nterm1, arj01, t ) else if ( x <= xhigh ) then t = five12 / ( x * x ) - one pib41 = pib411 / pib412 xmpi4 = ( x - pib41 ) - pib42 temp = cos ( xmpi4 ) * cheval ( nterm2, arj0a1, t ) / x temp = temp - sin ( xmpi4) * cheval ( nterm3, arj0a2, t ) bessel_j0_int = one - rt2bpi * temp / sqrt ( x ) else write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'BESSEL_J0_INT - Fatal error!' write ( *, '(a)' ) ' Argument magnitude too large.' bessel_j0_int = one end if if ( ind == -1 ) then bessel_j0_int = -bessel_j0_int end if return end subroutine bessel_j0_int_values ( n_data, x, fx ) !*****************************************************************************80 ! !! BESSEL_J0_INT_VALUES returns some values of the Bessel J0 integral. ! ! Discussion: ! ! The function is defined by: ! ! J0_INT(x) = Integral ( 0 <= t <= x ) J0(t) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 29 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.97656242238978822427D-03, & 0.39062450329491108875D-02, & -0.62479657927917933620D-01, & 0.12483733492120479139D+00, & -0.48968050664604505505D+00, & 0.91973041008976023931D+00, & -0.14257702931970265690D+01, & 0.10247341594606064818D+01, & -0.12107468348304501655D+01, & 0.11008652032736190799D+01, & -0.10060334829904124192D+01, & 0.81330572662485953519D+00, & -0.10583788214211277585D+01, & 0.87101492116545875169D+00, & -0.88424908882547488420D+00, & 0.11257761503599914603D+01, & -0.90141212258183461184D+00, & 0.91441344369647797803D+00, & -0.94482281938334394886D+00, & 0.92266255696016607257D+00 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0009765625D+00, & 0.0039062500D+00, & -0.0625000000D+00, & 0.1250000000D+00, & -0.5000000000D+00, & 1.0000000000D+00, & -2.0000000000D+00, & 4.0000000000D+00, & -8.0000000000D+00, & 16.0000000000D+00, & -16.5000000000D+00, & 18.0000000000D+00, & -20.0000000000D+00, & 25.0000000000D+00, & -30.0000000000D+00, & 40.0000000000D+00, & -50.0000000000D+00, & 75.0000000000D+00, & -80.0000000000D+00, & 100.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function bessel_k0_int ( xvalue ) !*****************************************************************************80 ! !! BESSEL_K0_INT calculates the integral of the modified Bessel function K0(X). ! ! Discussion: ! ! The function is defined by: ! ! K0_INT(x) = Integral ( 0 <= t <= x ) K0(t) dt ! ! The code uses Chebyshev expansions, whose coefficients are ! given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 29 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) BESSEL_K0_INT, the value of the function. ! implicit none real ( kind = 8 ) bessel_k0_int real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ), parameter :: nterm1 = 14 integer ( kind = 4 ), parameter :: nterm2 = 14 integer ( kind = 4 ), parameter :: nterm3 = 23 real ( kind = 8 ), parameter :: six = 6.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) ak0in1(0:15),ak0in2(0:15),ak0ina(0:27), & const1,const2,eightn,fval, & piby2,rt2bpi,t,temp,twelve, & xhigh,xlow data twelve,eightn / 12.0d0 , 18.0d0 / data const1/1.11593151565841244881d0/ data const2/-0.11593151565841244881d0/ data piby2/1.57079632679489661923d0/ data rt2bpi/0.79788456080286535588d0/ data ak0in1/16.79702714464710959477d0, & 9.79134687676889407070d0, & 2.80501316044337939300d0, & 0.45615620531888502068d0, & 0.4716224457074760784d-1, & 0.335265148269698289d-2, & 0.17335181193874727d-3, & 0.679951889364702d-5, & 0.20900268359924d-6, & 0.516603846976d-8, & 0.10485708331d-9, & 0.177829320d-11, & 0.2556844d-13, & 0.31557d-15, & 0.338d-17, & 0.3d-19/ data ak0in2/10.76266558227809174077d0, & 5.62333479849997511550d0, & 1.43543664879290867158d0, & 0.21250410143743896043d0, & 0.2036537393100009554d-1, & 0.136023584095623632d-2, & 0.6675388699209093d-4, & 0.250430035707337d-5, & 0.7406423741728d-7, & 0.176974704314d-8, & 0.3485775254d-10, & 0.57544785d-12, & 0.807481d-14, & 0.9747d-16, & 0.102d-17, & 0.1d-19/ data ak0ina(0)/ 1.91172065445060453895d0/ data ak0ina(1)/ -0.4183064565769581085d-1/ data ak0ina(2)/ 0.213352508068147486d-2/ data ak0ina(3)/ -0.15859497284504181d-3/ data ak0ina(4)/ 0.1497624699858351d-4/ data ak0ina(5)/ -0.167955955322241d-5/ data ak0ina(6)/ 0.21495472478804d-6/ data ak0ina(7)/ -0.3058356654790d-7/ data ak0ina(8)/ 0.474946413343d-8/ data ak0ina(9)/ -0.79424660432d-9/ data ak0ina(10)/ 0.14156555325d-9/ data ak0ina(11)/-0.2667825359d-10/ data ak0ina(12)/ 0.528149717d-11/ data ak0ina(13)/-0.109263199d-11/ data ak0ina(14)/ 0.23518838d-12/ data ak0ina(15)/-0.5247991d-13/ data ak0ina(16)/ 0.1210191d-13/ data ak0ina(17)/-0.287632d-14/ data ak0ina(18)/ 0.70297d-15/ data ak0ina(19)/-0.17631d-15/ data ak0ina(20)/ 0.4530d-16/ data ak0ina(21)/-0.1190d-16/ data ak0ina(22)/ 0.319d-17/ data ak0ina(23)/-0.87d-18/ data ak0ina(24)/ 0.24d-18/ data ak0ina(25)/-0.7d-19/ data ak0ina(26)/ 0.2d-19/ data ak0ina(27)/-0.1d-19/ ! ! Machine-dependent values (suitable for IEEE machines) ! data xlow,xhigh/4.47034836d-8,36.0436534d0/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'BESSEL_K0_INT - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' bessel_k0_int = zero else if ( x == zero ) then bessel_k0_int = zero else if ( x < xlow ) then bessel_k0_int = x * ( const1 - log ( x ) ) else if ( x <= six ) then t = ( ( x * x ) / eightn - half ) - half fval = ( const2 + log ( x ) ) * cheval ( nterm2, ak0in2, t ) bessel_k0_int = x * ( cheval ( nterm1, ak0in1, t ) - fval ) else if ( x < xhigh ) then fval = piby2 t = ( twelve / x - half ) - half temp = exp ( -x ) * cheval ( nterm3, ak0ina, t ) fval = fval - temp / ( sqrt ( x ) * rt2bpi ) bessel_k0_int = fval else bessel_k0_int = piby2 end if return end subroutine bessel_k0_int_values ( n_data, x, fx ) !*****************************************************************************80 ! !! BESSEL_K0_INT_VALUES returns some values of the Bessel K0 integral. ! ! Discussion: ! ! The function is defined by: ! ! K0_INT(x) = Integral ( 0 <= t <= x ) K0(t) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 29 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.78587929563466784589D-02, & 0.26019991617330578111D-01, & 0.24311842237541167904D+00, & 0.39999633750480508861D+00, & 0.92710252093114907345D+00, & 0.12425098486237782662D+01, & 0.14736757343168286825D+01, & 0.15606495706051741364D+01, & 0.15673873907283660493D+01, & 0.15696345532693743714D+01, & 0.15701153443250786355D+01, & 0.15706574852894436220D+01, & 0.15707793116159788598D+01, & 0.15707942066465767196D+01, & 0.15707962315469192247D+01, & 0.15707963262340149876D+01, & 0.15707963267948756308D+01, & 0.15707963267948966192D+01, & 0.15707963267948966192D+01, & 0.15707963267948966192D+01 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0009765625D+00, & 0.0039062500D+00, & 0.0625000000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 2.0000000000D+00, & 4.0000000000D+00, & 5.0000000000D+00, & 6.0000000000D+00, & 6.5000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 12.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00, & 80.0000000000D+00, & 100.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function bessel_y0_int ( xvalue ) !*****************************************************************************80 ! !! BESSEL_Y0_INT calculates the integral of the Bessel function Y0. ! ! Discussion: ! ! The function is defined by: ! ! Y0_INT(x) = Integral ( 0 <= t <= x ) Y0(t) dt ! ! The code uses Chebyshev expansions whose coefficients are ! given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 23 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) BESSEL_Y0_INT, the value of the function. ! implicit none real ( kind = 8 ) bessel_y0_int real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: nine = 9.0D+00 integer ( kind = 4 ), parameter :: nterm1 = 22 integer ( kind = 4 ), parameter :: nterm2 = 22 integer ( kind = 4 ), parameter :: nterm3 = 17 integer ( kind = 4 ), parameter :: nterm4 = 15 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: sixten = 16.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) arj01(0:23),ary01(0:24),ary0a1(0:21), & ary0a2(0:18),five12,gal2m1,gamln2, & one28,pib41,pib411,pib412, & pib42,rt2bpi,t,temp,twobpi,xhigh, & xlow,xmpi4 data one28,five12/ 128.0d0 , 512.0d0 / data rt2bpi/0.79788456080286535588d0/ data pib411,pib412/ 201.0d0, 256.0d0/ data pib42/0.24191339744830961566d-3/ data twobpi/0.63661977236758134308d0/ data gal2m1/-1.11593151565841244881d0/ data gamln2/-0.11593151565841244881d0/ data arj01(0)/ 0.38179279321690173518d0/ data arj01(1)/ -0.21275636350505321870d0/ data arj01(2)/ 0.16754213407215794187d0/ data arj01(3)/ -0.12853209772196398954d0/ data arj01(4)/ 0.10114405455778847013d0/ data arj01(5)/ -0.9100795343201568859d-1/ data arj01(6)/ 0.6401345264656873103d-1/ data arj01(7)/ -0.3066963029926754312d-1/ data arj01(8)/ 0.1030836525325064201d-1/ data arj01(9)/ -0.255670650399956918d-2/ data arj01(10)/ 0.48832755805798304d-3/ data arj01(11)/-0.7424935126036077d-4/ data arj01(12)/ 0.922260563730861d-5/ data arj01(13)/-0.95522828307083d-6/ data arj01(14)/ 0.8388355845986d-7/ data arj01(15)/-0.633184488858d-8/ data arj01(16)/ 0.41560504221d-9/ data arj01(17)/-0.2395529307d-10/ data arj01(18)/ 0.122286885d-11/ data arj01(19)/-0.5569711d-13/ data arj01(20)/ 0.227820d-14/ data arj01(21)/-0.8417d-16/ data arj01(22)/ 0.282d-17/ data arj01(23)/-0.9d-19/ data ary01(0)/ 0.54492696302724365490d0/ data ary01(1)/ -0.14957323588684782157d0/ data ary01(2)/ 0.11085634486254842337d0/ data ary01(3)/ -0.9495330018683777109d-1/ data ary01(4)/ 0.6820817786991456963d-1/ data ary01(5)/ -0.10324653383368200408d0/ data ary01(6)/ 0.10625703287534425491d0/ data ary01(7)/ -0.6258367679961681990d-1/ data ary01(8)/ 0.2385645760338293285d-1/ data ary01(9)/ -0.644864913015404481d-2/ data ary01(10)/ 0.131287082891002331d-2/ data ary01(11)/-0.20988088174989640d-3/ data ary01(12)/ 0.2716042484138347d-4/ data ary01(13)/-0.291199114014694d-5/ data ary01(14)/ 0.26344333093795d-6/ data ary01(15)/-0.2041172069780d-7/ data ary01(16)/ 0.137124781317d-8/ data ary01(17)/-0.8070680792d-10/ data ary01(18)/ 0.419883057d-11/ data ary01(19)/-0.19459104d-12/ data ary01(20)/ 0.808782d-14/ data ary01(21)/-0.30329d-15/ data ary01(22)/ 0.1032d-16/ data ary01(23)/-0.32d-18/ data ary01(24)/ 0.1d-19/ data ary0a1(0)/ 1.24030133037518970827d0/ data ary0a1(1)/ -0.478125353632280693d-2/ data ary0a1(2)/ 0.6613148891706678d-4/ data ary0a1(3)/ -0.186042740486349d-5/ data ary0a1(4)/ 0.8362735565080d-7/ data ary0a1(5)/ -0.525857036731d-8/ data ary0a1(6)/ 0.42606363251d-9/ data ary0a1(7)/ -0.4211761024d-10/ data ary0a1(8)/ 0.488946426d-11/ data ary0a1(9)/ -0.64834929d-12/ data ary0a1(10)/ 0.9617234d-13/ data ary0a1(11)/-0.1570367d-13/ data ary0a1(12)/ 0.278712d-14/ data ary0a1(13)/-0.53222d-15/ data ary0a1(14)/ 0.10844d-15/ data ary0a1(15)/-0.2342d-16/ data ary0a1(16)/ 0.533d-17/ data ary0a1(17)/-0.127d-17/ data ary0a1(18)/ 0.32d-18/ data ary0a1(19)/-0.8d-19/ data ary0a1(20)/ 0.2d-19/ data ary0a1(21)/-0.1d-19/ data ary0a2(0)/ 1.99616096301341675339d0/ data ary0a2(1)/ -0.190379819246668161d-2/ data ary0a2(2)/ 0.1539710927044226d-4/ data ary0a2(3)/ -0.31145088328103d-6/ data ary0a2(4)/ 0.1110850971321d-7/ data ary0a2(5)/ -0.58666787123d-9/ data ary0a2(6)/ 0.4139926949d-10/ data ary0a2(7)/ -0.365398763d-11/ data ary0a2(8)/ 0.38557568d-12/ data ary0a2(9)/ -0.4709800d-13/ data ary0a2(10)/ 0.650220d-14/ data ary0a2(11)/-0.99624d-15/ data ary0a2(12)/ 0.16700d-15/ data ary0a2(13)/-0.3028d-16/ data ary0a2(14)/ 0.589d-17/ data ary0a2(15)/-0.122d-17/ data ary0a2(16)/ 0.27d-18/ data ary0a2(17)/-0.6d-19/ data ary0a2(18)/ 0.1d-19/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow,xhigh/3.16101364d-8,9.007199256d15/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'BESSEL_Y0_INT - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' bessel_y0_int = zero else if ( x == zero ) then bessel_y0_int = zero else if ( x < xlow ) then bessel_y0_int = ( log ( x ) + gal2m1 ) * twobpi * x else if ( x <= sixten ) then t = x * x / one28 - one temp = ( log ( x ) + gamln2 ) * cheval ( nterm1, arj01, t ) temp = temp - cheval ( nterm2, ary01, t ) bessel_y0_int = twobpi * x * temp else if ( x <= xhigh ) then t = five12 / ( x * x ) - one pib41 = pib411 / pib412 xmpi4 = ( x - pib41 ) - pib42 temp = sin ( xmpi4 ) * cheval ( nterm3, ary0a1, t ) / x temp = temp + cos ( xmpi4 ) * cheval ( nterm4, ary0a2, t ) bessel_y0_int = - rt2bpi * temp / sqrt ( x ) else write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'BESSEL_Y0_INT - Fatal error!' write ( *, '(a)' ) ' Argument too large.' bessel_y0_int = zero end if return end subroutine bessel_y0_int_values ( n_data, x, fx ) !*****************************************************************************80 ! !! BESSEL_Y0_INT_VALUES returns some values of the Bessel Y0 integral. ! ! Discussion: ! ! The function is defined by: ! ! Y0_INT(x) = Integral ( 0 <= t <= x ) Y0(t) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 30 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & -0.91442642860172110926D-02, & -0.29682047390397591290D-01, & -0.25391431276585388961D+00, & -0.56179545591464028187D+00, & -0.63706937660742309754D+00, & -0.28219285008510084123D+00, & 0.38366964785312561103D+00, & -0.12595061285798929390D+00, & 0.24129031832266684828D+00, & 0.17138069757627037938D+00, & 0.18958142627134083732D+00, & 0.17203846136449706946D+00, & -0.16821597677215029611D+00, & -0.93607927351428988679D-01, & 0.88229711948036648408D-01, & -0.89324662736274161841D-02, & -0.54814071000063488284D-01, & -0.94958246003466381588D-01, & -0.19598064853404969850D-01, & -0.83084772357154773468D-02 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0078125000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 2.0000000000D+00, & 4.0000000000D+00, & 6.0000000000D+00, & 10.0000000000D+00, & 16.0000000000D+00, & 16.2500000000D+00, & 17.0000000000D+00, & 20.0000000000D+00, & 25.0000000000D+00, & 30.0000000000D+00, & 40.0000000000D+00, & 50.0000000000D+00, & 70.0000000000D+00, & 100.0000000000D+00, & 125.0000000000D+00 /) ! if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function cheval ( n, a, t ) !*****************************************************************************80 ! !! CHEVAL evaluates a Chebyshev series. ! ! Discussion: ! ! This function evaluates a Chebyshev series, using the ! Clenshaw method with Reinsch modification, as analysed ! in the paper by Oliver. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! J Oliver, ! An error analysis of the modified Clenshaw method for ! evaluating Chebyshev and Fourier series, ! Journal of the IMA, ! Volume 20, 1977, pages 379-391. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of terms in the sequence. ! ! Input, real ( kind = 8 ) A(0:N), the coefficients of the Chebyshev series. ! ! Input, real ( kind = 8 ) T, the value at which the series is ! to be evaluated. ! ! Output, real ( kind = 8 ) CHEVAL, the value of the Chebyshev series at T. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) :: a(0:n) real ( kind = 8 ) :: cheval real ( kind = 8 ) :: d1 real ( kind = 8 ) :: d2 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) i real ( kind = 8 ) :: t real ( kind = 8 ), parameter :: test = 0.6D+00 real ( kind = 8 ) :: tt real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) :: u0 real ( kind = 8 ) :: u1 real ( kind = 8 ) :: u2 real ( kind = 8 ), parameter :: zero = 0.0D+00 u1 = zero ! ! T <= -0.6, Reinsch modification. ! if ( t <= -test ) then d1 = zero tt = ( t + half ) + half tt = tt + tt do i = n, 0, -1 d2 = d1 u2 = u1 d1 = tt * u2 + a(i) - d2 u1 = d1 - u2 end do cheval = ( d1 - d2 ) / two ! ! -0.6 < T < 0.6, Standard Clenshaw method. ! else if ( t < test ) then u0 = zero tt = t + t do i = n, 0, -1 u2 = u1 u1 = u0 u0 = tt * u1 + a(i) - u2 end do cheval = ( u0 - u2 ) / two ! ! 0.6 <= T, Reinsch modification. ! else d1 = zero tt = ( t - half ) - half tt = tt + tt do i = n, 0, -1 d2 = d1 u2 = u1 d1 = tt * u2 + a(i) + d2 u1 = d1 + u2 end do cheval = ( d1 + d2 ) / two end if return end function clausen ( xvalue ) !*****************************************************************************80 ! !! CLAUSEN calculates Clausen's integral. ! ! Discussion: ! ! The function is defined by: ! ! CLAUSEN(x) = Integral ( 0 <= t <= x ) -ln ( 2 * sin ( t / 2 ) ) dt ! ! The code uses Chebyshev expansions with the coefficients ! given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) CLAUSEN, the value of the function. ! implicit none real ( kind = 8 ) aclaus(0:15) real ( kind = 8 ) cheval real ( kind = 8 ) clausen real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) indx integer ( kind = 4 ), parameter :: nterms = 13 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: pi = 3.1415926535897932385D+00 real ( kind = 8 ), parameter :: pisq = 9.8696044010893586188D+00 real ( kind = 8 ) t real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) twopi,twopia,twopib,xhigh,xsmall data twopi/6.2831853071795864769d0/ data twopia,twopib/6.28125d0, 0.19353071795864769253d-2/ data aclaus/2.14269436376668844709d0, & 0.7233242812212579245d-1, & 0.101642475021151164d-2, & 0.3245250328531645d-4, & 0.133315187571472d-5, & 0.6213240591653d-7, & 0.313004135337d-8, & 0.16635723056d-9, & 0.919659293d-11, & 0.52400462d-12, & 0.3058040d-13, & 0.181969d-14, & 0.11004d-15, & 0.675d-17, & 0.42d-18, & 0.3d-19/ ! ! Set machine-dependent constants (suitable for IEEE machines) ! data xsmall,xhigh/2.3406689d-8,4.5036d15/ x = xvalue if ( xhigh < abs ( x ) ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'CLAUSEN - Warning!' write ( *, '(a)' ) & ' Argument magnitude too large for accurate computation.' clausen = zero return end if indx = 1 if ( x < zero ) then x = -x indx = -1 end if ! ! Argument reduced using simulated extra precision ! if ( twopi < x ) then t = aint ( x / twopi ) x = ( x - t * twopia ) - t * twopib end if if ( pi < x ) then x = ( twopia - x ) + twopib indx = -indx end if if ( x == zero ) then clausen = zero else if ( x < xsmall ) then clausen = x * ( one - log ( x ) ) else t = ( x * x ) / pisq - half t = t + t if ( one < t ) then t = one end if clausen = x * cheval ( nterms, aclaus, t ) - x * log ( x ) end if if ( indx < 0 ) then clausen = -clausen end if return end subroutine clausen_values ( n_data, x, fx ) !*****************************************************************************80 ! !! CLAUSEN_VALUES returns some values of the Clausen's integral. ! ! Discussion: ! ! The function is defined by: ! ! CLAUSEN(x) = Integral ( 0 <= t <= x ) -ln ( 2 * sin ( t / 2 ) ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 25 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.14137352886760576684D-01, & 0.13955467081981281934D+00, & -0.38495732156574238507D+00, & 0.84831187770367927099D+00, & 0.10139591323607685043D+01, & -0.93921859275409211003D+00, & 0.72714605086327924743D+00, & 0.43359820323553277936D+00, & -0.98026209391301421161D-01, & -0.56814394442986978080D+00, & -0.70969701784448921625D+00, & 0.99282013254695671871D+00, & -0.98127747477447367875D+00, & -0.64078266570172320959D+00, & 0.86027963733231192456D+00, & 0.39071647608680211043D+00, & 0.47574793926539191502D+00, & 0.10105014481412878253D+01, & 0.96332089044363075154D+00, & -0.61782699481929311757D+00 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & -0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & -1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & -3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & -5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & -10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & -30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function debye1 ( xvalue ) !*****************************************************************************80 ! !! DEBYE1 calculates the Debye function of order 1. ! ! Discussion: ! ! The function is defined by: ! ! DEBYE1(x) = 1 / x * Integral ( 0 <= t <= x ) t / ( exp ( t ) - 1 ) dt ! ! The code uses Chebyshev series whose coefficients ! are given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) DEBYE1, the value of the function. ! implicit none real ( kind = 8 ) adeb1(0:18) real ( kind = 8 ) cheval real ( kind = 8 ) debye1 real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) i integer ( kind = 4 ) nexp integer ( kind = 4 ), parameter :: nterms = 15 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: quart = 0.25D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) debinf,expmx, & nine,rk,sum1,t,thirt6,xk,xlim,xlow, & xupper data nine,thirt6 /9.0d0, 36.0d0 / data debinf/0.60792710185402662866d0/ data adeb1/2.40065971903814101941d0, & 0.19372130421893600885d0, & -0.623291245548957703d-2, & 0.35111747702064800d-3, & -0.2282224667012310d-4, & 0.158054678750300d-5, & -0.11353781970719d-6, & 0.835833611875d-8, & -0.62644247872d-9, & 0.4760334890d-10, & -0.365741540d-11, & 0.28354310d-12, & -0.2214729d-13, & 0.174092d-14, & -0.13759d-15, & 0.1093d-16, & -0.87d-18, & 0.7d-19, & -0.1d-19/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow,xupper,xlim/0.298023d-7,35.35051d0,708.39642d0/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'DEBYE1 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' debye1 = zero else if ( x < xlow ) then debye1 = ( ( x - nine ) * x + thirt6 ) / thirt6 else if ( x <= four ) then t = ( ( x * x / eight ) - half ) - half debye1 = cheval ( nterms, adeb1, t ) - quart * x else debye1 = one / ( x * debinf ) if ( x < xlim ) then expmx = exp ( -x ) if ( xupper < x ) then debye1 = debye1 - expmx * ( one + one / x ) else sum1 = zero rk = aint ( xlim / x ) nexp = int ( rk ) xk = rk * x do i = nexp, 1, -1 t = ( one + one / xk ) / rk sum1 = sum1 * expmx + t rk = rk - one xk = xk - x end do debye1 = debye1 - sum1 * expmx end if end if end if return end subroutine debye1_values ( n_data, x, fx ) !*****************************************************************************80 ! !! DEBYE1_VALUES returns some values of Debye's function of order 1. ! ! Discussion: ! ! The function is defined by: ! ! DEBYE1(x) = 1 / x * Integral ( 0 <= t <= x ) t / ( exp ( t ) - 1 ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 27 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.99951182471380889183D+00, & 0.99221462647120597836D+00, & 0.96918395997895308324D+00, & 0.88192715679060552968D+00, & 0.77750463411224827642D+00, & 0.68614531078940204342D+00, & 0.60694728460981007205D+00, & 0.53878956907785587703D+00, & 0.48043521957304283829D+00, & 0.38814802129793784501D+00, & 0.36930802829242526815D+00, & 0.32087619770014612104D+00, & 0.29423996623154246701D+00, & 0.27126046678502189985D+00, & 0.20523930310221503723D+00, & 0.16444346567994602563D+00, & 0.10966194482735821276D+00, & 0.82246701178200016086D-01, & 0.54831135561510852445D-01, & 0.32898681336964528729D-01 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) ! if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function debye2 ( xvalue ) !*****************************************************************************80 ! !! DEBYE2 calculates the Debye function of order 2. ! ! Discussion: ! ! The function is defined by: ! ! DEBYE2(x) = 2 / x^2 * Integral ( 0 <= t <= x ) t^2 / ( exp ( t ) - 1 ) dt ! ! The code uses Chebyshev series whose coefficients ! are given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 24 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) DEBYE2, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ) debye2 real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) i integer ( kind = 4 ) nexp integer ( kind = 4 ), parameter :: nterms = 17 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) adeb2(0:18),debinf,expmx, & rk,sum1,t,twent4,xk,xlim1, & xlim2,xlow,xupper data twent4/24.0d0/ data debinf/4.80822761263837714160d0/ data adeb2/2.59438102325707702826d0, & 0.28633572045307198337d0, & -0.1020626561580467129d-1, & 0.60491097753468435d-3, & -0.4052576589502104d-4, & 0.286338263288107d-5, & -0.20863943030651d-6, & 0.1552378758264d-7, & -0.117312800866d-8, & 0.8973585888d-10, & -0.693176137d-11, & 0.53980568d-12, & -0.4232405d-13, & 0.333778d-14, & -0.26455d-15, & 0.2106d-16, & -0.168d-17, & 0.13d-18, & -0.1d-19/ ! ! Machine-dependent constants ! data xlow,xupper/0.298023d-7,35.35051d0/ data xlim1,xlim2/708.39642d0,2.1572317d154/ ! x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'DEBYE2 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' debye2 = zero else if ( x < xlow ) then debye2 = ( ( x - eight ) * x + twent4 ) / twent4 else if ( x <= four ) then t = ( ( x * x / eight ) - half ) - half debye2 = cheval ( nterms, adeb2, t ) - x / three else if ( x <= xupper ) then expmx = exp ( -x ) sum1 = zero rk = aint ( xlim1 / x ) nexp = int ( rk ) xk = rk * x do i = nexp, 1, -1 t = ( one + two / xk + two / ( xk * xk ) ) / rk sum1 = sum1 * expmx + t rk = rk - one xk = xk - x end do debye2 = debinf / ( x * x ) - two * sum1 * expmx else if ( x < xlim1 ) then expmx = exp ( -x ) sum1 = ( ( x + two ) * x + two ) / ( x * x ) debye2 = debinf / ( x * x ) - two * sum1 * expmx else if ( x <= xlim2 ) then debye2 = debinf / ( x * x ) else debye2 = zero end if return end subroutine debye2_values ( n_data, x, fx ) !*****************************************************************************80 ! !! DEBYE2_VALUES returns some values of Debye's function of order 2. ! ! Discussion: ! ! The function is defined by: ! ! DEBYE2(x) = 2 / x^2 * Integral ( 0 <= t <= x ) t^2 / ( exp ( t ) - 1 ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 27 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.99934911727904599738D+00, & 0.98962402299599181205D+00, & 0.95898426200345986743D+00, & 0.84372119334725358934D+00, & 0.70787847562782928288D+00, & 0.59149637225671282917D+00, & 0.49308264399053185014D+00, & 0.41079413579749669069D+00, & 0.34261396060786351671D+00, & 0.24055368752127897660D+00, & 0.22082770061202308232D+00, & 0.17232915939014138975D+00, & 0.14724346738730182894D+00, & 0.12666919046715789982D+00, & 0.74268805954862854626D-01, & 0.47971498020121871622D-01, & 0.21369201683658373846D-01, & 0.12020564476446432799D-01, & 0.53424751249537071952D-02, & 0.19232910450553508562D-02 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) ! if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function debye3 ( xvalue ) !*****************************************************************************80 ! !! DEBYE3 calculates the Debye function of order 3. ! ! Discussion: ! ! The function is defined by: ! ! DEBYE3(x) = 3 / x^3 * Integral ( 0 <= t <= x ) t^3 / ( exp ( t ) - 1 ) dt ! ! The code uses Chebyshev series whose coefficients ! are given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) DEBYE3, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ) debye3 real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) i integer ( kind = 4 ) nexp integer ( kind = 4 ), parameter :: nterms = 16 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: six = 6.0D+00 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) adeb3(0:18),debinf,expmx, & pt375,rk,sevp5,sum1,t,twenty, & xk,xki,xlim1,xlim2,xlow,xupper data pt375/0.375d0/ data sevp5,twenty/7.5d0 , 20.0d0/ data debinf/0.51329911273421675946d-1/ data adeb3/2.70773706832744094526d0, & 0.34006813521109175100d0, & -0.1294515018444086863d-1, & 0.79637553801738164d-3, & -0.5463600095908238d-4, & 0.392430195988049d-5, & -0.28940328235386d-6, & 0.2173176139625d-7, & -0.165420999498d-8, & 0.12727961892d-9, & -0.987963459d-11, & 0.77250740d-12, & -0.6077972d-13, & 0.480759d-14, & -0.38204d-15, & 0.3048d-16, & -0.244d-17, & 0.20d-18, & -0.2d-19/ ! ! Machine-dependent constants ! data xlow,xupper/0.298023d-7,35.35051d0/ data xlim1,xlim2/708.39642d0,0.9487163d103/ ! x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'DEBYE3 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' debye3 = zero return end if if ( x < xlow ) then debye3 = ( ( x - sevp5 ) * x + twenty ) / twenty else if ( x <= 4 ) then t = ( ( x * x / eight ) - half ) - half debye3 = cheval ( nterms, adeb3, t ) - pt375 * x else ! ! Code for x > 4.0 ! if ( xlim2 < x ) then debye3 = zero else debye3 = one / ( debinf * x * x * x ) if ( x < xlim1 ) then expmx = exp ( -x ) if ( xupper < x ) then sum1 = ((( x + three ) * x + six ) * x + six ) / ( x * x * x ) else sum1 = zero rk = aint ( xlim1 / x ) nexp = int ( rk ) xk = rk * x do i = nexp, 1, -1 xki = one / xk t = ((( six * xki + six ) * xki + three ) * xki + one ) / rk sum1 = sum1 * expmx + t rk = rk - one xk = xk - x end do end if debye3 = debye3 - three * sum1 * expmx end if end if end if return end subroutine debye3_values ( n_data, x, fx ) !*****************************************************************************80 ! !! DEBYE3_VALUES returns some values of Debye's function of order 3. ! ! Discussion: ! ! The function is defined by: ! ! DEBYE3(x) = 3 / x^3 * Integral ( 0 <= t <= x ) t^3 / ( exp ( t ) - 1 ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 28 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.99926776885985461940D+00, & 0.98833007755734698212D+00, & 0.95390610472023510237D+00, & 0.82496296897623372315D+00, & 0.67441556407781468010D+00, & 0.54710665141286285468D+00, & 0.44112847372762418113D+00, & 0.35413603481042394211D+00, & 0.28357982814342246206D+00, & 0.18173691382177474795D+00, & 0.16277924385112436877D+00, & 0.11759741179993396450D+00, & 0.95240802723158889887D-01, & 0.77581324733763020269D-01, & 0.36560295673194845002D-01, & 0.19295765690345489563D-01, & 0.57712632276188798621D-02, & 0.24352200674805479827D-02, & 0.72154882216335666096D-03, & 0.15585454565440389896D-03 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) ! if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function debye4 ( xvalue ) !*****************************************************************************80 ! !! DEBYE4 calculates the Debye function of order 4. ! ! Discussion: ! ! The function is defined by: ! ! DEBYE4(x) = 4 / x^4 * Integral ( 0 <= t <= x ) t^4 / ( exp ( t ) - 1 ) dt ! ! The code uses Chebyshev series whose coefficients ! are given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) DEBYE4, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ) debye4 real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) i integer ( kind = 4 ) nexp integer ( kind = 4 ), parameter :: nterms = 16 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) adeb4(0:18),debinf,eightn,expmx, & five,forty5,rk,sum1,t,twelve,twent4, & twopt5,xk,xki,xlim1,xlim2,xlow,xupper data twopt5,five/2.5d0, 5.0d0/ data twelve,eightn/ 12.0d0 , 18.0d0/ data twent4,forty5 /24.0d0 , 45.0d0 / data debinf/99.54506449376351292781d0/ data adeb4/2.78186941502052346008d0, & 0.37497678352689286364d0, & -0.1494090739903158326d-1, & 0.94567981143704274d-3, & -0.6613291613893255d-4, & 0.481563298214449d-5, & -0.35880839587593d-6, & 0.2716011874160d-7, & -0.208070991223d-8, & 0.16093838692d-9, & -0.1254709791d-10, & 0.98472647d-12, & -0.7772369d-13, & 0.616483d-14, & -0.49107d-15, & 0.3927d-16, & -0.315d-17, & 0.25d-18, & -0.2d-19/ ! ! Machine-dependent constants ! data xlow,xupper/0.298023d-7,35.35051d0/ data xlim1,xlim2/708.39642d0,2.5826924d77/ x = xvalue ! ! Check XVALUE >= 0.0 ! if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'DEBYE4 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' debye4 = zero return end if if ( x < xlow ) then debye4 = ( ( twopt5 * x - eightn ) * x + forty5 ) / forty5 else if ( x <= four ) then t = ( ( x * x / eight ) - half ) - half debye4 = cheval ( nterms, adeb4, t ) - ( x + x ) / five else ! ! Code for x > 4.0 ! if ( xlim2 < x ) then debye4 = zero else t = x * x debye4 = ( debinf / t ) / t if ( x < xlim1 ) then expmx = exp ( -x ) if ( xupper < x ) then sum1 = ( ( ( ( x + four ) * x + twelve ) * x + & twent4 ) * x + twent4 ) / ( x * x * x * x ) else sum1 = zero rk = aint ( xlim1 / x ) nexp = int ( rk ) xk = rk * x do i = nexp, 1, -1 xki = one / xk t = ( ( ( ( twent4 * xki + twent4 ) * xki + & twelve ) * xki + four ) * xki + one ) / rk sum1 = sum1 * expmx + t rk = rk - one xk = xk - x end do end if debye4 = debye4 - four * sum1 * expmx end if end if end if return end subroutine debye4_values ( n_data, x, fx ) !*****************************************************************************80 ! !! DEBYE4_VALUES returns some values of Debye's function of order 4. ! ! Discussion: ! ! The function is defined by: ! ! DEBYE4(x) = 4 / x^4 * Integral ( 0 <= t <= x ) t^4 / ( exp ( t ) - 1 ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 28 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.99921896192761576256D+00, & 0.98755425280996071022D+00, & 0.95086788606389739976D+00, & 0.81384569172034042516D+00, & 0.65487406888673697092D+00, & 0.52162830964878715188D+00, & 0.41189273671788528876D+00, & 0.32295434858707304628D+00, & 0.25187863642883314410D+00, & 0.15185461258672022043D+00, & 0.13372661145921413299D+00, & 0.91471377664481164749D-01, & 0.71227828197462523663D-01, & 0.55676547822738862783D-01, & 0.21967566525574960096D-01, & 0.96736755602711590082D-02, & 0.19646978158351837850D-02, & 0.62214648623965450200D-03, & 0.12289514092077854510D-03, & 0.15927210319002161231D-04 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) ! if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function exp3_int ( xvalue ) !*****************************************************************************80 ! !! EXP3_INT calculates the integral of exp(-t^3). ! ! Discussion: ! ! The function is defined by: ! ! EXP3_INT(x) = Integral ( 0 <= t <= x ) exp ( -t^3 ) dt ! ! The code uses Chebyshev expansions, whose coefficients are ! given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) EXP3_INT, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ) exp3_int real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ), parameter :: nterm1 = 22 integer ( kind = 4 ), parameter :: nterm2 = 20 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) aexp3(0:24),aexp3a(0:24), & funinf,sixten,t, & xlow,xupper data sixten /16.0d0 / data funinf/0.89297951156924921122d0/ data aexp3(0)/ 1.26919841422112601434d0/ data aexp3(1)/ -0.24884644638414098226d0/ data aexp3(2)/ 0.8052622071723104125d-1/ data aexp3(3)/ -0.2577273325196832934d-1/ data aexp3(4)/ 0.759987887307377429d-2/ data aexp3(5)/ -0.203069558194040510d-2/ data aexp3(6)/ 0.49083458669932917d-3/ data aexp3(7)/ -0.10768223914202077d-3/ data aexp3(8)/ 0.2155172626428984d-4/ data aexp3(9)/ -0.395670513738429d-5/ data aexp3(10)/ 0.66992409338956d-6/ data aexp3(11)/-0.10513218080703d-6/ data aexp3(12)/ 0.1536258019825d-7/ data aexp3(13)/-0.209909603636d-8/ data aexp3(14)/ 0.26921095381d-9/ data aexp3(15)/-0.3251952422d-10/ data aexp3(16)/ 0.371148157d-11/ data aexp3(17)/-0.40136518d-12/ data aexp3(18)/ 0.4123346d-13/ data aexp3(19)/-0.403375d-14/ data aexp3(20)/ 0.37658d-15/ data aexp3(21)/-0.3362d-16/ data aexp3(22)/ 0.288d-17/ data aexp3(23)/-0.24d-18/ data aexp3(24)/ 0.2d-19/ data aexp3a(0)/ 1.92704649550682737293d0/ data aexp3a(1)/ -0.3492935652048138054d-1/ data aexp3a(2)/ 0.145033837189830093d-2/ data aexp3a(3)/ -0.8925336718327903d-4/ data aexp3a(4)/ 0.705423921911838d-5/ data aexp3a(5)/ -0.66717274547611d-6/ data aexp3a(6)/ 0.7242675899824d-7/ data aexp3a(7)/ -0.878258256056d-8/ data aexp3a(8)/ 0.116722344278d-8/ data aexp3a(9)/ -0.16766312812d-9/ data aexp3a(10)/ 0.2575501577d-10/ data aexp3a(11)/-0.419578881d-11/ data aexp3a(12)/ 0.72010412d-12/ data aexp3a(13)/-0.12949055d-12/ data aexp3a(14)/ 0.2428703d-13/ data aexp3a(15)/-0.473311d-14/ data aexp3a(16)/ 0.95531d-15/ data aexp3a(17)/-0.19914d-15/ data aexp3a(18)/ 0.4277d-16/ data aexp3a(19)/-0.944d-17/ data aexp3a(20)/ 0.214d-17/ data aexp3a(21)/-0.50d-18/ data aexp3a(22)/ 0.12d-18/ data aexp3a(23)/-0.3d-19/ data aexp3a(24)/ 0.1d-19/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow,xupper/0.762939d-5,3.3243018d0/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'EXP3_INT - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' exp3_int = zero else if ( x < xlow ) then exp3_int = x else if ( x <= two ) then t = ( ( x * x * x / four ) - half ) - half exp3_int = x * cheval ( nterm1, aexp3, t ) else if ( x <= xupper ) then t = ( ( sixten / ( x * x * x ) ) - half ) - half t = cheval ( nterm2, aexp3a, t ) t = t * exp ( -x * x * x ) / ( three * x * x ) exp3_int = funinf - t else exp3_int = funinf end if return end subroutine exp3_int_values ( n_data, x, fx ) !*****************************************************************************80 ! !! EXP3_INT_VALUES returns some values of the EXP3 integral function. ! ! Discussion: ! ! The function is defined by: ! ! EXP3_INT(x) = Integral ( 0 <= t <= x ) exp ( -t^3 ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 28 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.19531249963620212007D-02, & 0.78124990686775522671D-02, & 0.31249761583499728667D-01, & 0.12493899888803079984D+00, & 0.48491714311363971332D+00, & 0.80751118213967145286D+00, & 0.86889265412623270696D+00, & 0.88861722235357162648D+00, & 0.89286018500218176869D+00, & 0.89295351429387631138D+00, & 0.89297479112737843939D+00, & 0.89297880579798112220D+00, & 0.89297950317496621294D+00, & 0.89297951152951902903D+00, & 0.89297951156918122102D+00, & 0.89297951156924734716D+00, & 0.89297951156924917298D+00, & 0.89297951156924921121D+00, & 0.89297951156924921122D+00, & 0.89297951156924921122D+00 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0078125000D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.2500000000D+00, & 1.5000000000D+00, & 1.8750000000D+00, & 2.0000000000D+00, & 2.1250000000D+00, & 2.2500000000D+00, & 2.5000000000D+00, & 2.7500000000D+00, & 3.0000000000D+00, & 3.1250000000D+00, & 3.2500000000D+00, & 3.5000000000D+00, & 3.7500000000D+00, & 4.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function goodwin ( xvalue ) !*****************************************************************************80 ! !! GOODWIN calculates the integral of exp(-t^2/(t+x)). ! ! Discussion: ! ! The function is defined by: ! ! GOODWIN(x) = Integral ( 0 <= t < infinity ) exp ( -t^2 ) / ( t + x ) dt ! ! The code uses Chebyshev expansions whose coefficients are ! given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 29 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) GOODWIN, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ) goodwin real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ), parameter :: nterm1 = 26 integer ( kind = 4 ), parameter :: nterm2 = 20 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: six = 6.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) agost(0:28),agosta(0:23), & fval,gamby2,rtpib2, & t,xhigh,xlow data gamby2/0.28860783245076643030d0/ data rtpib2/0.88622692545275801365d0/ data agost(0)/ 0.63106560560398446247d0/ data agost(1)/ 0.25051737793216708827d0/ data agost(2)/ -0.28466205979018940757d0/ data agost(3)/ 0.8761587523948623552d-1/ data agost(4)/ 0.682602267221252724d-2/ data agost(5)/ -0.1081129544192254677d-1/ data agost(6)/ 0.169101244117152176d-2/ data agost(7)/ 0.50272984622615186d-3/ data agost(8)/ -0.18576687204100084d-3/ data agost(9)/ -0.428703674168474d-5/ data agost(10)/ 0.1009598903202905d-4/ data agost(11)/-0.86529913517382d-6/ data agost(12)/-0.34983874320734d-6/ data agost(13)/ 0.6483278683494d-7/ data agost(14)/ 0.757592498583d-8/ data agost(15)/-0.277935424362d-8/ data agost(16)/-0.4830235135d-10/ data agost(17)/ 0.8663221283d-10/ data agost(18)/-0.394339687d-11/ data agost(19)/-0.209529625d-11/ data agost(20)/ 0.21501759d-12/ data agost(21)/ 0.3959015d-13/ data agost(22)/-0.692279d-14/ data agost(23)/-0.54829d-15/ data agost(24)/ 0.17108d-15/ data agost(25)/ 0.376d-17/ data agost(26)/-0.349d-17/ data agost(27)/ 0.7d-19/ data agost(28)/ 0.6d-19/ data agosta(0)/ 1.81775467984718758767d0/ data agosta(1)/ -0.9921146570744097467d-1/ data agosta(2)/ -0.894058645254819243d-2/ data agosta(3)/ -0.94955331277726785d-3/ data agosta(4)/ -0.10971379966759665d-3/ data agosta(5)/ -0.1346694539578590d-4/ data agosta(6)/ -0.172749274308265d-5/ data agosta(7)/ -0.22931380199498d-6/ data agosta(8)/ -0.3127844178918d-7/ data agosta(9)/ -0.436197973671d-8/ data agosta(10)/-0.61958464743d-9/ data agosta(11)/-0.8937991276d-10/ data agosta(12)/-0.1306511094d-10/ data agosta(13)/-0.193166876d-11/ data agosta(14)/-0.28844270d-12/ data agosta(15)/-0.4344796d-13/ data agosta(16)/-0.659518d-14/ data agosta(17)/-0.100801d-14/ data agosta(18)/-0.15502d-15/ data agosta(19)/-0.2397d-16/ data agosta(20)/-0.373d-17/ data agosta(21)/-0.58d-18/ data agosta(22)/-0.9d-19/ data agosta(23)/-0.1d-19/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow,xhigh/1.11022303d-16,1.80144d16/ x = xvalue if ( x <= zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'GOODWIN - Fatal error!' write ( *, '(a)' ) ' Argument X <= 0.' goodwin = zero else if ( x < xlow ) then goodwin = - gamby2 - log ( x ) else if ( x <= two ) then t = ( x - half ) - half goodwin = cheval ( nterm1, agost, t ) - exp ( -x * x ) * log ( x ) else if ( x <= xhigh ) then fval = rtpib2 / x t = ( six - x ) / ( two + x ) goodwin = fval * cheval ( nterm2, agosta, t ) else goodwin = rtpib2 / x end if return end subroutine goodwin_values ( n_data, x, fx ) !*****************************************************************************80 ! !! GOODWIN_VALUES returns some values of the Goodwin and Staton function. ! ! Discussion: ! ! The function is defined by: ! ! GOODWIN(x) = Integral ( 0 <= t < infinity ) exp ( -t^2 ) / ( t + x ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 29 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.59531540040441651584D+01, & 0.45769601268624494109D+01, & 0.32288921331902217638D+01, & 0.19746110873568719362D+01, & 0.96356046208697728563D+00, & 0.60513365250334458174D+00, & 0.51305506459532198016D+00, & 0.44598602820946133091D+00, & 0.37344458206879749357D+00, & 0.35433592884953063055D+00, & 0.33712156518881920994D+00, & 0.29436170729362979176D+00, & 0.25193499644897222840D+00, & 0.22028778222123939276D+00, & 0.19575258237698917033D+00, & 0.17616303166670699424D+00, & 0.16015469479664778673D+00, & 0.14096116876193391066D+00, & 0.13554987191049066274D+00, & 0.11751605060085098084D+00 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0078125000D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.2500000000D+00, & 1.5000000000D+00, & 1.8750000000D+00, & 2.0000000000D+00, & 2.1250000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 3.5000000000D+00, & 4.0000000000D+00, & 4.5000000000D+00, & 5.0000000000D+00, & 5.7500000000D+00, & 6.0000000000D+00, & 7.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function i0ml0 ( xvalue ) !*****************************************************************************80 ! !! I0ML0 calculates difference between the Bessel I0 and Struve L0 functions. ! ! Discussion: ! ! The function is defined by: ! ! I0ML0(x) = I0(x) - L0(x) ! ! I0(x) is the modified Bessel function of the first kind of order 0, ! L0(x) is the modified Struve function of order 0. ! ! The code uses Chebyshev expansions with the coefficients ! given to an accuracy of 20D. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 29 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) I0ML0, the value of the function. ! implicit none real ( kind = 8 ) ai0l0(0:23) real ( kind = 8 ) ai0l0a(0:23) real ( kind = 8 ) cheval real ( kind = 8 ) i0ml0 integer ( kind = 4 ), parameter :: nterm1 = 21 integer ( kind = 4 ), parameter :: nterm2 = 21 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: six = 6.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) atehun, & forty,sixten,t,twobpi,two88,xhigh, & xlow,xsq data sixten/ 16.0d0 / data forty / 40.0d0 / data two88,atehun/ 288.0d0, 800.0d0 / data twobpi/0.63661977236758134308d0/ data ai0l0(0)/ 0.52468736791485599138d0/ data ai0l0(1)/ -0.35612460699650586196d0/ data ai0l0(2)/ 0.20487202864009927687d0/ data ai0l0(3)/ -0.10418640520402693629d0/ data ai0l0(4)/ 0.4634211095548429228d-1/ data ai0l0(5)/ -0.1790587192403498630d-1/ data ai0l0(6)/ 0.597968695481143177d-2/ data ai0l0(7)/ -0.171777547693565429d-2/ data ai0l0(8)/ 0.42204654469171422d-3/ data ai0l0(9)/ -0.8796178522094125d-4/ data ai0l0(10)/ 0.1535434234869223d-4/ data ai0l0(11)/-0.219780769584743d-5/ data ai0l0(12)/ 0.24820683936666d-6/ data ai0l0(13)/-0.2032706035607d-7/ data ai0l0(14)/ 0.90984198421d-9/ data ai0l0(15)/ 0.2561793929d-10/ data ai0l0(16)/-0.710609790d-11/ data ai0l0(17)/ 0.32716960d-12/ data ai0l0(18)/ 0.2300215d-13/ data ai0l0(19)/-0.292109d-14/ data ai0l0(20)/-0.3566d-16/ data ai0l0(21)/ 0.1832d-16/ data ai0l0(22)/-0.10d-18/ data ai0l0(23)/-0.11d-18/ data ai0l0a(0)/ 2.00326510241160643125d0/ data ai0l0a(1)/ 0.195206851576492081d-2/ data ai0l0a(2)/ 0.38239523569908328d-3/ data ai0l0a(3)/ 0.7534280817054436d-4/ data ai0l0a(4)/ 0.1495957655897078d-4/ data ai0l0a(5)/ 0.299940531210557d-5/ data ai0l0a(6)/ 0.60769604822459d-6/ data ai0l0a(7)/ 0.12399495544506d-6/ data ai0l0a(8)/ 0.2523262552649d-7/ data ai0l0a(9)/ 0.504634857332d-8/ data ai0l0a(10)/0.97913236230d-9/ data ai0l0a(11)/0.18389115241d-9/ data ai0l0a(12)/0.3376309278d-10/ data ai0l0a(13)/0.611179703d-11/ data ai0l0a(14)/0.108472972d-11/ data ai0l0a(15)/0.18861271d-12/ data ai0l0a(16)/0.3280345d-13/ data ai0l0a(17)/0.565647d-14/ data ai0l0a(18)/0.93300d-15/ data ai0l0a(19)/0.15881d-15/ data ai0l0a(20)/0.2791d-16/ data ai0l0a(21)/0.389d-17/ data ai0l0a(22)/0.70d-18/ data ai0l0a(23)/0.16d-18/ ! ! MACHINE-DEPENDENT CONSTANTS (suitable for IEEE-arithmetic machines) ! data xlow,xhigh/1.11022303d-16,1.8981253d9/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'I0ML0 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' i0ml0 = zero else if ( x < xlow ) then i0ml0 = one else if ( x <= sixten ) then t = ( six * x - forty ) / ( x + forty ) i0ml0 = cheval ( nterm1, ai0l0, t ) else if ( x <= xhigh ) then xsq = x * x t = ( atehun - xsq ) / ( two88 + xsq ) i0ml0 = cheval ( nterm2, ai0l0a, t ) * twobpi / x else i0ml0 = twobpi / x end if return end subroutine i0ml0_values ( n_data, x, fx ) !*****************************************************************************80 ! !! I0ML0_VALUES returns some values of the I0ML0 function. ! ! Discussion: ! ! The function is defined by: ! ! I0ML0(x) = I0(x) - L0(x) ! ! I0(x) is the modified Bessel function of the first kind of order 0, ! L0(x) is the modified Struve function of order 0. ! ! The data was reported by McLeod. ! ! Modified: ! ! 30 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.99875755515461749793D+00, & 0.99011358230706643807D+00, & 0.92419435310023947018D+00, & 0.73624267134714273902D+00, & 0.55582269181411744686D+00, & 0.34215154434462160628D+00, & 0.17087174888774706539D+00, & 0.81081008709219208918D-01, & 0.53449421441089580702D-01, & 0.39950321008923244846D-01, & 0.39330637437584921392D-01, & 0.37582274342808670750D-01, & 0.31912486554480390343D-01, & 0.25506146883504738403D-01, & 0.21244480317825292412D-01, & 0.15925498348551684335D-01, & 0.12737506927242585015D-01, & 0.84897750814784916847D-02, & 0.63668349178454469153D-02, & 0.50932843163122551114D-02 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0156250000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 2.0000000000D+00, & 4.0000000000D+00, & 8.0000000000D+00, & 12.0000000000D+00, & 16.0000000000D+00, & 16.2500000000D+00, & 17.0000000000D+00, & 20.0000000000D+00, & 25.0000000000D+00, & 30.0000000000D+00, & 40.0000000000D+00, & 50.0000000000D+00, & 75.0000000000D+00, & 100.0000000000D+00, & 125.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function i1ml1 ( xvalue ) !*****************************************************************************80 ! !! I1ML1 calculates difference between the Bessel I1 and Struve L1 functions. ! ! Discussion: ! ! The function is defined by: ! ! I1ML1(x) = I1(x) - L1(x) ! ! I1(x) is the modified Bessel function of the first kind of order 1, ! L1(x) is the modified Struve function of order 1. ! ! The code uses Chebyshev expansions with the coefficients ! given to an accuracy of 20D. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 29 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! 0 <= XVALUE is required. ! ! Output, real ( kind = 8 ) I1ML1, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ) i1ml1 integer ( kind = 4 ), parameter :: nterm1 = 20 integer ( kind = 4 ), parameter :: nterm2 = 22 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: six = 6.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) ai1l1(0:23),ai1l1a(0:25),atehun, & forty,sixten,t,twobpi,two88, & xhigh,xlow,xsq data sixten,forty/ 16.0d0 , 40.0d0 / data two88,atehun/ 288.0d0 , 800.0d0 / data twobpi/0.63661977236758134308d0/ data ai1l1(0)/ 0.67536369062350576137d0/ data ai1l1(1)/ -0.38134971097266559040d0/ data ai1l1(2)/ 0.17452170775133943559d0/ data ai1l1(3)/ -0.7062105887235025061d-1/ data ai1l1(4)/ 0.2517341413558803702d-1/ data ai1l1(5)/ -0.787098561606423321d-2/ data ai1l1(6)/ 0.214814368651922006d-2/ data ai1l1(7)/ -0.50862199717906236d-3/ data ai1l1(8)/ 0.10362608280442330d-3/ data ai1l1(9)/ -0.1795447212057247d-4/ data ai1l1(10)/ 0.259788274515414d-5/ data ai1l1(11)/-0.30442406324667d-6/ data ai1l1(12)/ 0.2720239894766d-7/ data ai1l1(13)/-0.158126144190d-8/ data ai1l1(14)/ 0.1816209172d-10/ data ai1l1(15)/ 0.647967659d-11/ data ai1l1(16)/-0.54113290d-12/ data ai1l1(17)/-0.308311d-14/ data ai1l1(18)/ 0.305638d-14/ data ai1l1(19)/-0.9717d-16/ data ai1l1(20)/-0.1422d-16/ data ai1l1(21)/ 0.84d-18/ data ai1l1(22)/ 0.7d-19/ data ai1l1(23)/-0.1d-19/ data ai1l1a(0)/ 1.99679361896789136501d0/ data ai1l1a(1)/ -0.190663261409686132d-2/ data ai1l1a(2)/ -0.36094622410174481d-3/ data ai1l1a(3)/ -0.6841847304599820d-4/ data ai1l1a(4)/ -0.1299008228509426d-4/ data ai1l1a(5)/ -0.247152188705765d-5/ data ai1l1a(6)/ -0.47147839691972d-6/ data ai1l1a(7)/ -0.9020819982592d-7/ data ai1l1a(8)/ -0.1730458637504d-7/ data ai1l1a(9)/ -0.332323670159d-8/ data ai1l1a(10)/-0.63736421735d-9/ data ai1l1a(11)/-0.12180239756d-9/ data ai1l1a(12)/-0.2317346832d-10/ data ai1l1a(13)/-0.439068833d-11/ data ai1l1a(14)/-0.82847110d-12/ data ai1l1a(15)/-0.15562249d-12/ data ai1l1a(16)/-0.2913112d-13/ data ai1l1a(17)/-0.543965d-14/ data ai1l1a(18)/-0.101177d-14/ data ai1l1a(19)/-0.18767d-15/ data ai1l1a(20)/-0.3484d-16/ data ai1l1a(21)/-0.643d-17/ data ai1l1a(22)/-0.118d-17/ data ai1l1a(23)/-0.22d-18/ data ai1l1a(24)/-0.4d-19/ data ai1l1a(25)/-0.1d-19/ ! ! MACHINE-DEPENDENT CONSTANTS (suitable for IEEE machines) ! data xlow,xhigh/2.22044605d-16,1.8981253d9/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'I1ML1 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' i1ml1 = zero else if ( x < xlow ) then i1ml1 = x / two else if ( x <= sixten ) then t = ( six * x - forty ) / ( x + forty ) i1ml1 = cheval ( nterm1, ai1l1, t ) * x / two else if ( x <= xhigh ) then xsq = x * x t = ( atehun - xsq ) / ( two88 + xsq ) i1ml1 = cheval ( nterm2, ai1l1a, t ) * twobpi else i1ml1 = twobpi end if return end subroutine i1ml1_values ( n_data, x, fx ) !*****************************************************************************80 ! !! I1ML1_VALUES returns some values of the I1ML1 function. ! ! Discussion: ! ! The function is defined by: ! ! I1ML1(x) = I1(x) - L1(x) ! ! I1(x) is the modified Bessel function of the first kind of order 1, ! L1(x) is the modified Struve function of order 1. ! ! The data was reported by McLeod. ! ! Modified: ! ! 30 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.97575346155386267134D-03, & 0.77609293280609272733D-02, & 0.59302966404545373770D-01, & 0.20395212276737365307D+00, & 0.33839472293667639038D+00, & 0.48787706726961324579D+00, & 0.59018734196576517506D+00, & 0.62604539530312149476D+00, & 0.63209315274909764698D+00, & 0.63410179313235359215D+00, & 0.63417966797578128188D+00, & 0.63439268632392089434D+00, & 0.63501579073257770690D+00, & 0.63559616677359459337D+00, & 0.63591001826697110312D+00, & 0.63622113181751073643D+00, & 0.63636481702133606597D+00, & 0.63650653499619902120D+00, & 0.63655609126300261851D+00, & 0.63657902087183929223D+00 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0156250000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 2.0000000000D+00, & 4.0000000000D+00, & 8.0000000000D+00, & 12.0000000000D+00, & 16.0000000000D+00, & 16.2500000000D+00, & 17.0000000000D+00, & 20.0000000000D+00, & 25.0000000000D+00, & 30.0000000000D+00, & 40.0000000000D+00, & 50.0000000000D+00, & 75.0000000000D+00, & 100.0000000000D+00, & 125.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function lobachevsky ( xvalue ) !*****************************************************************************80 ! !! LOBACHEVSKY calculates the Lobachevsky function. ! ! Discussion: ! ! The function is defined by: ! ! LOBACHEVSKY(x) = Integral ( 0 <= t <= x ) -ln ( abs ( cos ( t ) ) dt ! ! The code uses Chebyshev expansions whose coefficients are given ! to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) LOBACHEVSKY, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) indpi2 integer ( kind = 4 ) indsgn integer ( kind = 4 ) npi integer ( kind = 4 ), parameter :: nterm1 = 13 integer ( kind = 4 ), parameter :: nterm2 = 9 real ( kind = 8 ) lobachevsky real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: six = 6.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) arlob1(0:15),arlob2(0:10), & fval,fval1,lbpb21,lbpb22,lobpia,lobpib, & lobpi1,lobpi2,pi,piby2,piby21,piby22,piby4,pi1, & pi11,pi12,pi2,t,tcon,xcub,xhigh,xlow1, & xlow2,xlow3,xr data lobpia,lobpib/ 1115.0d0 , 512.0d0 / data lobpi2/-1.48284696397869499311d-4/ data lbpb22/-7.41423481989347496556d-5/ data pi11,pi12/ 201.0d0 , 64.0d0 / data pi2/9.67653589793238462643d-4/ data piby22/4.83826794896619231322d-4/ data tcon/3.24227787655480868620d0/ data arlob1/0.34464884953481300507d0, & 0.584198357190277669d-2, & 0.19175029694600330d-3, & 0.787251606456769d-5, & 0.36507477415804d-6, & 0.1830287272680d-7, & 0.96890333005d-9, & 0.5339055444d-10, & 0.303408025d-11, & 0.17667875d-12, & 0.1049393d-13, & 0.63359d-15, & 0.3878d-16, & 0.240d-17, & 0.15d-18, & 0.1d-19/ data arlob2/2.03459418036132851087d0, & 0.1735185882027407681d-1, & 0.5516280426090521d-4, & 0.39781646276598d-6, & 0.369018028918d-8, & 0.3880409214d-10, & 0.44069698d-12, & 0.527674d-14, & 0.6568d-16, & 0.84d-18, & 0.1d-19/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow1,xlow2/5.11091385d-103,4.71216091d-8/ data xlow3,xhigh/6.32202727d-8,4.5035996d15/ x = abs ( xvalue ) indsgn = 1 if ( xvalue < zero ) then indsgn = -1 end if if ( xhigh < x ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'LOBACHEVSKY - Fatal error!' write ( *, '(a)' ) ' Argument magnitude too large.' lobachevsky = zero return end if ! ! Reduce argument to [0,pi] ! pi1 = pi11 / pi12 pi = pi1 + pi2 piby2 = pi / two piby21 = pi1 / two piby4 = piby2 / two npi = int ( x / pi ) xr = ( x - npi * pi1 ) - npi * pi2 ! ! Reduce argument to [0,pi/2] ! indpi2 = 0 if ( piby2 < xr ) then indpi2 = 1 xr = ( pi1 - xr ) + pi2 end if ! ! Code for argument in [0,pi/4] ! if ( xr <= piby4 ) then if ( xr < xlow1 ) then fval = zero else xcub = xr * xr * xr if ( xr < xlow2 ) then fval = xcub / six else t = ( tcon * xr * xr - half ) - half fval = xcub * cheval ( nterm1, arlob1, t ) end if end if else ! ! Code for argument in [pi/4,pi/2] ! xr = ( piby21 - xr ) + piby22 if ( xr == zero ) then fval1 = zero else if ( xr < xlow3 ) then fval1 = xr * ( one - log ( xr ) ) else t = ( tcon * xr * xr - half ) - half fval1 = xr * ( cheval ( nterm2, arlob2, t ) - log ( xr ) ) end if end if lbpb21 = lobpia / ( lobpib + lobpib ) fval = ( lbpb21 - fval1 ) + lbpb22 end if lobpi1 = lobpia / lobpib ! ! Compute value for argument in [pi/2,pi] ! if ( indpi2 == 1 ) then fval = ( lobpi1 - fval ) + lobpi2 end if if ( npi <= 0 ) then lobachevsky = fval else lobachevsky = ( fval + npi * lobpi2 ) + npi * lobpi1 end if if ( indsgn == -1 ) then lobachevsky = -lobachevsky end if return end subroutine lobachevsky_values ( n_data, x, fx ) !*****************************************************************************80 ! !! LOBACHEVSKY_VALUES returns some values of the Lobachevsky function. ! ! Discussion: ! ! The function is defined by: ! ! LOBACHEVSKY(x) = Integral ( 0 <= t <= x ) -ln ( abs ( cos ( t ) ) dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 31 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.12417639065161393857D-08, & 0.79473344770001088225D-07, & 0.50867598186208834198D-05, & 0.32603097901207200319D-03, & 0.21380536815408214419D-01, & 0.18753816902083824050D+00, & 0.83051199971883645115D+00, & 0.18854362426679034904D+01, & 0.21315988986516411053D+01, & 0.21771120185613427221D+01, & 0.22921027921896650849D+01, & 0.39137195028784495586D+01, & 0.43513563983836427904D+01, & 0.44200644968478185898D+01, & 0.65656013133623829156D+01, & 0.10825504661504599479D+02, & 0.13365512855474227325D+02, & 0.21131002685639959927D+02, & 0.34838236589449117389D+02, & 0.69657062437837394278D+02 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0078125000D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 5.0000000000D+00, & 6.0000000000D+00, & 7.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00, & 100.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function stromgen ( xvalue ) !*****************************************************************************80 ! !! STROMGEN calculates Stromgen's integral. ! ! Discussion: ! ! The function is defined by: ! ! STROMGEN(X) = Integral ( 0 <= t <= X ) t^7 * exp(2*t) / (exp(t)-1)^3 dt ! ! The code uses a Chebyshev series, the coefficients of which are ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) STROMGEN, the value of the function. ! implicit none real ( kind = 8 ) astrom(0:26) real ( kind = 8 ) cheval real ( kind = 8 ) epngln real ( kind = 8 ) epsln real ( kind = 8 ) f15bp4 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) k1 integer ( kind = 4 ) k2 integer ( kind = 4 ), parameter :: nterms = 23 integer ( kind = 4 ) numexp real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) one5ln real ( kind = 8 ) pi4b3 real ( kind = 8 ) rk real ( kind = 8 ) stromgen real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) seven,sumexp,sum2,t,valinf,xhigh, & xk,xk1,xlow0,xlow1 data seven/ 7.0d0 / data one5ln/ 0.4055d0 / data f15bp4/0.38497433455066256959d-1 / data pi4b3/1.29878788045336582982d2 / data valinf/196.51956920868988261257d0/ data astrom(0)/ 0.56556120872539155290d0/ data astrom(1)/ 0.4555731969101785525d-1/ data astrom(2)/ -0.4039535875936869170d-1/ data astrom(3)/ -0.133390572021486815d-2/ data astrom(4)/ 0.185862506250538030d-2/ data astrom(5)/ -0.4685555868053659d-4/ data astrom(6)/ -0.6343475643422949d-4/ data astrom(7)/ 0.572548708143200d-5/ data astrom(8)/ 0.159352812216822d-5/ data astrom(9)/ -0.28884328431036d-6/ data astrom(10)/-0.2446633604801d-7/ data astrom(11)/ 0.1007250382374d-7/ data astrom(12)/-0.12482986104d-9/ data astrom(13)/-0.26300625283d-9/ data astrom(14)/ 0.2490407578d-10/ data astrom(15)/ 0.485454902d-11/ data astrom(16)/-0.105378913d-11/ data astrom(17)/-0.3604417d-13/ data astrom(18)/ 0.2992078d-13/ data astrom(19)/-0.163971d-14/ data astrom(20)/-0.61061d-15/ data astrom(21)/ 0.9335d-16/ data astrom(22)/ 0.709d-17/ data astrom(23)/-0.291d-17/ data astrom(24)/ 0.8d-19/ data astrom(25)/ 0.6d-19/ data astrom(26)/-0.1d-19/ ! ! Machine-dependent constants ! data xlow0,xlow1/7.80293d-62,2.22045d-16/ data epsln,epngln/-36.0436534d0,-36.7368006d0/ data xhigh/3.1525197d16/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'STROMGEN - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' stromgen = zero return end if if ( x < xlow0 ) then stromgen = zero else if ( x < xlow1 ) then stromgen = x**5 / pi4b3 else if ( x <= four ) then t = ( ( x / two ) - half ) - half stromgen = x**5 * cheval ( nterms, astrom, t ) * f15bp4 else ! ! Code for x > 4.0 ! if ( xhigh < x ) then sumexp = one else numexp = int ( epsln / ( one5ln - x ) ) + 1 if ( 1 < numexp ) then t = exp ( -x ) else t = one end if rk = zero do k1 = 1, numexp rk = rk + one end do sumexp = zero do k1 = 1, numexp sum2 = one xk = one / ( rk * x ) xk1 = one do k2 = 1, 7 sum2 = sum2 * xk1 * xk + one xk1 = xk1 + one end do sum2 = sum2 * ( rk + one ) / two sumexp = sumexp * t + sum2 rk = rk - one end do end if t = seven * log ( x ) - x + log ( sumexp ) if ( t < epngln ) then stromgen = valinf else stromgen = valinf - exp ( t ) * f15bp4 end if end if return end subroutine stromgen_values ( n_data, x, fx ) !*****************************************************************************80 ! !! STROMGEN_VALUES returns some values of the Stromgen function. ! ! Discussion: ! ! The function is defined by: ! ! STROMGEN(X) = Integral ( 0 <= t <= X ) t^7 * exp(2*t) / (exp(t)-1)^3 dt ! ! The data was reported by McLeod. ! ! Modified: ! ! 31 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.21901065985698662316D-15, & 0.22481399438625244761D-12, & 0.23245019579558857124D-09, & 0.24719561475975007037D-06, & 0.28992610989833245669D-03, & 0.10698146390809715091D-01, & 0.89707650964424730705D-01, & 0.40049605719592888440D+00, & 0.30504104398079096598D+01, & 0.11367704858439426431D+02, & 0.12960679405324786954D+02, & 0.18548713944748505675D+02, & 0.27866273821903121400D+02, & 0.51963334071699323351D+02, & 0.10861016747891228129D+03, & 0.15378903316556621624D+03, & 0.19302665532558721516D+03, & 0.19636850166006541482D+03, & 0.19651946766008214217D+03, & 0.19651956920868316152D+03 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0078125000D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.1250000000D+00, & 4.5000000000D+00, & 5.0000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function struve_h0 ( xvalue ) !*****************************************************************************80 ! !! STRUVE_H0 calculates the Struve function of order 0. ! ! Discussion: ! ! The function is defined by: ! ! HO(x) = (2/pi) Integral ( 0 <= t <= pi/2 ) sin ( x * cos ( t ) ) dt ! ! H0 also satisfies the second-order equation ! ! x*D(Df) + Df + x * f = 2 * x / pi ! ! The code uses Chebyshev expansions whose coefficients are ! given to 20D. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) STRUVE_H0, the value of the function. ! implicit none real ( kind = 8 ) arrh0(0:19) real ( kind = 8 ) arrh0a(0:20) real ( kind = 8 ) ay0asp(0:12) real ( kind = 8 ) ay0asq(0:13) real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) indsgn integer ( kind = 4 ), parameter :: nterm1 = 18 integer ( kind = 4 ), parameter :: nterm2 = 18 integer ( kind = 4 ), parameter :: nterm3 = 11 integer ( kind = 4 ), parameter :: nterm4 = 11 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) struve_h0 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) eleven,h0as, & piby4,rt2bpi,sixtp5,t,thr2p5,twenty, & twobpi,two62,xhigh,xlow,xmp4,xsq, & y0p,y0q,y0val data eleven/ 11.0d0/ data twenty /20.0d0 / data sixtp5,two62,thr2p5/60.5d0, 262.0d0, 302.5d0/ data piby4/0.78539816339744830962d0/ data rt2bpi/0.79788456080286535588d0/ data twobpi/0.63661977236758134308d0/ data arrh0(0)/ 0.28696487399013225740d0/ data arrh0(1)/ -0.25405332681618352305d0/ data arrh0(2)/ 0.20774026739323894439d0/ data arrh0(3)/ -0.20364029560386585140d0/ data arrh0(4)/ 0.12888469086866186016d0/ data arrh0(5)/ -0.4825632815622261202d-1/ data arrh0(6)/ 0.1168629347569001242d-1/ data arrh0(7)/ -0.198118135642418416d-2/ data arrh0(8)/ 0.24899138512421286d-3/ data arrh0(9)/ -0.2418827913785950d-4/ data arrh0(10)/ 0.187437547993431d-5/ data arrh0(11)/-0.11873346074362d-6/ data arrh0(12)/ 0.626984943346d-8/ data arrh0(13)/-0.28045546793d-9/ data arrh0(14)/ 0.1076941205d-10/ data arrh0(15)/-0.35904793d-12/ data arrh0(16)/ 0.1049447d-13/ data arrh0(17)/-0.27119d-15/ data arrh0(18)/ 0.624d-17/ data arrh0(19)/-0.13d-18/ data arrh0a(0)/ 1.99291885751992305515d0/ data arrh0a(1)/ -0.384232668701456887d-2/ data arrh0a(2)/ -0.32871993712353050d-3/ data arrh0a(3)/ -0.2941181203703409d-4/ data arrh0a(4)/ -0.267315351987066d-5/ data arrh0a(5)/ -0.24681031075013d-6/ data arrh0a(6)/ -0.2295014861143d-7/ data arrh0a(7)/ -0.215682231833d-8/ data arrh0a(8)/ -0.20303506483d-9/ data arrh0a(9)/ -0.1934575509d-10/ data arrh0a(10)/-0.182773144d-11/ data arrh0a(11)/-0.17768424d-12/ data arrh0a(12)/-0.1643296d-13/ data arrh0a(13)/-0.171569d-14/ data arrh0a(14)/-0.13368d-15/ data arrh0a(15)/-0.2077d-16/ data arrh0a(16)/ 0.2d-19/ data arrh0a(17)/-0.55d-18/ data arrh0a(18)/ 0.10d-18/ data arrh0a(19)/-0.4d-19/ data arrh0a(20)/ 0.1d-19/ data ay0asp/1.99944639402398271568d0, & -0.28650778647031958d-3, & -0.1005072797437620d-4, & -0.35835941002463d-6, & -0.1287965120531d-7, & -0.46609486636d-9, & -0.1693769454d-10, & -0.61852269d-12, & -0.2261841d-13, & -0.83268d-15, & -0.3042d-16, & -0.115d-17, & -0.4d-19/ data ay0asq/1.99542681386828604092d0, & -0.236013192867514472d-2, & -0.7601538908502966d-4, & -0.256108871456343d-5, & -0.8750292185106d-7, & -0.304304212159d-8, & -0.10621428314d-9, & -0.377371479d-11, & -0.13213687d-12, & -0.488621d-14, & -0.15809d-15, & -0.762d-17, & -0.3d-19, & -0.3d-19/ ! ! MACHINE-DEPENDENT CONSTANTS (Suitable for IEEE-arithmetic machines) ! data xlow,xhigh/3.1610136d-8,4.50359963d15/ x = xvalue indsgn = 1 if ( x < zero ) then x = -x indsgn = -1 end if if ( x < xlow ) then struve_h0 = twobpi * x else if ( x <= eleven ) then t = ( ( x * x ) / sixtp5 - half ) - half struve_h0 = twobpi * x * cheval ( nterm1, arrh0, t ) else if ( x <= xhigh ) then xsq = x * x t = ( two62 - xsq ) / ( twenty + xsq ) y0p = cheval ( nterm3, ay0asp, t ) y0q = cheval ( nterm4, ay0asq, t ) / ( eight * x ) xmp4 = x - piby4 y0val = y0p * sin ( xmp4 ) - y0q * cos ( xmp4 ) y0val = y0val * rt2bpi / sqrt ( x ) t = ( thr2p5 - xsq ) / ( sixtp5 + xsq ) h0as = twobpi * cheval ( nterm2, arrh0a, t ) / x struve_h0 = y0val + h0as else write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'STRUVE_H0 - Fatal error!' write ( *, '(a)' ) ' Argument magnitude too large.' struve_h0 = zero end if if ( indsgn == -1 ) then struve_h0 = -struve_h0 end if return end subroutine struve_h0_values ( n_data, x, fx ) !*****************************************************************************80 ! !! STRUVE_H0_VALUES returns some values of the Struve H0 function. ! ! Discussion: ! ! The function is defined by: ! ! HO(x) = (2/pi) * Integral ( 0 <= t <= pi/2 ) sin ( x * cos ( t ) ) dt ! ! In Mathematica, the function can be evaluated by: ! ! StruveH[0,x] ! ! The data was reported by McLeod. ! ! Modified: ! ! 01 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.12433974658847434366D-02, & -0.49735582423748415045D-02, & 0.39771469054536941564D-01, & -0.15805246001653314198D+00, & 0.56865662704828795099D+00, & 0.66598399314899916605D+00, & 0.79085884950809589255D+00, & -0.13501457342248639716D+00, & 0.20086479668164503137D+00, & -0.11142097800261991552D+00, & -0.17026804865989885869D+00, & -0.13544931808186467594D+00, & 0.94393698081323450897D-01, & -0.10182482016001510271D+00, & 0.96098421554162110012D-01, & -0.85337674826118998952D-01, & -0.76882290637052720045D-01, & 0.47663833591418256339D-01, & -0.70878751689647343204D-01, & 0.65752908073352785368D-01 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & -0.0078125000D+00, & 0.0625000000D+00, & -0.2500000000D+00, & 1.0000000000D+00, & 1.2500000000D+00, & 2.0000000000D+00, & -4.0000000000D+00, & 7.5000000000D+00, & 11.0000000000D+00, & 11.5000000000D+00, & -16.0000000000D+00, & 20.0000000000D+00, & 25.0000000000D+00, & -30.0000000000D+00, & 50.0000000000D+00, & 75.0000000000D+00, & -80.0000000000D+00, & 100.0000000000D+00, & -125.0000000000D+00 /) ! if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function struve_h1 ( xvalue ) !*****************************************************************************80 ! !! STRUVE_H1 calculates the Struve function of order 1. ! ! Discussion: ! ! The function is defined by: ! ! H1(x) = 2*x/pi * Integral ( 0 <= t <= pi/2 ) ! sin ( x * cos ( t ) )^2 * sin ( t ) dt ! ! H1 also satisfies the second-order differential equation ! ! x^2 * D^2 f + x * Df + (x^2 - 1)f = 2x^2 / pi ! ! The code uses Chebyshev expansions with the coefficients ! given to 20D. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) STRUVE_H1, the value of the function. ! implicit none real ( kind = 8 ) arrh1(0:17) real ( kind = 8 ) arrh1a(0:21) real ( kind = 8 ) ay1asp(0:14) real ( kind = 8 ) ay1asq(0:15) real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ), parameter :: nterm1 = 15 integer ( kind = 4 ), parameter :: nterm2 = 17 integer ( kind = 4 ), parameter :: nterm3 = 12 integer ( kind = 4 ), parameter :: nterm4 = 13 real ( kind = 8 ) struve_h1 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) fortp5, & h1as,nine,one82,rt2bpi,t,thpby4, & twenty,twobpi,tw02p5,xhigh,xlow1,xlow2, & xm3p4,xsq,y1p,y1q,y1val data nine / 9.0d0 / data twenty / 20.0d0 / data fortp5,one82,tw02p5/40.5d0, 182.0d0 , 202.5d0/ data rt2bpi/0.79788456080286535588d0/ data thpby4/2.35619449019234492885d0/ data twobpi/0.63661977236758134308d0/ data arrh1/0.17319061083675439319d0, & -0.12606917591352672005d0, & 0.7908576160495357500d-1, & -0.3196493222321870820d-1, & 0.808040581404918834d-2, & -0.136000820693074148d-2, & 0.16227148619889471d-3, & -0.1442352451485929d-4, & 0.99219525734072d-6, & -0.5441628049180d-7, & 0.243631662563d-8, & -0.9077071338d-10, & 0.285926585d-11, & -0.7716975d-13, & 0.180489d-14, & -0.3694d-16, & 0.67d-18, & -0.1d-19/ data arrh1a(0)/ 2.01083504951473379407d0/ data arrh1a(1)/ 0.592218610036099903d-2/ data arrh1a(2)/ 0.55274322698414130d-3/ data arrh1a(3)/ 0.5269873856311036d-4/ data arrh1a(4)/ 0.506374522140969d-5/ data arrh1a(5)/ 0.49028736420678d-6/ data arrh1a(6)/ 0.4763540023525d-7/ data arrh1a(7)/ 0.465258652283d-8/ data arrh1a(8)/ 0.45465166081d-9/ data arrh1a(9)/ 0.4472462193d-10/ data arrh1a(10)/ 0.437308292d-11/ data arrh1a(11)/ 0.43568368d-12/ data arrh1a(12)/ 0.4182190d-13/ data arrh1a(13)/ 0.441044d-14/ data arrh1a(14)/ 0.36391d-15/ data arrh1a(15)/ 0.5558d-16/ data arrh1a(16)/-0.4d-19/ data arrh1a(17)/ 0.163d-17/ data arrh1a(18)/-0.34d-18/ data arrh1a(19)/ 0.13d-18/ data arrh1a(20)/-0.4d-19/ data arrh1a(21)/ 0.1d-19/ data ay1asp/2.00135240045889396402d0, & 0.71104241596461938d-3, & 0.3665977028232449d-4, & 0.191301568657728d-5, & 0.10046911389777d-6, & 0.530401742538d-8, & 0.28100886176d-9, & 0.1493886051d-10, & 0.79578420d-12, & 0.4252363d-13, & 0.227195d-14, & 0.12216d-15, & 0.650d-17, & 0.36d-18, & 0.2d-19/ data ay1asq/5.99065109477888189116d0, & -0.489593262336579635d-2, & -0.23238321307070626d-3, & -0.1144734723857679d-4, & -0.57169926189106d-6, & -0.2895516716917d-7, & -0.147513345636d-8, & -0.7596537378d-10, & -0.390658184d-11, & -0.20464654d-12, & -0.1042636d-13, & -0.57702d-15, & -0.2550d-16, & -0.210d-17, & 0.2d-19, & -0.2d-19/ ! ! MACHINE-DEPENDENT CONSTANTS (Suitable for IEEE-arithmetic machines) ! data xlow1,xlow2/2.23750222d-154,4.08085106d-8/ data xhigh/4.50359963d15/ ! x = abs ( xvalue ) if ( x < xlow1 ) then struve_h1 = zero else if ( x < xlow2 ) then xsq = x * x struve_h1 = twobpi * xsq else if ( x <= nine ) then xsq = x * x t = ( xsq / fortp5 - half ) - half struve_h1 = twobpi * xsq * cheval ( nterm1, arrh1, t ) else if ( x <= xhigh ) then xsq = x * x t = ( one82 - xsq ) / ( twenty + xsq ) y1p = cheval ( nterm3, ay1asp, t ) y1q = cheval ( nterm4, ay1asq, t ) / ( eight * x) xm3p4 = x - thpby4 y1val = y1p * sin ( xm3p4 ) + y1q * cos ( xm3p4 ) y1val = y1val * rt2bpi / sqrt ( x ) t = ( tw02p5 - xsq ) / ( fortp5 + xsq ) h1as = twobpi * cheval ( nterm2, arrh1a, t ) struve_h1 = y1val + h1as else write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'STRUVE_H1 - Fatal error!' write ( *, '(a)' ) ' Argument magnitude too large.' struve_h1 = zero end if return end subroutine struve_h1_values ( n_data, x, fx ) !*****************************************************************************80 ! !! STRUVE_H1_VALUES returns some values of the Struve H1 function. ! ! Discussion: ! ! The function is defined by: ! ! H1(x) = 2*x/pi * Integral ( 0 <= t <= pi/2 ) ! sin ( x * cos ( t ) )^2 * sin ( t ) dt ! ! In Mathematica, the function can be evaluated by: ! ! StruveH[1,x] ! ! The data was reported by McLeod. ! ! Modified: ! ! 02 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.80950369576367526071D-06, & 0.12952009724113229165D-04, & 0.82871615165407083021D-03, & 0.13207748375849572564D-01, & 0.19845733620194439894D+00, & 0.29853823231804706294D+00, & 0.64676372828356211712D+00, & 0.10697266613089193593D+01, & 0.38831308000420560970D+00, & 0.74854243745107710333D+00, & 0.84664854642567359993D+00, & 0.58385732464244384564D+00, & 0.80600584524215772824D+00, & 0.53880362132692947616D+00, & 0.72175037834698998506D+00, & 0.58007844794544189900D+00, & 0.60151910385440804463D+00, & 0.70611511147286827018D+00, & 0.61631110327201338454D+00, & 0.62778480765443656489D+00 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & -0.0078125000D+00, & 0.0625000000D+00, & -0.2500000000D+00, & 1.0000000000D+00, & 1.2500000000D+00, & 2.0000000000D+00, & -4.0000000000D+00, & 7.5000000000D+00, & 11.0000000000D+00, & 11.5000000000D+00, & -16.0000000000D+00, & 20.0000000000D+00, & 25.0000000000D+00, & -30.0000000000D+00, & 50.0000000000D+00, & 75.0000000000D+00, & -80.0000000000D+00, & 100.0000000000D+00, & -125.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function struve_l0 ( xvalue ) !*****************************************************************************80 ! !! STRUVE_L0 calculates the modified Struve function of order 0. ! ! Discussion: ! ! This function calculates the modified Struve function of ! order 0, denoted L0(x), defined as the solution of the ! second-order equation ! ! x*D(Df) + Df - x*f = 2x/pi ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) STRUVE_L0, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: four = 4.0D+00 integer ( kind = 4 ) indsgn integer ( kind = 4 ), parameter :: nterm1 = 25 integer ( kind = 4 ), parameter :: nterm2 = 14 integer ( kind = 4 ), parameter :: nterm3 = 21 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) struve_l0 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) arl0(0:27),arl0as(0:15),ai0ml0(0:23), & atehun,ch1,ch2,lnr2pi, & sixten,t,test,twent4,twent8,twobpi,two88, & xhigh1,xhigh2,xlow,xmax,xsq data sixten/16.0d0/ data twent4,twent8/24.0d0 , 28.0d0 / data two88,atehun/288.0d0 , 800.0d0/ data lnr2pi/0.91893853320467274178d0/ data twobpi/0.63661977236758134308d0/ data arl0(0)/ 0.42127458349979924863d0/ data arl0(1)/ -0.33859536391220612188d0/ data arl0(2)/ 0.21898994812710716064d0/ data arl0(3)/ -0.12349482820713185712d0/ data arl0(4)/ 0.6214209793866958440d-1/ data arl0(5)/ -0.2817806028109547545d-1/ data arl0(6)/ 0.1157419676638091209d-1/ data arl0(7)/ -0.431658574306921179d-2/ data arl0(8)/ 0.146142349907298329d-2/ data arl0(9)/ -0.44794211805461478d-3/ data arl0(10)/ 0.12364746105943761d-3/ data arl0(11)/-0.3049028334797044d-4/ data arl0(12)/ 0.663941401521146d-5/ data arl0(13)/-0.125538357703889d-5/ data arl0(14)/ 0.20073446451228d-6/ data arl0(15)/-0.2588260170637d-7/ data arl0(16)/ 0.241143742758d-8/ data arl0(17)/-0.10159674352d-9/ data arl0(18)/-0.1202430736d-10/ data arl0(19)/ 0.262906137d-11/ data arl0(20)/-0.15313190d-12/ data arl0(21)/-0.1574760d-13/ data arl0(22)/ 0.315635d-14/ data arl0(23)/-0.4096d-16/ data arl0(24)/-0.3620d-16/ data arl0(25)/ 0.239d-17/ data arl0(26)/ 0.36d-18/ data arl0(27)/-0.4d-19/ data arl0as(0)/ 2.00861308235605888600d0/ data arl0as(1)/ 0.403737966500438470d-2/ data arl0as(2)/ -0.25199480286580267d-3/ data arl0as(3)/ 0.1605736682811176d-4/ data arl0as(4)/ -0.103692182473444d-5/ data arl0as(5)/ 0.6765578876305d-7/ data arl0as(6)/ -0.444999906756d-8/ data arl0as(7)/ 0.29468889228d-9/ data arl0as(8)/ -0.1962180522d-10/ data arl0as(9)/ 0.131330306d-11/ data arl0as(10)/-0.8819190d-13/ data arl0as(11)/ 0.595376d-14/ data arl0as(12)/-0.40389d-15/ data arl0as(13)/ 0.2651d-16/ data arl0as(14)/-0.208d-17/ data arl0as(15)/ 0.11d-18/ data ai0ml0(0)/ 2.00326510241160643125d0/ data ai0ml0(1)/ 0.195206851576492081d-2/ data ai0ml0(2)/ 0.38239523569908328d-3/ data ai0ml0(3)/ 0.7534280817054436d-4/ data ai0ml0(4)/ 0.1495957655897078d-4/ data ai0ml0(5)/ 0.299940531210557d-5/ data ai0ml0(6)/ 0.60769604822459d-6/ data ai0ml0(7)/ 0.12399495544506d-6/ data ai0ml0(8)/ 0.2523262552649d-7/ data ai0ml0(9)/ 0.504634857332d-8/ data ai0ml0(10)/0.97913236230d-9/ data ai0ml0(11)/0.18389115241d-9/ data ai0ml0(12)/0.3376309278d-10/ data ai0ml0(13)/0.611179703d-11/ data ai0ml0(14)/0.108472972d-11/ data ai0ml0(15)/0.18861271d-12/ data ai0ml0(16)/0.3280345d-13/ data ai0ml0(17)/0.565647d-14/ data ai0ml0(18)/0.93300d-15/ data ai0ml0(19)/0.15881d-15/ data ai0ml0(20)/0.2791d-16/ data ai0ml0(21)/0.389d-17/ data ai0ml0(22)/0.70d-18/ data ai0ml0(23)/0.16d-18/ ! ! MACHINE-DEPENDENT VALUES (Suitable for IEEE-arithmetic machines) ! data xlow,xmax/4.4703484d-8,1.797693d308/ data xhigh1,xhigh2/5.1982303d8,2.5220158d17/ x = xvalue indsgn = 1 if ( x < zero ) then x = -x indsgn = -1 end if if ( x < xlow ) then struve_l0 = twobpi * x else if ( x <= sixten ) then t = ( four * x - twent4 ) / ( x + twent4 ) struve_l0 = twobpi * x * cheval ( nterm1, arl0, t ) * exp ( x ) else ! ! Code for |xvalue| > 16 ! if ( xhigh2 < x ) then ch1 = one else t = ( x - twent8 ) / ( four - x ) ch1 = cheval ( nterm2, arl0as, t ) end if if ( xhigh1 < x ) then ch2 = one else xsq = x * x t = ( atehun - xsq ) / ( two88 + xsq ) ch2 = cheval ( nterm3, ai0ml0, t ) end if test = log ( ch1 ) - lnr2pi - log ( x ) / two + x if ( log ( xmax ) < test ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'STRUVE_L0 - Fatal error!' write ( *, '(a)' ) ' Argument would cause overflow.' struve_l0 = xmax else struve_l0 = exp ( test ) - twobpi * ch2 / x end if end if if ( indsgn == -1 ) then struve_l0 = -struve_l0 end if return end subroutine struve_l0_values ( n_data, x, fx ) !*****************************************************************************80 ! !! STRUVE_L0_VALUES returns some values of the Struve L0 function. ! ! Discussion: ! ! In Mathematica, the function can be evaluated by: ! ! StruveL[0,x] ! ! The data was reported by McLeod. ! ! Modified: ! ! 03 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.12433985199262820188D-02, & -0.19896526647882937004D-01, & 0.79715713253115014945D-01, & -0.32724069939418078025D+00, & 0.71024318593789088874D+00, & 0.19374337579914456612D+01, & -0.11131050203248583431D+02, & 0.16850062034703267148D+03, & -0.28156522493745948555D+04, & 0.89344618796978400815D+06, & 0.11382025002851451057D+07, & -0.23549701855860190304D+07, & 0.43558282527641046718D+08, & 0.49993516476037957165D+09, & -0.57745606064408041689D+10, & 0.78167229782395624524D+12, & -0.14894774793419899908D+17, & 0.29325537838493363267D+21, & 0.58940770556098011683D+25, & -0.12015889579125463605D+30 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & -0.0312500000D+00, & 0.1250000000D+00, & -0.5000000000D+00, & 1.0000000000D+00, & 2.0000000000D+00, & -4.0000000000D+00, & 7.0000000000D+00, & -10.0000000000D+00, & 16.0000000000D+00, & 16.2500000000D+00, & -17.0000000000D+00, & 20.0000000000D+00, & 22.5000000000D+00, & -25.0000000000D+00, & 30.0000000000D+00, & -40.0000000000D+00, & 50.0000000000D+00, & 60.0000000000D+00, & -70.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function struve_l1 ( xvalue ) !*****************************************************************************80 ! !! STRUVE_L1 calculates the modified Struve function of order 1. ! ! Discussion: ! ! This function calculates the modified Struve function of ! order 1, denoted L1(x), defined as the solution of ! ! x*x*D(Df) + x*Df - (x*x+1)f = 2 * x * x / pi ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) STRUVE_L1, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: four = 4.0D+00 integer ( kind = 4 ), parameter :: nterm1 = 24 integer ( kind = 4 ), parameter :: nterm2 = 13 integer ( kind = 4 ), parameter :: nterm3 = 22 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ), parameter :: sixten = 16.0D+00 real ( kind = 8 ) struve_l1 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) arl1(0:26),arl1as(0:16),ai1ml1(0:25), & atehun,ch1,ch2,lnr2pi, & pi3by2,t,test,thirty,twent4, & twobpi,two88,xhigh1,xhigh2,xlow1,xlow2, & xmax,xsq data twent4,thirty/24.0d0, 30.0d0/ data two88,atehun/288.0d0, 800.0d0/ data lnr2pi/0.91893853320467274178d0/ data pi3by2/4.71238898038468985769d0/ data twobpi/0.63661977236758134308d0/ data arl1(0)/ 0.38996027351229538208d0/ data arl1(1)/ -0.33658096101975749366d0/ data arl1(2)/ 0.23012467912501645616d0/ data arl1(3)/ -0.13121594007960832327d0/ data arl1(4)/ 0.6425922289912846518d-1/ data arl1(5)/ -0.2750032950616635833d-1/ data arl1(6)/ 0.1040234148637208871d-1/ data arl1(7)/ -0.350532294936388080d-2/ data arl1(8)/ 0.105748498421439717d-2/ data arl1(9)/ -0.28609426403666558d-3/ data arl1(10)/ 0.6925708785942208d-4/ data arl1(11)/-0.1489693951122717d-4/ data arl1(12)/ 0.281035582597128d-5/ data arl1(13)/-0.45503879297776d-6/ data arl1(14)/ 0.6090171561770d-7/ data arl1(15)/-0.623543724808d-8/ data arl1(16)/ 0.38430012067d-9/ data arl1(17)/ 0.790543916d-11/ data arl1(18)/-0.489824083d-11/ data arl1(19)/ 0.46356884d-12/ data arl1(20)/ 0.684205d-14/ data arl1(21)/-0.569748d-14/ data arl1(22)/ 0.35324d-15/ data arl1(23)/ 0.4244d-16/ data arl1(24)/-0.644d-17/ data arl1(25)/-0.21d-18/ data arl1(26)/ 0.9d-19/ data arl1as(0)/ 1.97540378441652356868d0/ data arl1as(1)/ -0.1195130555088294181d-1/ data arl1as(2)/ 0.33639485269196046d-3/ data arl1as(3)/ -0.1009115655481549d-4/ data arl1as(4)/ 0.30638951321998d-6/ data arl1as(5)/ -0.953704370396d-8/ data arl1as(6)/ 0.29524735558d-9/ data arl1as(7)/ -0.951078318d-11/ data arl1as(8)/ 0.28203667d-12/ data arl1as(9)/ -0.1134175d-13/ data arl1as(10)/ 0.147d-17/ data arl1as(11)/-0.6232d-16/ data arl1as(12)/-0.751d-17/ data arl1as(13)/-0.17d-18/ data arl1as(14)/ 0.51d-18/ data arl1as(15)/ 0.23d-18/ data arl1as(16)/ 0.5d-19/ data ai1ml1(0)/ 1.99679361896789136501d0/ data ai1ml1(1)/ -0.190663261409686132d-2/ data ai1ml1(2)/ -0.36094622410174481d-3/ data ai1ml1(3)/ -0.6841847304599820d-4/ data ai1ml1(4)/ -0.1299008228509426d-4/ data ai1ml1(5)/ -0.247152188705765d-5/ data ai1ml1(6)/ -0.47147839691972d-6/ data ai1ml1(7)/ -0.9020819982592d-7/ data ai1ml1(8)/ -0.1730458637504d-7/ data ai1ml1(9)/ -0.332323670159d-8/ data ai1ml1(10)/-0.63736421735d-9/ data ai1ml1(11)/-0.12180239756d-9/ data ai1ml1(12)/-0.2317346832d-10/ data ai1ml1(13)/-0.439068833d-11/ data ai1ml1(14)/-0.82847110d-12/ data ai1ml1(15)/-0.15562249d-12/ data ai1ml1(16)/-0.2913112d-13/ data ai1ml1(17)/-0.543965d-14/ data ai1ml1(18)/-0.101177d-14/ data ai1ml1(19)/-0.18767d-15/ data ai1ml1(20)/-0.3484d-16/ data ai1ml1(21)/-0.643d-17/ data ai1ml1(22)/-0.118d-17/ data ai1ml1(23)/-0.22d-18/ data ai1ml1(24)/-0.4d-19/ data ai1ml1(25)/-0.1d-19/ ! ! MACHINE-DEPENDENT VALUES (Suitable for IEEE-arithmetic machines) ! data xlow1,xlow2,xmax/5.7711949d-8,3.3354714d-154,1.797693d308/ data xhigh1,xhigh2/5.19823025d8,2.7021597d17/ x = abs ( xvalue ) if ( x <= xlow2 ) then struve_l1 = zero else if ( x < xlow1 ) then xsq = x * x struve_l1 = xsq / pi3by2 else if ( x <= sixten ) then xsq = x * x t = ( four * x - twent4 ) / ( x + twent4 ) struve_l1 = xsq * cheval ( nterm1, arl1, t ) * exp ( x ) / pi3by2 else if ( xhigh2 < x ) then ch1 = one else t = ( x - thirty ) / ( two - x ) ch1 = cheval ( nterm2, arl1as, t ) end if if ( xhigh1 < x ) then ch2 = one else xsq = x * x t = ( atehun - xsq ) / ( two88 + xsq ) ch2 = cheval ( nterm3, ai1ml1, t ) end if test = log ( ch1 ) - lnr2pi - log ( x ) / two + x if ( log ( xmax ) < test ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'STRUVE_L1 - Fatal error!' write ( *, '(a)' ) ' Argument would cause overflow.' struve_l1 = xmax else struve_l1 = exp ( test ) - twobpi * ch2 end if end if return end subroutine struve_l1_values ( n_data, x, fx ) !*****************************************************************************80 ! !! STRUVE_L1_VALUES returns some values of the Struve L1 function. ! ! Discussion: ! ! In Mathematica, the function can be evaluated by: ! ! StruveL[1,x] ! ! The data was reported by McLeod. ! ! Modified: ! ! 06 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.80950410749865126939d-06, & 0.20724649092571514607d-03, & 0.33191834066894516744d-02, & 0.53942182623522663292d-01, & 0.22676438105580863683d+00, & 0.11027597873677158176d+01, & 0.91692778117386847344d+01, & 0.15541656652426660966d+03, & 0.26703582852084829694d+04, & 0.86505880175304633906d+06, & 0.11026046613094942620d+07, & 0.22846209494153934787d+07, & 0.42454972750111979449d+08, & 0.48869614587997695539d+09, & 0.56578651292431051863d+10, & 0.76853203893832108948d+12, & 0.14707396163259352103d+17, & 0.29030785901035567967d+21, & 0.58447515883904682813d+25, & 0.11929750788892311875d+30 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & -0.0312500000D+00, & 0.1250000000D+00, & -0.5000000000D+00, & 1.0000000000D+00, & 2.0000000000D+00, & -4.0000000000D+00, & 7.0000000000D+00, & -10.0000000000D+00, & 16.0000000000D+00, & 16.2500000000D+00, & -17.0000000000D+00, & 20.0000000000D+00, & 22.5000000000D+00, & -25.0000000000D+00, & 30.0000000000D+00, & -40.0000000000D+00, & 50.0000000000D+00, & 60.0000000000D+00, & -70.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function synch1 ( xvalue ) !*****************************************************************************80 ! !! SYNCH1 calculates the synchrotron radiation function. ! ! Discussion: ! ! The function is defined by: ! ! SYNCH1(x) = x * Integral ( x <= t < infinity ) K(5/3)(t) dt ! ! where K(5/3) is a modified Bessel function of order 5/3. ! ! The code uses Chebyshev expansions, the coefficients of which ! are given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 September 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) SYNCH1, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ), parameter :: nterm1 = 12 integer ( kind = 4 ), parameter :: nterm2 = 10 integer ( kind = 4 ), parameter :: nterm3 = 21 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) synch1 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) async1(0:13),async2(0:11),asynca(0:24), & cheb1,cheb2,conlow, & lnrtp2,pibrt3,t,twelve,xhigh1, & xhigh2,xlow,xpowth data twelve/ 12.0d0 / data conlow/2.14952824153447863671d0/ data pibrt3/1.81379936423421785059d0/ data lnrtp2/0.22579135264472743236d0/ data async1/30.36468298250107627340d0, & 17.07939527740839457449d0, & 4.56013213354507288887d0, & 0.54928124673041997963d0, & 0.3729760750693011724d-1, & 0.161362430201041242d-2, & 0.4819167721203707d-4, & 0.105124252889384d-5, & 0.1746385046697d-7, & 0.22815486544d-9, & 0.240443082d-11, & 0.2086588d-13, & 0.15167d-15, & 0.94d-18/ data async2/0.44907216235326608443d0, & 0.8983536779941872179d-1, & 0.810445737721512894d-2, & 0.42617169910891619d-3, & 0.1476096312707460d-4, & 0.36286336153998d-6, & 0.666348074984d-8, & 0.9490771655d-10, & 0.107912491d-11, & 0.1002201d-13, & 0.7745d-16, & 0.51d-18/ data asynca(0)/ 2.13293051613550009848d0/ data asynca(1)/ 0.7413528649542002401d-1/ data asynca(2)/ 0.869680999099641978d-2/ data asynca(3)/ 0.117038262487756921d-2/ data asynca(4)/ 0.16451057986191915d-3/ data asynca(5)/ 0.2402010214206403d-4/ data asynca(6)/ 0.358277563893885d-5/ data asynca(7)/ 0.54477476269837d-6/ data asynca(8)/ 0.8388028561957d-7/ data asynca(9)/ 0.1306988268416d-7/ data asynca(10)/0.205309907144d-8/ data asynca(11)/0.32518753688d-9/ data asynca(12)/0.5179140412d-10/ data asynca(13)/0.830029881d-11/ data asynca(14)/0.133527277d-11/ data asynca(15)/0.21591498d-12/ data asynca(16)/0.3499673d-13/ data asynca(17)/0.569942d-14/ data asynca(18)/0.92906d-15/ data asynca(19)/0.15222d-15/ data asynca(20)/0.2491d-16/ data asynca(21)/0.411d-17/ data asynca(22)/0.67d-18/ data asynca(23)/0.11d-18/ data asynca(24)/0.2d-19/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow/2.98023224d-8/ data xhigh1,xhigh2/809.595907d0,-708.396418d0/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'SYNCH1 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' synch1 = zero else if ( x < xlow ) then xpowth = x ** ( one / three ) synch1 = conlow * xpowth else if ( x <= four ) then xpowth = x ** ( one / three ) t = ( x * x / eight - half ) - half cheb1 = cheval ( nterm1, async1, t ) cheb2 = cheval ( nterm2, async2, t ) t = xpowth * cheb1 - xpowth**11 * cheb2 synch1 = t - pibrt3 * x else if ( x <= xhigh1 ) then t = ( twelve - x ) / ( x + four ) cheb1 = cheval ( nterm3, asynca, t ) t = lnrtp2 - x + log ( sqrt ( x ) * cheb1 ) if ( t < xhigh2 ) then synch1 = zero else synch1 = exp ( t ) end if else synch1 = zero end if return end subroutine synch1_values ( n_data, x, fx ) !*****************************************************************************80 ! !! SYNCH1_VALUES returns some values of the synchrotron radiation function. ! ! Discussion: ! ! The function is defined by: ! ! SYNCH1(x) = x * Integral ( x <= t < infinity ) K(5/3)(t) dt ! ! where K(5/3) is a modified Bessel function of order 5/3. ! ! Modified: ! ! 05 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.26514864547487397044D+00, & 0.62050129979079045645D+00, & 0.85112572132368011206D+00, & 0.87081914687546885094D+00, & 0.65142281535536396975D+00, & 0.45064040920322354579D+00, & 0.30163590285073940285D+00, & 0.19814490804441305867D+00, & 0.12856571000906381300D+00, & 0.52827396697866818297D-01, & 0.42139298471720305542D-01, & 0.21248129774981984268D-01, & 0.13400258907505536491D-01, & 0.84260797314108699935D-02, & 0.12884516186754671469D-02, & 0.19223826430086897418D-03, & 0.28221070834007689394D-04, & 0.15548757973038189372D-05, & 0.11968634456097453636D-07, & 0.89564246772237127742D-10 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 12.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 25.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function synch2 ( xvalue ) !*****************************************************************************80 ! !! SYNCH2 calculates the synchrotron radiation function. ! ! Discussion: ! ! The function is defined by: ! ! SYNCH2(x) = x * K(2/3)(x) ! ! where K(2/3) is a modified Bessel function of order 2/3. ! ! The code uses Chebyshev expansions, the coefficients of which ! are given to 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) SYNCH2, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ), parameter :: nterm1 = 13 integer ( kind = 4 ), parameter :: nterm2 = 12 integer ( kind = 4 ), parameter :: nterm3 = 16 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) synch2 real ( kind = 8 ), parameter :: three = 3.0D+00 real ( kind = 8 ), parameter :: two = 2.0D+00 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) asyn21(0:14),asyn22(0:13),asyn2a(0:18), & cheb1,cheb2,conlow, & lnrtp2,t,ten,xhigh1, & xhigh2,xlow,xpowth data ten/ 10.0d0 / data conlow/1.07476412076723931836d0/ data lnrtp2/0.22579135264472743236d0/ data asyn21/38.61783992384308548014d0, & 23.03771559496373459697d0, & 5.38024998683357059676d0, & 0.61567938069957107760d0, & 0.4066880046688955843d-1, & 0.172962745526484141d-2, & 0.5106125883657699d-4, & 0.110459595022012d-5, & 0.1823553020649d-7, & 0.23707698034d-9, & 0.248872963d-11, & 0.2152868d-13, & 0.15607d-15, & 0.96d-18, & 0.1d-19/ data asyn22/7.90631482706608042875d0, & 3.13534636128534256841d0, & 0.48548794774537145380d0, & 0.3948166758272372337d-1, & 0.196616223348088022d-2, & 0.6590789322930420d-4, & 0.158575613498559d-5, & 0.2868653011233d-7, & 0.40412023595d-9, & 0.455684443d-11, & 0.4204590d-13, & 0.32326d-15, & 0.210d-17, & 0.1d-19/ data asyn2a/2.02033709417071360032d0, & 0.1095623712180740443d-1, & 0.85423847301146755d-3, & 0.7234302421328222d-4, & 0.631244279626992d-5, & 0.56481931411744d-6, & 0.5128324801375d-7, & 0.471965329145d-8, & 0.43807442143d-9, & 0.4102681493d-10, & 0.386230721d-11, & 0.36613228d-12, & 0.3480232d-13, & 0.333010d-14, & 0.31856d-15, & 0.3074d-16, & 0.295d-17, & 0.29d-18, & 0.3d-19/ ! ! Machine-dependent constants (suitable for IEEE machines) ! data xlow/2.98023224d-8/ data xhigh1,xhigh2/809.595907d0,-708.396418d0/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'SYNCH2 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' synch2 = zero else if ( x < xlow ) then xpowth = x ** ( one / three ) synch2 = conlow * xpowth else if ( x <= four ) then xpowth = x ** ( one / three ) t = ( x * x / eight - half ) - half cheb1 = cheval ( nterm1, asyn21, t ) cheb2 = cheval ( nterm2, asyn22, t ) synch2 = xpowth * cheb1 - xpowth**5 * cheb2 else if ( x <= xhigh1 ) then t = ( ten - x ) / ( x + two ) cheb1 = cheval ( nterm3, asyn2a, t ) t = lnrtp2 - x + log ( sqrt ( x ) * cheb1 ) if ( t < xhigh2 ) then synch2 = zero else synch2 = exp ( t ) end if else synch2 = zero end if return end subroutine synch2_values ( n_data, x, fx ) !*****************************************************************************80 ! !! SYNCH2_VALUES returns some values of the synchrotron radiation function. ! ! Discussion: ! ! The function is defined by: ! ! SYNCH2(x) = x * K(2/3)(x) ! ! where K(2/3) is a modified Bessel function of order 2/3. ! ! Modified: ! ! 05 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.13430727275667378338D+00, & 0.33485265272424176976D+00, & 0.50404224110911078651D+00, & 0.60296523236016785113D+00, & 0.49447506210420826699D+00, & 0.36036067860473360389D+00, & 0.24967785497625662113D+00, & 0.16813830542905833533D+00, & 0.11117122348556549832D+00, & 0.46923205826101330711D-01, & 0.37624545861980001482D-01, & 0.19222123172484106436D-01, & 0.12209535343654701398D-01, & 0.77249644268525771866D-02, & 0.12029044213679269639D-02, & 0.18161187569530204281D-03, & 0.26884338006629353506D-04, & 0.14942212731345828759D-05, & 0.11607696854385161390D-07, & 0.87362343746221526073D-10 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 12.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 25.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end subroutine timestamp ( ) !*****************************************************************************80 ! !! TIMESTAMP prints the current YMDHMS date as a time stamp. ! ! Example: ! ! 31 May 2001 9:45:54.872 AM ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 May 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! None ! implicit none character ( len = 8 ) ampm integer ( kind = 4 ) d integer ( kind = 4 ) h integer ( kind = 4 ) m integer ( kind = 4 ) mm character ( len = 9 ), parameter, dimension(12) :: month = (/ & 'January ', 'February ', 'March ', 'April ', & 'May ', 'June ', 'July ', 'August ', & 'September', 'October ', 'November ', 'December ' /) integer ( kind = 4 ) n integer ( kind = 4 ) s integer ( kind = 4 ) values(8) integer ( kind = 4 ) y call date_and_time ( values = values ) y = values(1) m = values(2) d = values(3) h = values(5) n = values(6) s = values(7) mm = values(8) if ( h < 12 ) then ampm = 'AM' else if ( h == 12 ) then if ( n == 0 .and. s == 0 ) then ampm = 'Noon' else ampm = 'PM' end if else h = h - 12 if ( h < 12 ) then ampm = 'PM' else if ( h == 12 ) then if ( n == 0 .and. s == 0 ) then ampm = 'Midnight' else ampm = 'AM' end if end if end if write ( *, '(i2,1x,a,1x,i4,2x,i2,a1,i2.2,a1,i2.2,a1,i3.3,1x,a)' ) & d, trim ( month(m) ), y, h, ':', n, ':', s, '.', mm, trim ( ampm ) return end function tran02 ( xvalue ) !*****************************************************************************80 ! !! TRAN02 calculates the transport integral of order 2. ! ! Discussion: ! ! The function is defined by: ! ! TRAN02(x) = Integral ( 0 <= t <= x ) t^2 exp(t) / ( exp(t) - 1 )^2 dt ! ! The program uses a Chebyshev series, the coefficients of which are ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) TRAN02, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) k1 integer ( kind = 4 ) k2 integer ( kind = 4 ), parameter :: nterms = 17 integer ( kind = 4 ) numexp integer ( kind = 4 ), parameter :: numjn = 2 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) tran02 real ( kind = 8 ), parameter :: valinf = 0.32898681336964528729D+01 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) atran(0:19),rk, & rnumjn,sumexp,sum2,t,xhigh1,xhigh2, & xhigh3,xk,xk1,xlow1 data rnumjn/ 2.0d0 / data atran/1.67176044643453850301d0, & -0.14773535994679448986d0, & 0.1482138199469363384d-1, & -0.141953303263056126d-2, & 0.13065413244157083d-3, & -0.1171557958675790d-4, & 0.103334984457557d-5, & -0.9019113042227d-7, & 0.781771698331d-8, & -0.67445656840d-9, & 0.5799463945d-10, & -0.497476185d-11, & 0.42596097d-12, & -0.3642189d-13, & 0.311086d-14, & -0.26547d-15, & 0.2264d-16, & -0.193d-17, & 0.16d-18, & -0.1d-19/ ! ! Machine-dependent constants ! data xlow1/2.98023224d-8/ data xhigh1,xhigh3/36.04365668d0,-36.73680056d0/ data xhigh2/9.00719925d15/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'TRAN02 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' tran02 = zero else if ( x < xlow1 ) then tran02 = ( x ** ( numjn - 1 ) ) / ( rnumjn - one ) else if ( x <= four ) then t = ( ( ( x * x ) / eight ) - half ) - half tran02 = ( x ** ( numjn - 1 ) ) * cheval ( nterms, atran, t ) else if ( xhigh2 < x ) then sumexp = one else if ( x <= xhigh1 ) then numexp = int ( xhigh1 / x ) + 1 t = exp ( -x ) else numexp = 1 t = one end if rk = zero do k1 = 1, numexp rk = rk + one end do sumexp = zero do k1 = 1, numexp sum2 = one xk = one / ( rk * x ) xk1 = one do k2 = 1, numjn sum2 = sum2 * xk1 * xk + one xk1 = xk1 + one end do sumexp = sumexp * t + sum2 rk = rk - one end do end if t = rnumjn * log ( x ) - x + log ( sumexp ) if ( t < xhigh3 ) then tran02 = valinf else tran02 = valinf - exp ( t ) end if end if return end subroutine tran02_values ( n_data, x, fx ) !*****************************************************************************80 ! !! TRAN02_VALUES returns some values of the order 2 transportation function. ! ! Discussion: ! ! The function is defined by: ! ! TRAN02(x) = Integral ( 0 <= t <= x ) t^2 exp(t) / ( exp(t) - 1 )^2 dt ! ! Modified: ! ! 06 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 ! before the first call. On each call, the routine increments N_DATA by 1, ! and returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.19531247930394515480D-02, & 0.31249152314331109004D-01, & 0.12494577194783451032D+00, & 0.49655363615640595865D+00, & 0.97303256135517012845D+00, & 0.14121978695932525805D+01, & 0.18017185674405776809D+01, & 0.21350385339277043015D+01, & 0.24110500490169534620D+01, & 0.28066664045631179931D+01, & 0.28777421863296234131D+01, & 0.30391706043438554330D+01, & 0.31125074928667355940D+01, & 0.31656687817738577185D+01, & 0.32623520367816009184D+01, & 0.32843291144979517358D+01, & 0.32897895167775788137D+01, & 0.32898672226665499687D+01, & 0.32898681336064325400D+01, & 0.32898681336964528724D+01 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function tran03 ( xvalue ) !*****************************************************************************80 ! !! TRAN03 calculates the transport integral of order 3. ! ! Discussion: ! ! The function is defined by: ! ! TRAN03(x) = Integral ( 0 <= t <= x ) t^3 * exp(t) / ( exp(t) - 1 )^2 dt ! ! The program uses a Chebyshev series, the coefficients of which are ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) TRAN03, the value of the function. ! implicit none real ( kind = 8 ) atran(0:19) real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) k1 integer ( kind = 4 ) k2 integer ( kind = 4 ), parameter :: nterms = 17 integer ( kind = 4 ) numexp integer ( kind = 4 ) numjn real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) tran03 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) rk, & rnumjn,sumexp,sum2,t,valinf,xhigh1,xhigh2, & xhigh3,xk,xk1,xlow1,xlow2 data numjn,rnumjn/ 3 , 3.0d0 / data valinf/0.72123414189575657124d1/ data atran/0.76201254324387200657d0, & -0.10567438770505853250d0, & 0.1197780848196578097d-1, & -0.121440152036983073d-2, & 0.11550997693928547d-3, & -0.1058159921244229d-4, & 0.94746633853018d-6, & -0.8362212128581d-7, & 0.731090992775d-8, & -0.63505947788d-9, & 0.5491182819d-10, & -0.473213954d-11, & 0.40676948d-12, & -0.3489706d-13, & 0.298923d-14, & -0.25574d-15, & 0.2186d-16, & -0.187d-17, & 0.16d-18, & -0.1d-19/ ! ! Machine-dependent constants ! data xlow1,xlow2/2.98023224d-8,2.10953733d-154/ data xhigh1,xhigh3/36.04365668d0,-36.73680056d0/ data xhigh2/1.35107988d16/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'TRAN03 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' tran03 = zero else if ( x < xlow2 ) then tran03 = zero else if ( x < xlow1 ) then tran03 = ( x**( numjn - 1 ) ) / ( rnumjn - one ) else if ( x <= four ) then t = ( ( ( x*x ) / eight ) - half ) - half tran03 = ( x**( numjn - 1 ) ) * cheval ( nterms, atran, t ) else if ( xhigh2 < x ) then sumexp = one else if ( x <= xhigh1 ) then numexp = int ( xhigh1 / x ) + 1 t = exp ( -x ) else numexp = 1 t = one end if rk = zero do k1 = 1, numexp rk = rk + one end do sumexp = zero do k1 = 1, numexp sum2 = one xk = one / ( rk * x ) xk1 = one do k2 = 1, numjn sum2 = sum2 * xk1 * xk + one xk1 = xk1 + one end do sumexp = sumexp * t + sum2 rk = rk - one end do end if t = rnumjn * log ( x ) - x + log ( sumexp ) if ( t < xhigh3 ) then tran03 = valinf else tran03 = valinf - exp ( t ) end if end if return end subroutine tran03_values ( n_data, x, fx ) !*****************************************************************************80 ! !! TRAN03_VALUES returns some values of the order 3 transportation function. ! ! Discussion: ! ! The function is defined by: ! ! TRAN03(x) = Integral ( 0 <= t <= x ) t^3 * exp(t) / ( exp(t) - 1 )^2 dt ! ! Modified: ! ! 06 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.19073483296476379584D-05, & 0.48826138243180786081D-03, & 0.78074163848431205820D-02, & 0.12370868718812031049D+00, & 0.47984100657241749994D+00, & 0.10269431622039754738D+01, & 0.17063547219458658863D+01, & 0.24539217444475937661D+01, & 0.32106046629422467723D+01, & 0.45792174372291563703D+01, & 0.48722022832940370805D+01, & 0.56143866138422732286D+01, & 0.59984455864575470009D+01, & 0.63033953673480961120D+01, & 0.69579908688361166266D+01, & 0.71503227120085929750D+01, & 0.72110731475871876393D+01, & 0.72123221966388461839D+01, & 0.72123414161609465119D+01, & 0.72123414189575656868D+01 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function tran04 ( xvalue ) !*****************************************************************************80 ! !! TRAN04 calculates the transport integral of order 4. ! ! Discussion: ! ! The function is defined by: ! ! TRAN04(x) = Integral ( 0 <= t <= x ) t^4 * exp(t) / ( exp(t) - 1 )^2 dt ! ! The program uses a Chebyshev series, the coefficients of which are ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) TRAN04, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) k1 integer ( kind = 4 ) k2 integer ( kind = 4 ), parameter :: nterms = 17 integer ( kind = 4 ) numexp integer ( kind = 4 ) numjn real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) tran04 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) atran(0:19),rk, & rnumjn,sumexp,sum2,t,valinf,xhigh1,xhigh2, & xhigh3,xk,xk1,xlow1,xlow2 data numjn,rnumjn/ 4 , 4.0d0 / data valinf/0.25975757609067316596d2/ data atran/0.48075709946151105786d0, & -0.8175378810321083956d-1, & 0.1002700665975162973d-1, & -0.105993393598201507d-2, & 0.10345062450304053d-3, & -0.964427054858991d-5, & 0.87455444085147d-6, & -0.7793212079811d-7, & 0.686498861410d-8, & -0.59995710764d-9, & 0.5213662413d-10, & -0.451183819d-11, & 0.38921592d-12, & -0.3349360d-13, & 0.287667d-14, & -0.24668d-15, & 0.2113d-16, & -0.181d-17, & 0.15d-18, & -0.1d-19/ ! ! Machine-dependent constants ! data xlow1,xlow2/2.98023224d-8,4.05653502d-103/ data xhigh1,xhigh3/36.04365668d0,-36.73680056d0/ data xhigh2/1.80143985d16/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'TRAN04 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' tran04 = zero return end if ! ! Code for x < = 4.0 ! if ( x <= four ) then if ( x < xlow2 ) then tran04 = zero else if ( x < xlow1 ) then tran04 = ( x ** ( numjn-1 ) ) / ( rnumjn - one ) else t = ( ( ( x * x ) / eight ) - half ) - half tran04 = ( x ** ( numjn-1 ) ) * cheval ( nterms, atran, t ) end if end if else ! ! Code for x > 4.0 ! if ( xhigh2 < x ) then sumexp = one else if ( x <= xhigh1 ) then numexp = int ( xhigh1 / x ) + 1 t = exp ( -x ) else numexp = 1 t = one end if rk = zero do k1 = 1, numexp rk = rk + one end do sumexp = zero do k1 = 1, numexp sum2 = one xk = one / ( rk * x ) xk1 = one do k2 = 1, numjn sum2 = sum2 * xk1 * xk + one xk1 = xk1 + one end do sumexp = sumexp * t + sum2 rk = rk - one end do end if t = rnumjn * log ( x ) - x + log ( sumexp ) if ( t < xhigh3 ) then tran04 = valinf else tran04 = valinf - exp ( t ) end if end if return end subroutine tran04_values ( n_data, x, fx ) !*****************************************************************************80 ! !! TRAN04_VALUES returns some values of the order 4 transportation function. ! ! Discussion: ! ! The function is defined by: ! ! TRAN04(x) = Integral ( 0 <= t <= x ) t^4 * exp(t) / ( exp(t) - 1 )^2 dt ! ! Modified: ! ! 06 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.24835263919461834041D-08, & 0.10172029353616724881D-04, & 0.65053332405940765479D-03, & 0.41150448004155727767D-01, & 0.31724404523442648241D+00, & 0.10079442901142373591D+01, & 0.22010881024333408363D+01, & 0.38846508619156545210D+01, & 0.59648223973714765245D+01, & 0.10731932392998622219D+02, & 0.11940028876819364777D+02, & 0.15359784316882182982D+02, & 0.17372587633093742893D+02, & 0.19122976016053166969D+02, & 0.23583979156921941515D+02, & 0.25273667677030441733D+02, & 0.25955198214572256372D+02, & 0.25975350935212241910D+02, & 0.25975757522084093747D+02, & 0.25975757609067315288D+02 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function tran05 ( xvalue ) !*****************************************************************************80 ! !! TRAN05 calculates the transport integral of order 5. ! ! Discussion: ! ! The function is defined by: ! ! TRAN05(x) = Integral ( 0 <= t <= x ) t^5 * exp(t) / ( exp(t) - 1 )^2 dt ! ! The program uses a Chebyshev series, the coefficients of which are ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) TRAN05, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) k1 integer ( kind = 4 ) k2 integer ( kind = 4 ), parameter :: nterms = 17 integer ( kind = 4 ) numexp integer ( kind = 4 ) numjn real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) tran05 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) atran(0:19),rk, & rnumjn,sumexp,sum2,t,valinf,xhigh1,xhigh2, & xhigh3,xk,xk1,xlow1,xlow2 data numjn,rnumjn/ 5 , 5.0d0 / data valinf/0.12443133061720439116d3/ data atran/0.34777777713391078928d0, & -0.6645698897605042801d-1, & 0.861107265688330882d-2, & -0.93966822237555384d-3, & 0.9363248060815134d-4, & -0.885713193408328d-5, & 0.81191498914503d-6, & -0.7295765423277d-7, & 0.646971455045d-8, & -0.56849028255d-9, & 0.4962559787d-10, & -0.431093996d-11, & 0.37310094d-12, & -0.3219769d-13, & 0.277220d-14, & -0.23824d-15, & 0.2044d-16, & -0.175d-17, & 0.15d-18, & -0.1d-19/ ! ! Machine-dependent constants ! data xlow1,xlow2/2.98023224d-8,1.72723372d-77/ data xhigh1,xhigh3/36.04365668d0,-36.73680056d0/ data xhigh2/2.25179981d16/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'TRAN05 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' tran05 = zero return end if ! ! Code for x < = 4.0 ! if ( x <= four ) then if ( x < xlow2 ) then tran05 = zero else if ( x < xlow1 ) then tran05 = ( x ** ( numjn - 1 ) ) / ( rnumjn - one ) else t = ( ( ( x * x ) / eight ) - half ) - half tran05 = ( x ** ( numjn-1 ) ) * cheval ( nterms, atran, t ) end if end if else ! ! Code for x > 4.0 ! if ( xhigh2 < x ) then sumexp = one else if ( x <= xhigh1 ) then numexp = int ( xhigh1 / x ) + 1 t = exp ( -x ) else numexp = 1 t = one end if rk = zero do k1 = 1, numexp rk = rk + one end do sumexp = zero do k1 = 1, numexp sum2 = one xk = one / ( rk * x ) xk1 = one do k2 = 1, numjn sum2 = sum2 * xk1 * xk + one xk1 = xk1 + one end do sumexp = sumexp * t + sum2 rk = rk - one end do end if t = rnumjn * log ( x ) - x + log ( sumexp ) if ( t < xhigh3 ) then tran05 = valinf else tran05 = valinf - exp ( t ) end if end if return end subroutine tran05_values ( n_data, x, fx ) !*****************************************************************************80 ! !! TRAN05_VALUES returns some values of the order 5 transportation function. ! ! Discussion: ! ! The function is defined by: ! ! TRAN05(x) = Integral ( 0 <= t <= x ) t^5 * exp(t) / ( exp(t) - 1 )^2 dt ! ! Modified: ! ! 06 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.36379780361036116971D-11, & 0.23840564453948442379D-06, & 0.60982205372226969189D-04, & 0.15410004586376649337D-01, & 0.23661587923909478926D+00, & 0.11198756851307629651D+01, & 0.32292901663684049171D+01, & 0.70362973105160654056D+01, & 0.12770557691044159511D+02, & 0.29488339015245845447D+02, & 0.34471340540362254586D+02, & 0.50263092218175187785D+02, & 0.60819909101127165207D+02, & 0.70873334429213460498D+02, & 0.10147781242977788097D+03, & 0.11638074540242071077D+03, & 0.12409623901262967878D+03, & 0.12442270155632550228D+03, & 0.12443132790838589548D+03, & 0.12443133061720432435D+03 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function tran06 ( xvalue ) !*****************************************************************************80 ! !! TRAN06 calculates the transport integral of order 6. ! ! Discussion: ! ! The function is defined by: ! ! TRAN06(x) = Integral ( 0 <= t <= x ) t^6 * exp(t) / ( exp(t) - 1 )^2 dt ! ! The program uses a Chebyshev series, the coefficients of which are ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) TRAN06, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) k1 integer ( kind = 4 ) k2 integer ( kind = 4 ), parameter :: nterms = 17 integer ( kind = 4 ) numexp integer ( kind = 4 ) numjn real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) tran06 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) atran(0:19),rk, & rnumjn,sumexp,sum2,t,valinf,xhigh1,xhigh2, & xhigh3,xk,xk1,xlow1,xlow2 data numjn,rnumjn/ 6 , 6.0d0 / data valinf/0.73248700462880338059d3/ data atran/0.27127335397840008227d0, & -0.5588610553191453393d-1, & 0.753919513290083056d-2, & -0.84351138579211219d-3, & 0.8549098079676702d-4, & -0.818715493293098d-5, & 0.75754240427986d-6, & -0.6857306541831d-7, & 0.611700376031d-8, & -0.54012707024d-9, & 0.4734306435d-10, & -0.412701055d-11, & 0.35825603d-12, & -0.3099752d-13, & 0.267501d-14, & -0.23036d-15, & 0.1980d-16, & -0.170d-17, & 0.15d-18, & -0.1d-19/ ! ! Machine-dependent constants ! data xlow1,xlow2/2.98023224d-8,4.06689432d-62/ data xhigh1,xhigh3/36.04365668d0,-36.73680056d0/ data xhigh2/2.70215977d16/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'TRAN06 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' tran06 = zero return end if ! ! Code for x < = 4 .0 ! if ( x <= four ) then if ( x < xlow2 ) then tran06 = zero else if ( x < xlow1 ) then tran06 = ( x ** ( numjn-1 ) ) / ( rnumjn - one ) else t = ( ( ( x * x ) / eight ) - half ) - half tran06 = ( x ** ( numjn-1 ) ) * cheval ( nterms, atran, t ) end if end if else ! ! Code for x > 4 .0 ! if ( xhigh2 < x ) then sumexp = one else if ( x <= xhigh1 ) then numexp = int ( xhigh1 / x ) + 1 t = exp ( - x ) else numexp = 1 t = one end if rk = zero do k1 = 1, numexp rk = rk + one end do sumexp = zero do k1 = 1, numexp sum2 = one xk = one / ( rk * x ) xk1 = one do k2 = 1, numjn sum2 = sum2 * xk1 * xk + one xk1 = xk1 + one end do sumexp = sumexp * t + sum2 rk = rk - one end do end if t = rnumjn * log ( x ) - x + log ( sumexp ) if ( t < xhigh3 ) then tran06 = valinf else tran06 = valinf - exp ( t ) end if end if return end subroutine tran06_values ( n_data, x, fx ) !*****************************************************************************80 ! !! TRAN06_VALUES returns some values of the order 6 transportation function. ! ! Discussion: ! ! The function is defined by: ! ! TRAN06(x) = Integral ( 0 <= t <= x ) t^6 * exp(t) / ( exp(t) - 1 )^2 dt ! ! Modified: ! ! 06 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.56843405953641209574D-14, & 0.59601180165247401484D-08, & 0.60978424397580572815D-05, & 0.61578909866319494394D-02, & 0.18854360275680840514D+00, & 0.13319251347921659134D+01, & 0.50857202271697616755D+01, & 0.13729222365466557122D+02, & 0.29579592481641441292D+02, & 0.88600835706899853768D+02, & 0.10916037113373004909D+03, & 0.18224323749575359518D+03, & 0.23765383125586756031D+03, & 0.29543246745959381136D+03, & 0.50681244381280455592D+03, & 0.63878231134946125623D+03, & 0.72699203556994876111D+03, & 0.73230331643146851717D+03, & 0.73248692015882096369D+03, & 0.73248700462879996604D+03 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function tran07 ( xvalue ) !*****************************************************************************80 ! !! TRAN07 calculates the transport integral of order 7. ! ! Discussion: ! ! The function is defined by: ! ! TRAN07(x) = Integral ( 0 <= t <= x ) t^7 * exp(t) / ( exp(t) - 1 )^2 dt ! ! The program uses a Chebyshev series, the coefficients of which are ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) TRAN07, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) k1 integer ( kind = 4 ) k2 integer ( kind = 4 ), parameter :: nterms = 17 integer ( kind = 4 ) numexp integer ( kind = 4 ) numjn real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) tran07 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) atran(0:19),rk, & rnumjn,sumexp,sum2,t,valinf,xhigh1,xhigh2, & xhigh3,xk,xk1,xlow1,xlow2 data numjn,rnumjn/ 7 , 7.0d0/ data valinf/0.50820803580048910473d4/ data atran/0.22189250734010404423d0, & -0.4816751061177993694d-1, & 0.670092448103153629d-2, & -0.76495183443082557d-3, & 0.7863485592348690d-4, & -0.761025180887504d-5, & 0.70991696299917d-6, & -0.6468025624903d-7, & 0.580039233960d-8, & -0.51443370149d-9, & 0.4525944183d-10, & -0.395800363d-11, & 0.34453785d-12, & -0.2988292d-13, & 0.258434d-14, & -0.22297d-15, & 0.1920d-16, & -0.165d-17, & 0.14d-18, & -0.1d-19/ ! ! Machine-dependent constants ! data xlow1,xlow2/2.98023224d-8,7.14906557d-52/ data xhigh1,xhigh3/36.04365668d0,-36.73680056d0/ data xhigh2/3.15251973d16/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'TRAN07 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' tran07 = zero return end if ! ! Code for x <= 4.0 ! if ( x <= four ) then if ( x < xlow2 ) then tran07 = zero else if ( x < xlow1 ) then tran07 = ( x**(numjn-1) ) / ( rnumjn - one ) else t = ( ( ( x * x ) / eight ) - half ) - half tran07 = ( x**(numjn-1) ) * cheval ( nterms, atran, t ) end if end if else ! ! Code for x > 4.0 ! if ( xhigh2 < x ) then sumexp = one else if ( x <= xhigh1 ) then numexp = int ( xhigh1 / x ) + 1 t = exp ( -x ) else numexp = 1 t = one end if rk = zero do k1 = 1, numexp rk = rk + one end do sumexp = zero do k1 = 1, numexp sum2 = one xk = one / ( rk * x ) xk1 = one do k2 = 1, numjn sum2 = sum2 * xk1 * xk + one xk1 = xk1 + one end do sumexp = sumexp * t + sum2 rk = rk - one end do end if t = rnumjn * log ( x ) - x + log ( sumexp ) if ( t < xhigh3 ) then tran07 = valinf else tran07 = valinf - exp ( t ) end if end if return end subroutine tran07_values ( n_data, x, fx ) !*****************************************************************************80 ! !! TRAN07_VALUES returns some values of the order 7 transportation function. ! ! Discussion: ! ! The function is defined by: ! ! TRAN07(x) = Integral ( 0 <= t <= x ) t^7 * exp(t) / ( exp(t) - 1 )^2 dt ! ! Modified: ! ! 06 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.92518563327283409427D-17, & 0.15521095556949867541D-09, & 0.63516238373841716290D-06, & 0.25638801246626135714D-02, & 0.15665328993811649746D+00, & 0.16538225039181097423D+01, & 0.83763085709508211054D+01, & 0.28078570717830763747D+02, & 0.72009676046751991365D+02, & 0.28174905701691911450D+03, & 0.36660227975327792529D+03, & 0.70556067982603601123D+03, & 0.99661927562755629434D+03, & 0.13288914430417403901D+04, & 0.27987640273169129925D+04, & 0.39721376409416504325D+04, & 0.49913492839319899726D+04, & 0.50781562639825019000D+04, & 0.50820777202028708434D+04, & 0.50820803580047164618D+04 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function tran08 ( xvalue ) !*****************************************************************************80 ! !! TRAN08 calculates the transport integral of order 8. ! ! Discussion: ! ! The function is defined by: ! ! TRAN08(x) = Integral ( 0 <= t <= x ) t^8 * exp(t) / ( exp(t) - 1 )^2 dt ! ! The program uses a Chebyshev series, the coefficients of which are ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) TRAN08, the value of the function. ! implicit none real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) k1 integer ( kind = 4 ) k2 integer ( kind = 4 ), parameter :: nterms = 17 integer ( kind = 4 ) numexp integer ( kind = 4 ) numjn real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) tran08 real ( kind = 8 ) x real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 real ( kind = 8 ) atran(0:19),rk, & rnumjn,sumexp,sum2,t,valinf,xhigh1,xhigh2, & xhigh3,xk,xk1,xlow1,xlow2 data numjn,rnumjn/ 8, 8.0d0 / data valinf/0.40484399001901115764d5/ data atran/0.18750695774043719233d0, & -0.4229527646093673337d-1, & 0.602814856929065592d-2, & -0.69961054811814776d-3, & 0.7278482421298789d-4, & -0.710846250050067d-5, & 0.66786706890115d-6, & -0.6120157501844d-7, & 0.551465264474d-8, & -0.49105307052d-9, & 0.4335000869d-10, & -0.380218700d-11, & 0.33182369d-12, & -0.2884512d-13, & 0.249958d-14, & -0.21605d-15, & 0.1863d-16, & -0.160d-17, & 0.14d-18, & -0.1d-19/ ! ! Machine-dependent constants ! data xlow1,xlow2/2.98023224d-8,1.48029723d-44/ data xhigh1,xhigh3/36.04365668d0,-36.73680056d0/ data xhigh2/3.6028797d16/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'TRAN08 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' tran08 = zero return end if ! ! Code for x < = 4.0 ! if ( x <= four ) then if ( x < xlow2 ) then tran08 = zero else if ( x < xlow1 ) then tran08 = ( x ** ( numjn - 1 ) ) / ( rnumjn - one ) else t = ( ( ( x * x ) / eight ) - half ) - half tran08 = ( x ** ( numjn - 1 ) ) * cheval ( nterms, atran, t ) end if end if else ! ! Code for x > 4.0 ! if ( xhigh2 < x ) then sumexp = one else if ( x <= xhigh1 ) then numexp = int ( xhigh1 / x ) + 1 t = exp ( - x ) else numexp = 1 t = one end if rk = zero do k1 = 1, numexp rk = rk + one end do sumexp = zero do k1 = 1, numexp sum2 = one xk = one / ( rk * x ) xk1 = one do k2 = 1, numjn sum2 = sum2 * xk1 * xk + one xk1 = xk1 + one end do sumexp = sumexp * t + sum2 rk = rk - one end do end if t = rnumjn * log ( x ) - x + log ( sumexp ) if ( t < xhigh3 ) then tran08 = valinf else tran08 = valinf - exp ( t ) end if end if return end subroutine tran08_values ( n_data, x, fx ) !*****************************************************************************80 ! !! TRAN08_VALUES returns some values of the order 8 transportation function. ! ! Discussion: ! ! The function is defined by: ! ! TRAN08(x) = Integral ( 0 <= t <= x ) t^8 * exp(t) / ( exp(t) - 1 )^2 dt ! ! Modified: ! ! 06 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.15488598634539359463D-19, & 0.41574269117845953797D-11, & 0.68050651245227411689D-07, & 0.10981703519563009836D-02, & 0.13396432776187883834D+00, & 0.21153387806998617182D+01, & 0.14227877028750735641D+02, & 0.59312061431647843226D+02, & 0.18139614577043147745D+03, & 0.93148001928992220863D+03, & 0.12817928112604611804D+04, & 0.28572838386329242218D+04, & 0.43872971687877730010D+04, & 0.62993229139406657611D+04, & 0.16589426277154888511D+05, & 0.27064780798797398935D+05, & 0.38974556062543661284D+05, & 0.40400240716905025786D+05, & 0.40484316504120655568D+05, & 0.40484399001892184901D+05 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end function tran09 ( xvalue ) !*****************************************************************************80 ! !! TRAN09 calculates the transport integral of order 9. ! ! Discussion: ! ! The function is defined by: ! ! TRAN09(x) = Integral ( 0 <= t <= x ) t^9 * exp(t) / ( exp(t) - 1 )^2 dt ! ! The program uses a Chebyshev series, the coefficients of which are ! given to an accuracy of 20 decimal places. ! ! This subroutine is set up to work on IEEE machines. ! ! Modified: ! ! 07 August 2004 ! ! Author: ! ! Allan McLeod, ! Department of Mathematics and Statistics, ! Paisley University, High Street, Paisley, Scotland, PA12BE ! macl_ms0@paisley.ac.uk ! ! Reference: ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Parameters: ! ! Input, real ( kind = 8 ) XVALUE, the argument of the function. ! ! Output, real ( kind = 8 ) TRAN09, the value of the function. ! implicit none real ( kind = 8 ) atran(0:19) real ( kind = 8 ) cheval real ( kind = 8 ), parameter :: eight = 8.0D+00 real ( kind = 8 ), parameter :: four = 4.0D+00 real ( kind = 8 ), parameter :: half = 0.5D+00 integer ( kind = 4 ) k1 integer ( kind = 4 ) k2 integer ( kind = 4 ), parameter :: nterms = 17 integer ( kind = 4 ) numexp integer ( kind = 4 ), parameter :: numjn = 9 real ( kind = 8 ), parameter :: one = 1.0D+00 real ( kind = 8 ) rk real ( kind = 8 ), parameter :: rnumjn = 9.0D+00 real ( kind = 8 ) sumexp real ( kind = 8 ) sum2 real ( kind = 8 ) t real ( kind = 8 ) tran09 real ( kind = 8 ), parameter :: valinf = 0.36360880558872871397d6 real ( kind = 8 ) x real ( kind = 8 ) xhigh1 real ( kind = 8 ) xhigh2 real ( kind = 8 ) xhigh3 real ( kind = 8 ) xk real ( kind = 8 ) xk1 real ( kind = 8 ) xlow1 real ( kind = 8 ) xlow2 real ( kind = 8 ) xvalue real ( kind = 8 ), parameter :: zero = 0.0D+00 data atran/0.16224049991949846835d0, & -0.3768351452195937773d-1, & 0.547669715917719770d-2, & -0.64443945009449521d-3, & 0.6773645285280983d-4, & -0.666813497582042d-5, & 0.63047560019047d-6, & -0.5807478663611d-7, & 0.525551305123d-8, & -0.46968861761d-9, & 0.4159395065d-10, & -0.365808491d-11, & 0.32000794d-12, & -0.2787651d-13, & 0.242017d-14, & -0.20953d-15, & 0.1810d-16, & -0.156d-17, & 0.13d-18, & -0.1d-19/ ! ! Machine-dependent constants (for IEEE machines) ! data xlow1,xlow2/2.98023224d-8,4.5321503d-39/ data xhigh1,xhigh3/36.04365668d0,-36.73680056d0/ data xhigh2/4.05323966d16/ x = xvalue if ( x < zero ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'TRAN09 - Fatal error!' write ( *, '(a)' ) ' Argument X < 0.' tran09 = zero return end if ! ! Code for x < = 4.0 ! if ( x <= four ) then if ( x < xlow2 ) then tran09 = zero else if ( x < xlow1 ) then tran09 = ( x ** ( numjn - 1 ) ) / ( rnumjn - one ) else t = ( ( ( x * x ) / eight ) - half ) - half tran09 = ( x ** ( numjn - 1 ) ) * cheval ( nterms, atran, t ) end if end if else ! ! Code for x > 4.0 ! if ( xhigh2 < x ) then sumexp = one else if ( x <= xhigh1 ) then numexp = int ( xhigh1 / x ) + 1 t = exp ( -x ) else numexp = 1 t = one end if rk = zero do k1 = 1, numexp rk = rk + one end do sumexp = zero do k1 = 1, numexp sum2 = one xk = one / ( rk * x ) xk1 = one do k2 = 1, numjn sum2 = sum2 * xk1 * xk + one xk1 = xk1 + one end do sumexp = sumexp * t + sum2 rk = rk - one end do end if t = rnumjn * log ( x ) - x + log ( sumexp ) if ( t < xhigh3 ) then tran09 = valinf else tran09 = valinf - exp ( t ) end if end if return end subroutine tran09_values ( n_data, x, fx ) !*****************************************************************************80 ! !! TRAN09_VALUES returns some values of the order 9 transportation function. ! ! Discussion: ! ! The function is defined by: ! ! TRAN09(x) = Integral ( 0 <= t <= x ) t^9 * exp(t) / ( exp(t) - 1 )^2 dt ! ! Modified: ! ! 06 September 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964. ! ! Allan McLeod, ! Algorithm 757, MISCFUN: A software package to compute uncommon ! special functions, ! ACM Transactions on Mathematical Software, ! Volume 22, Number 3, September 1996, pages 288-301. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Wolfram Media / Cambridge University Press, 1999. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 before the ! first call. On each call, the routine increments N_DATA by 1, and ! returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.26469772870084897671D-22, & 0.11367943653594246210D-12, & 0.74428246255329800255D-08, & 0.48022728485415366194D-03, & 0.11700243014358676725D+00, & 0.27648973910899914391D+01, & 0.24716631405829192997D+02, & 0.12827119828849828583D+03, & 0.46842894800662208986D+03, & 0.31673967371627895718D+04, & 0.46140886546630195390D+04, & 0.11952718545392302185D+05, & 0.20001612666477027728D+05, & 0.31011073271851366554D+05, & 0.10352949905541130133D+06, & 0.19743173017140591390D+06, & 0.33826030414658460679D+06, & 0.36179607036750755227D+06, & 0.36360622124777561525D+06, & 0.36360880558827162725D+06 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.0019531250D+00, & 0.0312500000D+00, & 0.1250000000D+00, & 0.5000000000D+00, & 1.0000000000D+00, & 1.5000000000D+00, & 2.0000000000D+00, & 2.5000000000D+00, & 3.0000000000D+00, & 4.0000000000D+00, & 4.2500000000D+00, & 5.0000000000D+00, & 5.5000000000D+00, & 6.0000000000D+00, & 8.0000000000D+00, & 10.0000000000D+00, & 15.0000000000D+00, & 20.0000000000D+00, & 30.0000000000D+00, & 50.0000000000D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end