subroutine ch_cap ( ch ) c*********************************************************************72 c cc CH_CAP capitalizes a single character. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 03 January 2007 c c Author: c c John Burkardt c c Parameters: c c Input/output, character CH, the character to capitalize. c implicit none character ch integer itemp itemp = ichar ( ch ) if ( 97 .le. itemp .and. itemp .le. 122 ) then ch = char ( itemp - 32 ) end if return end function ch_eqi ( c1, c2 ) c*********************************************************************72 c cc CH_EQI is a case insensitive comparison of two characters for equality. c c Example: c c CH_EQI ( 'A', 'a' ) is TRUE. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 03 January 2007 c c Author: c c John Burkardt c c Parameters: c c Input, character C1, C2, the characters to compare. c c Output, logical CH_EQI, the result of the comparison. c implicit none character c1 character c1_cap character c2 character c2_cap logical ch_eqi c1_cap = c1 c2_cap = c2 call ch_cap ( c1_cap ) call ch_cap ( c2_cap ) if ( c1_cap .eq. c2_cap ) then ch_eqi = .true. else ch_eqi = .false. end if return end subroutine ch_to_digit ( c, digit ) c*********************************************************************72 c cc CH_TO_DIGIT returns the integer value of a base 10 digit. c c Example: c c C DIGIT c --- ----- c '0' 0 c '1' 1 c ... ... c '9' 9 c ' ' 0 c 'X' -1 c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 04 August 1999 c c Author: c c John Burkardt c c Parameters: c c Input, character C, the decimal digit, '0' through '9' or blank c are legal. c c Output, integer DIGIT, the corresponding integer value. If C was c 'illegal', then DIGIT is -1. c implicit none character c integer digit if ( lge ( c, '0' ) .and. lle ( c, '9' ) ) then digit = ichar ( c ) - 48 else if ( c .eq. ' ' ) then digit = 0 else digit = -1 end if return end subroutine daxpy ( n, da, dx, incx, dy, incy ) c*********************************************************************72 c cc DAXPY computes constant times a vector plus a vector. c c Discussion: c c This routine uses double precision real arithmetic. c c This routine uses unrolled loops for increments equal to one. c c Modified: c c 18 December 2008 c c Author: c c Jack Dongarra c c Reference: c c Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, c LINPACK User's Guide, c SIAM, 1979, c ISBN13: 978-0-898711-72-1, c LC: QA214.L56. c c Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh, c Basic Linear Algebra Subprograms for FORTRAN usage, c ACM Transactions on Mathematical Software, c Volume 5, Number 3, pages 308-323, 1979. c c Parameters: c c Input, integer N, the number of elements in DX and DY. c c Input, double precision DA, the multiplier of DX. c c Input, double precision DX(*), the first vector. c c Input, integer INCX, the increment between successive entries of DX. c c Input/output, double precision DY(*), the second vector. c On output, DY(*) has been replaced by DY(*) + DA * DX(*). c c Input, integer INCY, the increment between successive entries of DY. c implicit none double precision da double precision dx(*) double precision dy(*) integer i integer incx integer incy integer ix integer iy integer m integer n if ( n .le. 0 ) then return end if if ( da .eq. 0.0d0 ) then return end if if ( incx .ne. 1 .or. incy .ne. 1 ) then ix = 1 iy = 1 if ( incx .lt. 0 ) ix = (-n+1)*incx + 1 if ( incy .lt. 0 ) iy = (-n+1)*incy + 1 do i = 1, n dy(iy) = dy(iy) + da*dx(ix) ix = ix + incx iy = iy + incy end do else m = mod(n,4) do i = 1, m dy(i) = dy(i) + da*dx(i) end do do i = m + 1, n, 4 dy(i) = dy(i) + da*dx(i) dy(i + 1) = dy(i + 1) + da*dx(i + 1) dy(i + 2) = dy(i + 2) + da*dx(i + 2) dy(i + 3) = dy(i + 3) + da*dx(i + 3) end do end if return end function ddot ( n, dx, incx, dy, incy ) c*********************************************************************72 c cc DDOT forms the dot product of two vectors. c c Discussion: c c This routine uses double precision real arithmetic. c c This routine uses unrolled loops for increments equal to one. c c Modified: c c 07 July 2007 c c Author: c c Jack Dongarra c c Reference: c c Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, c LINPACK User's Guide, c SIAM, 1979, c ISBN13: 978-0-898711-72-1, c LC: QA214.L56. c c Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh, c Basic Linear Algebra Subprograms for FORTRAN usage, c ACM Transactions on Mathematical Software, c Volume 5, Number 3, pages 308-323, 1979. c c Parameters: c c Input, integer N, the number of entries in the vectors. c c Input, double precision DX(*), the first vector. c c Input, integer INCX, the increment between successive entries in DX. c c Input, double precision DY(*), the second vector. c c Input, integer INCY, the increment between successive entries in DY. c c Output, double precision DDOT, the sum of the product of the c corresponding entries of DX and DY. c implicit none double precision ddot double precision dx(*) double precision dy(*) double precision dtemp integer i,incx,incy,ix,iy,m,n ddot = 0.0d0 dtemp = 0.0d0 if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do i = 1,n dtemp = dtemp + dx(ix)*dy(iy) ix = ix + incx iy = iy + incy end do ddot = dtemp return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do i = 1,m dtemp = dtemp + dx(i)*dy(i) end do if( n .lt. 5 ) go to 60 40 continue do i = m+1, n, 5 dtemp = dtemp + dx(i)*dy(i) + dx(i + 1)*dy(i + 1) + & dx(i + 2)*dy(i + 2) + dx(i + 3)*dy(i + 3) + dx(i + 4)*dy(i + 4) end do 60 ddot = dtemp return end function dnrm2 ( n, x, incx ) c*********************************************************************72 c cc DNRM2 returns the euclidean norm of a vector. c c Discussion: c c This routine uses double precision real arithmetic. c c DNRM2 ( X ) = sqrt ( X' * X ) c c Author: c c Sven Hammarling c c Reference: c c Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, c LINPACK User's Guide, c SIAM, 1979, c ISBN13: 978-0-898711-72-1, c LC: QA214.L56. c c Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh, c Basic Linear Algebra Subprograms for FORTRAN usage, c ACM Transactions on Mathematical Software, c Volume 5, Number 3, pages 308-323, 1979. c c Parameters: c c Input, integer N, the number of entries in the vector. c c Input, double precision X(*), the vector whose norm is to be computed. c c Input, integer INCX, the increment between successive entries of X. c c Output, double precision DNRM2, the Euclidean norm of X. c implicit none integer incx, n double precision dnrm2 double precision x( * ) double precision one , zero parameter ( one = 1.0d+0, zero = 0.0d+0 ) integer ix double precision absxi, norm, scale, ssq intrinsic abs, sqrt if( n.lt.1 .or. incx.lt.1 )then norm = zero else if( n.eq.1 )then norm = abs( x( 1 ) ) else scale = zero ssq = one c c The following loop is equivalent to this call to the LAPACK c auxiliary routine: c call dlassq( n, x, incx, scale, ssq ) c do ix = 1, 1 + ( n - 1 )*incx, incx if( x( ix ).ne.zero )then absxi = abs( x( ix ) ) if( scale.lt.absxi )then ssq = one + ssq*( scale/absxi )**2 scale = absxi else ssq = ssq + ( absxi/scale )**2 end if end if end do norm = scale * sqrt( ssq ) end if dnrm2 = norm return end subroutine drot ( n, dx, incx, dy, incy, c, s ) c*********************************************************************72 c cc DROT applies a plane rotation. c c Discussion: c c This routine uses double precision real arithmetic. c c Modified: c c 07 July 2007 c c Author: c c Jack Dongarra c c Reference: c c Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, c LINPACK User's Guide, c SIAM, 1979, c ISBN13: 978-0-898711-72-1, c LC: QA214.L56. c c Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh, c Basic Linear Algebra Subprograms for FORTRAN usage, c ACM Transactions on Mathematical Software, c Volume 5, Number 3, pages 308-323, 1979. c c Parameters: c c Input, integer N, the number of entries in the vectors. c c Input/output, double precision X(*), one of the vectors to be rotated. c c Input, integer INCX, the increment between successive entries of X. c c Input/output, double precision Y(*), one of the vectors to be rotated. c c Input, integer INCY, the increment between successive elements of Y. c c Input, double precision C, S, parameters (presumably the cosine and c sine of some angle) that define a plane rotation. c implicit none double precision dx(*),dy(*),dtemp,c,s integer i,incx,incy,ix,iy,n if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do i = 1,n dtemp = c*dx(ix) + s*dy(iy) dy(iy) = c*dy(iy) - s*dx(ix) dx(ix) = dtemp ix = ix + incx iy = iy + incy end do return c c code for both increments equal to 1 c 20 do i = 1,n dtemp = c*dx(i) + s*dy(i) dy(i) = c*dy(i) - s*dx(i) dx(i) = dtemp end do return end subroutine drotg ( da, db, c, s ) c*********************************************************************72 c cc DROTG constructs a Givens plane rotation. c c Discussion: c c This routine uses double precision real arithmetic. c c Given values A and B, this routine computes c c SIGMA = sign ( A ) if abs ( A ) > abs ( B ) c = sign ( B ) if abs ( A ) <= abs ( B ); c c R = SIGMA * ( A * A + B * B ); c c C = A / R if R is not 0 c = 1 if R is 0; c c S = B / R if R is not 0, c 0 if R is 0. c c The computed numbers then satisfy the equation c c ( C S ) ( A ) = ( R ) c ( -S C ) ( B ) = ( 0 ) c c The routine also computes c c Z = S if abs ( A ) > abs ( B ), c = 1 / C if abs ( A ) <= abs ( B ) and C is not 0, c = 1 if C is 0. c c The single value Z encodes C and S, and hence the rotation: c c If Z = 1, set C = 0 and S = 1; c If abs ( Z ) < 1, set C = sqrt ( 1 - Z * Z ) and S = Z; c if abs ( Z ) > 1, set C = 1/ Z and S = sqrt ( 1 - C * C ); c c Modified: c c 07 July 2007 c c Author: c c Jack Dongarra c c Reference: c c Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, c LINPACK User's Guide, c SIAM, 1979, c ISBN13: 978-0-898711-72-1, c LC: QA214.L56. c c Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh, c Basic Linear Algebra Subprograms for FORTRAN usage, c ACM Transactions on Mathematical Software, c Volume 5, Number 3, pages 308-323, 1979. c c Parameters: c c Input/output, double precision SA, SB. On input, SA and SB are the values c A and B. On output, SA is overwritten with R, and SB is c overwritten with Z. c c Output, double precision C, S, the cosine and sine of the c Givens rotation. c implicit none double precision da,db,c,s,roe,scale,r,z roe = db if( dabs(da) .gt. dabs(db) ) roe = da scale = dabs(da) + dabs(db) if( scale .eq. 0.0d0 ) then c = 1.0d0 s = 0.0d0 r = 0.0d0 z = 0.0d0 else r = scale*dsqrt((da/scale)**2 + (db/scale)**2) r = dsign(1.0d0,roe)*r c = da/r s = db/r z = 1.0d0 if( dabs(da) .gt. dabs(db) ) z = s if( dabs(db) .ge. dabs(da) .and. c .ne. 0.0d0 ) z = 1.0d0/c end if da = r db = z return end subroutine dscal ( n, da, dx, incx ) c*********************************************************************72 c cc DSCAL scales a vector by a constant. c c Discussion: c c This routine uses double precision real arithmetic. c c Modified: c c 07 July 2007 c c Author: c c Jack Dongarra c c Reference: c c Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, c LINPACK User's Guide, c SIAM, 1979, c ISBN13: 978-0-898711-72-1, c LC: QA214.L56. c c Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh, c Basic Linear Algebra Subprograms for FORTRAN usage, c ACM Transactions on Mathematical Software, c Volume 5, Number 3, pages 308-323, 1979. c c Parameters: c c Input, integer N, the number of entries in the vector. c c Input, double precision SA, the multiplier. c c Input/output, double precision X(*), the vector to be scaled. c c Input, integer INCX, the increment between successive entries of X. c implicit none double precision da double precision dx(*) integer i,incx,m,n,nincx if( n.le.0 .or. incx.le.0 )return if(incx.eq.1)go to 20 c c code for increment not equal to 1 c nincx = n*incx do i = 1,nincx,incx dx(i) = da*dx(i) end do return c c code for increment equal to 1 c c c clean-up loop c 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do i = 1,m dx(i) = da*dx(i) end do if( n .lt. 5 ) return 40 continue do i = m+1, n, 5 dx(i) = da*dx(i) dx(i + 1) = da*dx(i + 1) dx(i + 2) = da*dx(i + 2) dx(i + 3) = da*dx(i + 3) dx(i + 4) = da*dx(i + 4) end do return end subroutine dsvdc ( x, ldx, n, p, s, e, u, ldu, v, ldv, work, & job, info ) c*********************************************************************72 c cc DSVDC computes the singular value decomposition of a matrix. c c dsvdc is a subroutine to reduce a double precision nxp matrix x c by orthogonal transformations u and v to diagonal form. the c diagonal elements s(i) are the singular values of x. the c columns of u are the corresponding left singular vectors, c and the columns of v the right singular vectors. c c on entry c c x double precision(ldx,p), where ldx.ge.n. c x contains the matrix whose singular value c decomposition is to be computed. x is c destroyed by dsvdc. c c ldx integer. c ldx is the leading dimension of the array x. c c n integer. c n is the number of rows of the matrix x. c c p integer. c p is the number of columns of the matrix x. c c ldu integer. c ldu is the leading dimension of the array u. c (see below). c c ldv integer. c ldv is the leading dimension of the array v. c (see below). c c work double precision(n). c work is a scratch array. c c job integer. c job controls the computation of the singular c vectors. it has the decimal expansion ab c with the following meaning c c a.eq.0 do not compute the left singular c vectors. c a.eq.1 return the n left singular vectors c in u. c a.ge.2 return the first min(n,p) singular c vectors in u. c b.eq.0 do not compute the right singular c vectors. c b.eq.1 return the right singular vectors c in v. c c on return c c s double precision(mm), where mm=max(n+1,p). c the first min(n,p) entries of s contain the c singular values of x arranged in descending c order of magnitude. c c e double precision(mm), where mm=max(n+1,p). c e ordinarily contains zeros. however see the c discussion of info for exceptions. c c u double precision(ldu,k), where ldu.ge.n. if c joba.eq.1 then k.eq.n, if joba.ge.2 c then k.eq.min(n,p). c u contains the matrix of left singular vectors. c u is not referenced if joba.eq.0. if n.le.p c or if joba.eq.2, then u may be identified with x c in the subroutine call. c c v double precision(ldv,p), where ldv.ge.p. c v contains the matrix of right singular vectors. c v is not referenced if job.eq.0. if p.le.n, c then v may be identified with x in the c subroutine call. c c info integer. c the singular values (and their corresponding c singular vectors) s(info+1),s(info+2),...,s(m) c are correct (here m=min(n,p)). thus if c info.eq.0, all the singular values and their c vectors are correct. in any event, the matrix c b = trans(u)*x*v is the bidiagonal matrix c with the elements of s on its diagonal and the c elements of e on its super-diagonal (trans(u) c is the transpose of u). thus the singular c values of x and b are the same. c c linpack. this version dated 08/14/78 . c correction made to shift 2/84. c g.w. stewart, university of maryland, argonne national lab. c c dsvdc uses the following functions and subprograms. c c external drot c blas daxpy,ddot,dscal,dswap,dnrm2,drotg c fortran dabs,dmax1,max0,min0,mod,dsqrt c integer ldx,n,p,ldu,ldv,job,info double precision x(ldx,*),s(*),e(*),u(ldu,*),v(ldv,*),work(*) integer i,iter,j,jobu,k,kase,kk,l,ll,lls,lm1,lp1,ls,lu,m,maxit, * mm,mm1,mp1,nct,nctp1,ncu,nrt,nrtp1 double precision ddot,t,r double precision b,c,cs,el,emm1,f,g,dnrm2,scale,shift,sl,sm,sn, * smm1,t1,test,ztest logical wantu,wantv c c c set the maximum number of iterations. c maxit = 30 c c determine what is to be computed. c wantu = .false. wantv = .false. jobu = mod(job,100)/10 ncu = n if (jobu .gt. 1) ncu = min0(n,p) if (jobu .ne. 0) wantu = .true. if (mod(job,10) .ne. 0) wantv = .true. c c reduce x to bidiagonal form, storing the diagonal elements c in s and the super-diagonal elements in e. c info = 0 nct = min0(n-1,p) nrt = max0(0,min0(p-2,n)) lu = max0(nct,nrt) if (lu .lt. 1) go to 170 do 160 l = 1, lu lp1 = l + 1 if (l .gt. nct) go to 20 c c compute the transformation for the l-th column and c place the l-th diagonal in s(l). c s(l) = dnrm2(n-l+1,x(l,l),1) if (s(l) .eq. 0.0d0) go to 10 if (x(l,l) .ne. 0.0d0) s(l) = dsign(s(l),x(l,l)) call dscal(n-l+1,1.0d0/s(l),x(l,l),1) x(l,l) = 1.0d0 + x(l,l) 10 continue s(l) = -s(l) 20 continue if (p .lt. lp1) go to 50 do 40 j = lp1, p if (l .gt. nct) go to 30 if (s(l) .eq. 0.0d0) go to 30 c c apply the transformation. c t = -ddot(n-l+1,x(l,l),1,x(l,j),1)/x(l,l) call daxpy(n-l+1,t,x(l,l),1,x(l,j),1) 30 continue c c place the l-th row of x into e for the c subsequent calculation of the row transformation. c e(j) = x(l,j) 40 continue 50 continue if (.not.wantu .or. l .gt. nct) go to 70 c c place the transformation in u for subsequent back c multiplication. c do 60 i = l, n u(i,l) = x(i,l) 60 continue 70 continue if (l .gt. nrt) go to 150 c c compute the l-th row transformation and place the c l-th super-diagonal in e(l). c e(l) = dnrm2(p-l,e(lp1),1) if (e(l) .eq. 0.0d0) go to 80 if (e(lp1) .ne. 0.0d0) e(l) = dsign(e(l),e(lp1)) call dscal(p-l,1.0d0/e(l),e(lp1),1) e(lp1) = 1.0d0 + e(lp1) 80 continue e(l) = -e(l) if (lp1 .gt. n .or. e(l) .eq. 0.0d0) go to 120 c c apply the transformation. c do 90 i = lp1, n work(i) = 0.0d0 90 continue do 100 j = lp1, p call daxpy(n-l,e(j),x(lp1,j),1,work(lp1),1) 100 continue do 110 j = lp1, p call daxpy(n-l,-e(j)/e(lp1),work(lp1),1,x(lp1,j),1) 110 continue 120 continue if (.not.wantv) go to 140 c c place the transformation in v for subsequent c back multiplication. c do 130 i = lp1, p v(i,l) = e(i) 130 continue 140 continue 150 continue 160 continue 170 continue c c set up the final bidiagonal matrix or order m. c m = min0(p,n+1) nctp1 = nct + 1 nrtp1 = nrt + 1 if (nct .lt. p) s(nctp1) = x(nctp1,nctp1) if (n .lt. m) s(m) = 0.0d0 if (nrtp1 .lt. m) e(nrtp1) = x(nrtp1,m) e(m) = 0.0d0 c c if required, generate u. c if (.not.wantu) go to 300 if (ncu .lt. nctp1) go to 200 do 190 j = nctp1, ncu do 180 i = 1, n u(i,j) = 0.0d0 180 continue u(j,j) = 1.0d0 190 continue 200 continue if (nct .lt. 1) go to 290 do 280 ll = 1, nct l = nct - ll + 1 if (s(l) .eq. 0.0d0) go to 250 lp1 = l + 1 if (ncu .lt. lp1) go to 220 do 210 j = lp1, ncu t = -ddot(n-l+1,u(l,l),1,u(l,j),1)/u(l,l) call daxpy(n-l+1,t,u(l,l),1,u(l,j),1) 210 continue 220 continue call dscal(n-l+1,-1.0d0,u(l,l),1) u(l,l) = 1.0d0 + u(l,l) lm1 = l - 1 if (lm1 .lt. 1) go to 240 do 230 i = 1, lm1 u(i,l) = 0.0d0 230 continue 240 continue go to 270 250 continue do 260 i = 1, n u(i,l) = 0.0d0 260 continue u(l,l) = 1.0d0 270 continue 280 continue 290 continue 300 continue c c if it is required, generate v. c if (.not.wantv) go to 350 do 340 ll = 1, p l = p - ll + 1 lp1 = l + 1 if (l .gt. nrt) go to 320 if (e(l) .eq. 0.0d0) go to 320 do 310 j = lp1, p t = -ddot(p-l,v(lp1,l),1,v(lp1,j),1)/v(lp1,l) call daxpy(p-l,t,v(lp1,l),1,v(lp1,j),1) 310 continue 320 continue do 330 i = 1, p v(i,l) = 0.0d0 330 continue v(l,l) = 1.0d0 340 continue 350 continue c c main iteration loop for the singular values. c mm = m iter = 0 360 continue c c quit if all the singular values have been found. c c ...exit if (m .eq. 0) go to 620 c c if too many iterations have been performed, set c flag and return. c if (iter .lt. maxit) go to 370 info = m c ......exit go to 620 370 continue c c this section of the program inspects for c negligible elements in the s and e arrays. on c completion the variables kase and l are set as follows. c c kase = 1 if s(m) and e(l-1) are negligible and l.lt.m c kase = 2 if s(l) is negligible and l.lt.m c kase = 3 if e(l-1) is negligible, l.lt.m, and c s(l), ..., s(m) are not negligible (qr step). c kase = 4 if e(m-1) is negligible (convergence). c do 390 ll = 1, m l = m - ll c ...exit if (l .eq. 0) go to 400 test = dabs(s(l)) + dabs(s(l+1)) ztest = test + dabs(e(l)) if (ztest .ne. test) go to 380 e(l) = 0.0d0 c ......exit go to 400 380 continue 390 continue 400 continue if (l .ne. m - 1) go to 410 kase = 4 go to 480 410 continue lp1 = l + 1 mp1 = m + 1 do 430 lls = lp1, mp1 ls = m - lls + lp1 c ...exit if (ls .eq. l) go to 440 test = 0.0d0 if (ls .ne. m) test = test + dabs(e(ls)) if (ls .ne. l + 1) test = test + dabs(e(ls-1)) ztest = test + dabs(s(ls)) if (ztest .ne. test) go to 420 s(ls) = 0.0d0 c ......exit go to 440 420 continue 430 continue 440 continue if (ls .ne. l) go to 450 kase = 3 go to 470 450 continue if (ls .ne. m) go to 460 kase = 1 go to 470 460 continue kase = 2 l = ls 470 continue 480 continue l = l + 1 c c perform the task indicated by kase. c go to (490,520,540,570), kase c c deflate negligible s(m). c 490 continue mm1 = m - 1 f = e(m-1) e(m-1) = 0.0d0 do 510 kk = l, mm1 k = mm1 - kk + l t1 = s(k) call drotg(t1,f,cs,sn) s(k) = t1 if (k .eq. l) go to 500 f = -sn*e(k-1) e(k-1) = cs*e(k-1) 500 continue if (wantv) call drot(p,v(1,k),1,v(1,m),1,cs,sn) 510 continue go to 610 c c split at negligible s(l). c 520 continue f = e(l-1) e(l-1) = 0.0d0 do 530 k = l, m t1 = s(k) call drotg(t1,f,cs,sn) s(k) = t1 f = -sn*e(k) e(k) = cs*e(k) if (wantu) call drot(n,u(1,k),1,u(1,l-1),1,cs,sn) 530 continue go to 610 c c perform one qr step. c 540 continue c c calculate the shift. c scale = dmax1(dabs(s(m)),dabs(s(m-1)),dabs(e(m-1)), * dabs(s(l)),dabs(e(l))) sm = s(m)/scale smm1 = s(m-1)/scale emm1 = e(m-1)/scale sl = s(l)/scale el = e(l)/scale b = ((smm1 + sm)*(smm1 - sm) + emm1**2)/2.0d0 c = (sm*emm1)**2 shift = 0.0d0 if (b .eq. 0.0d0 .and. c .eq. 0.0d0) go to 550 shift = dsqrt(b**2+c) if (b .lt. 0.0d0) shift = -shift shift = c/(b + shift) 550 continue f = (sl + sm)*(sl - sm) + shift g = sl*el c c chase zeros. c mm1 = m - 1 do 560 k = l, mm1 call drotg(f,g,cs,sn) if (k .ne. l) e(k-1) = f f = cs*s(k) + sn*e(k) e(k) = cs*e(k) - sn*s(k) g = sn*s(k+1) s(k+1) = cs*s(k+1) if (wantv) call drot(p,v(1,k),1,v(1,k+1),1,cs,sn) call drotg(f,g,cs,sn) s(k) = f f = cs*e(k) + sn*s(k+1) s(k+1) = -sn*e(k) + cs*s(k+1) g = sn*e(k+1) e(k+1) = cs*e(k+1) if (wantu .and. k .lt. n) * call drot(n,u(1,k),1,u(1,k+1),1,cs,sn) 560 continue e(m-1) = f iter = iter + 1 go to 610 c c convergence. c 570 continue c c make the singular value positive. c if (s(l) .ge. 0.0d0) go to 580 s(l) = -s(l) if (wantv) call dscal(p,-1.0d0,v(1,l),1) 580 continue c c order the singular value. c 590 if (l .eq. mm) go to 600 c ...exit if (s(l) .ge. s(l+1)) go to 600 t = s(l) s(l) = s(l+1) s(l+1) = t if (wantv .and. l .lt. p) * call dswap(p,v(1,l),1,v(1,l+1),1) if (wantu .and. l .lt. n) * call dswap(n,u(1,l),1,u(1,l+1),1) l = l + 1 go to 590 600 continue iter = 0 m = m - 1 610 continue go to 360 620 continue return end subroutine dswap ( n, dx, incx, dy, incy ) c*********************************************************************72 c cc DSWAP interchanges two vectors. c c Discussion: c c This routine uses double precision real arithmetic. c c Modified: c c 07 July 2007 c c Author: c c Jack Dongarra c c Reference: c c Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, c LINPACK User's Guide, c SIAM, 1979, c ISBN13: 978-0-898711-72-1, c LC: QA214.L56. c c Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh, c Basic Linear Algebra Subprograms for FORTRAN usage, c ACM Transactions on Mathematical Software, c Volume 5, Number 3, pages 308-323, 1979. c c Parameters: c c Input, integer N, the number of entries in the vectors. c c Input/output, double precision X(*), one of the vectors to swap. c c Input, integer INCX, the increment between successive entries of X. c c Input/output, double precision Y(*), one of the vectors to swap. c c Input, integer INCY, the increment between successive elements of Y. c implicit none double precision dx(*),dy(*),dtemp integer i,incx,incy,ix,iy,m,n if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do i = 1,n dtemp = dx(ix) dx(ix) = dy(iy) dy(iy) = dtemp ix = ix + incx iy = iy + incy end do return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,3) if( m .eq. 0 ) go to 40 do i = 1,m dtemp = dx(i) dx(i) = dy(i) dy(i) = dtemp end do if( n .lt. 3 ) return 40 continue do i = m+1, n, 3 dtemp = dx(i) dx(i) = dy(i) dy(i) = dtemp dtemp = dx(i + 1) dx(i + 1) = dy(i + 1) dy(i + 1) = dtemp dtemp = dx(i + 2) dx(i + 2) = dy(i + 2) dy(i + 2) = dtemp end do return end subroutine file_column_count ( input_filename, column_num ) c*********************************************************************72 c cc FILE_COLUMN_COUNT counts the number of columns in the first line of a file. c c Discussion: c c The file is assumed to be a simple text file. c c Most lines of the file is presumed to consist of COLUMN_NUM words, c separated by spaces. There may also be some blank lines, and some c comment lines, c which have a "#" in column 1. c c The routine tries to find the first non-comment non-blank line and c counts the number of words in that line. c c If all lines are blanks or comments, it goes back and tries to analyze c a comment line. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 28 April 2008 c c Author: c c John Burkardt c c Parameters: c c Input, character * ( * ) INPUT_FILENAME, the name of the file. c c Output, integer COLUMN_NUM, the number of columns in the file. c implicit none integer column_num logical got_one character * ( * ) input_filename integer input_unit character * ( 255 ) line integer s_len_trim c c Open the file. c call get_unit ( input_unit ) open ( unit = input_unit, file = input_filename, & status = 'old', form = 'formatted', access = 'sequential' ) c c Read one line, but skip blank lines and comment lines. c got_one = .false. 10 continue read ( input_unit, '(a)', err = 20 ) line if ( s_len_trim ( line ) .eq. 0 ) then go to 10 end if if ( line(1:1) .eq. '#' ) then go to 10 end if got_one = .true. go to 20 go to 10 20 continue if ( .not. got_one ) then rewind ( input_unit ) 30 continue read ( input_unit, '(a)', err = 40 ) line if ( s_len_trim ( line ) .eq. 0 ) then go to 30 end if got_one = .true. go to 40 go to 30 40 continue end if close ( unit = input_unit ) if ( .not. got_one ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'FILE_COLUMN_COUNT - Warning.' write ( *, '(a)' ) ' The file does not contain any data.' column_num = -1 return end if call s_word_count ( line, column_num ) return end subroutine file_row_count ( input_filename, row_num ) c*********************************************************************72 c cc FILE_ROW_COUNT counts the number of row records in a file. c c Discussion: c c It does not count lines that are blank, or that begin with a c comment symbol '#'. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 28 April 2008 c c Author: c c John Burkardt c c Parameters: c c Input, character * ( * ) INPUT_FILENAME, the name of the input file. c c Output, integer ROW_NUM, the number of rows found. c implicit none integer bad_num integer comment_num integer ierror character * ( * ) input_filename integer input_status integer input_unit character * ( 255 ) line integer record_num integer row_num integer s_len_trim call get_unit ( input_unit ) open ( unit = input_unit, file = input_filename, & status = 'old' ) comment_num = 0 row_num = 0 record_num = 0 bad_num = 0 10 continue read ( input_unit, '(a)', err = 20, end = 20 ) line record_num = record_num + 1 if ( line(1:1) .eq. '#' ) then comment_num = comment_num + 1 go to 10 end if if ( s_len_trim ( line ) .eq. 0 ) then comment_num = comment_num + 1 go to 10 end if row_num = row_num + 1 go to 10 20 continue close ( unit = input_unit ) return end subroutine get_unit ( iunit ) c*********************************************************************72 c cc GET_UNIT returns a free FORTRAN unit number. c c Discussion: c c A "free" FORTRAN unit number is a value between 1 and 99 which c is not currently associated with an I/O device. A free FORTRAN unit c number is needed in order to open a file with the OPEN command. c c If IUNIT = 0, then no free FORTRAN unit could be found, although c all 99 units were checked (except for units 5, 6 and 9, which c are commonly reserved for console I/O). c c Otherwise, IUNIT is a value between 1 and 99, representing a c free FORTRAN unit. Note that GET_UNIT assumes that units 5 and 6 c are special, and will never return those values. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 02 September 2013 c c Author: c c John Burkardt c c Parameters: c c Output, integer IUNIT, the free unit number. c implicit none integer i integer iunit logical value iunit = 0 do i = 1, 99 if ( i .ne. 5 .and. i .ne. 6 .and. i .ne. 9 ) then inquire ( unit = i, opened = value, err = 10 ) if ( .not. value ) then iunit = i return end if end if 10 continue end do return end subroutine r8col_normalize_li ( m, n, a ) c*********************************************************************72 c cc R8COL_NORMALIZE_LI normalizes an R8COL with the column infinity norm. c c Discussion: c c Each column is scaled so that the entry of maximum norm has the value 1. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 08 February 2012 c c Author: c c John Burkardt c c Parameters: c c Input, integer M, N, the number of rows and columns. c c Input/output, double precision A(M,N), the array to be normalized. c implicit none integer m integer n double precision a(m,n) double precision c integer i integer j do j = 1, n c = a(1,j) do i = 2, m if ( abs ( c ) .lt. abs ( a(i,j) ) ) then c = a(i,j) end if end do if ( c .ne. 0.0D+00 ) then do i = 1, m a(i,j) = a(i,j) / c end do end if end do return end subroutine r8col_reverse ( m, n, a ) c*********************************************************************72 c cc R8COL_REVERSE reverses the order of columns in an R8COL. c c Discussion: c c To reverse the columns is to start with something like c c 11 12 13 14 15 c 21 22 23 24 25 c 31 32 33 34 35 c 41 42 43 44 45 c 51 52 53 54 55 c c and return c c 15 14 13 12 11 c 25 24 23 22 21 c 35 34 33 32 31 c 45 44 43 42 41 c 55 54 53 52 51 c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 06 May 2013 c c Author: c c John Burkardt c c Parameters: c c Input, integer M, N, the number of rows and columns. c c Input/output, double precision A(M,N), the matrix. c implicit none integer m integer n double precision a(m,n) integer i integer j integer jhi double precision t jhi = n / 2 do j = 1, jhi do i = 1, m t = a(i,j) a(i,j) = a(i,n+1-j) a(i,n+1-j) = t end do end do return end subroutine r8mat_copy ( m, n, a1, a2 ) c*********************************************************************72 c cc R8MAT_COPY copies an R8MAT. c c Discussion: c c An R8MAT is an array of R8's. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 26 July 2008 c c Author: c c John Burkardt c c Parameters: c c Input, integer M, N, the order of the matrix. c c Input, double precision A1(M,N), the matrix to be copied. c c Output, double precision A2(M,N), a copy of the matrix. c implicit none integer m integer n double precision a1(m,n) double precision a2(m,n) integer i integer j do j = 1, n do i = 1, m a2(i,j) = a1(i,j) end do end do return end subroutine r8mat_data_read ( input_filename, m, n, table ) c*********************************************************************72 c cc R8MAT_DATA_READ reads data from an R8MAT file. c c Discussion: c c An R8MAT is an array of R8's. c c The file may contain more than N points, but this routine will c return after reading N of them. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 28 April 2008 c c Author: c c John Burkardt c c Parameters: c c Input, character * ( * ) INPUT_FILENAME, the name of the input file. c c Input, integer M, the spatial dimension. c c Input, integer N, the number of points. c c Output, double precision TABLE(M,N), the data. c implicit none integer m integer n integer i integer ierror character * ( * ) input_filename integer input_status integer input_unit integer j character * ( 255 ) line integer s_len_trim double precision table(m,n) double precision x(m) ierror = 0 call get_unit ( input_unit ) open ( unit = input_unit, file = input_filename, & status = 'old' ) j = 0 10 continue if ( j .lt. n ) then read ( input_unit, '(a)' ) line if ( line(1:1) .eq. '#' .or. s_len_trim ( line ) .eq. 0 ) then go to 10 end if call s_to_r8vec ( line, m, x, ierror ) if ( ierror .ne. 0 ) then go to 10 end if j = j + 1 do i = 1, m table(i,j) = x(i) end do go to 10 end if close ( unit = input_unit ) return end subroutine r8mat_header_read ( input_filename, m, n ) c*********************************************************************72 c cc R8MAT_HEADER_READ reads the header from an R8MAT file. c c Discussion: c c An R8MAT is an array of R8's. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 28 April 2008 c c Author: c c John Burkardt c c Parameters: c c Input, character * ( * ) INPUT_FILENAME, the name of the input file. c c Output, integer M, spatial dimension. c c Output, integer N, the number of points. c implicit none character * ( * ) input_filename integer m integer n call file_column_count ( input_filename, m ) if ( m .le. 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_HEADER_READ - Fatal error!' write ( *, '(a)' ) ' There was an I/O problem while trying' write ( *, '(a)' ) ' to count the number of data columns in' write ( *, '(a,a,a)' ) & ' the file "', trim ( input_filename ), '".' stop end if call file_row_count ( input_filename, n ) if ( n .le. 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_HEADER_READ - Fatal error!' write ( *, '(a)' ) ' There was an I/O problem while trying' write ( *, '(a)' ) ' to count the number of data rows in' write ( *, '(a,a,a)' ) & ' the file "', trim ( input_filename ), '".' stop end if return end subroutine r8mat_mm ( n1, n2, n3, a, b, c ) c*********************************************************************72 c cc R8MAT_MM multiplies two R8MAT's. c c Discussion: c c An R8MAT is an array of R8 values. c c In FORTRAN90, this operation is more efficiently done by the c command: c c C(1:N1,1:N3) = MATMUL ( A(1:N1,1;N2), B(1:N2,1:N3) ) c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 03 March 2013 c c Author: c c John Burkardt c c Parameters: c c Input, integer N1, N2, N3, the order of the matrices. c c Input, double precision A(N1,N2), B(N2,N3), the matrices to multiply. c c Output, double precision C(N1,N3), the product matrix C = A * B. c implicit none integer n1 integer n2 integer n3 double precision a(n1,n2) double precision b(n2,n3) double precision c(n1,n3) double precision c1(n1,n3) integer i integer j integer k do i = 1, n1 do j = 1, n3 c1(i,j) = 0.0D+00 do k = 1, n2 c1(i,j) = c1(i,j) + a(i,k) * b(k,j) end do end do end do call r8mat_copy ( n1, n3, c1, c ) return end subroutine r8mat_mmt ( n1, n2, n3, a, b, c ) c*********************************************************************72 c cc R8MAT_MMT multiplies computes C = A * B' for two R8MAT's. c c Discussion: c c An R8MAT is an array of R8 values. c c In FORTRAN90, this operation is more efficiently done by the c command: c c C(1:N1,1:N3) = matmul ( A(1:N1,1;N2), transpose ( B(1:N3,1:N2) ) ) c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 13 November 2012 c c Author: c c John Burkardt c c Parameters: c c Input, integer N1, N2, N3, the order of the matrices. c c Input, double precision A(N1,N2), B(N3,N2), the matrices to multiply. c c Output, double precision C(N1,N3), the product matrix C = A * B'. c implicit none integer n1 integer n2 integer n3 double precision a(n1,n2) double precision b(n3,n2) double precision c(n1,n3) double precision c1(n1,n3) integer i integer j integer k do i = 1, n1 do j = 1, n3 c1(i,j) = 0.0D+00 do k = 1, n2 c1(i,j) = c1(i,j) + a(i,k) * b(j,k) end do end do end do call r8mat_copy ( n1, n3, c1, c ) return end subroutine r8mat_print ( m, n, a, title ) c*********************************************************************72 c cc R8MAT_PRINT prints an R8MAT. c c Discussion: c c An R8MAT is an array of R8's. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 20 May 2004 c c Author: c c John Burkardt c c Parameters: c c Input, integer M, the number of rows in A. c c Input, integer N, the number of columns in A. c c Input, double precision A(M,N), the matrix. c c Input, character ( len = * ) TITLE, a title. c implicit none integer m integer n double precision a(m,n) character ( len = * ) title call r8mat_print_some ( m, n, a, 1, 1, m, n, title ) return end subroutine r8mat_print_some ( m, n, a, ilo, jlo, ihi, jhi, & title ) c*********************************************************************72 c cc R8MAT_PRINT_SOME prints some of an R8MAT. c c Discussion: c c An R8MAT is an array of R8's. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 25 January 2007 c c Author: c c John Burkardt c c Parameters: c c Input, integer M, N, the number of rows and columns. c c Input, double precision A(M,N), an M by N matrix to be printed. c c Input, integer ILO, JLO, the first row and column to print. c c Input, integer IHI, JHI, the last row and column to print. c c Input, character ( len = * ) TITLE, a title. c implicit none integer incx parameter ( incx = 5 ) integer m integer n double precision a(m,n) character * ( 14 ) ctemp(incx) integer i integer i2hi integer i2lo integer ihi integer ilo integer inc integer j integer j2 integer j2hi integer j2lo integer jhi integer jlo character * ( * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) if ( m .le. 0 .or. n .le. 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) ' (None)' return end if do j2lo = max ( jlo, 1 ), min ( jhi, n ), incx j2hi = j2lo + incx - 1 j2hi = min ( j2hi, n ) j2hi = min ( j2hi, jhi ) inc = j2hi + 1 - j2lo write ( *, '(a)' ) ' ' do j = j2lo, j2hi j2 = j + 1 - j2lo write ( ctemp(j2), '(i7,7x)') j end do write ( *, '('' Col '',5a14)' ) ( ctemp(j), j = 1, inc ) write ( *, '(a)' ) ' Row' write ( *, '(a)' ) ' ' i2lo = max ( ilo, 1 ) i2hi = min ( ihi, m ) do i = i2lo, i2hi do j2 = 1, inc j = j2lo - 1 + j2 write ( ctemp(j2), '(g14.6)' ) a(i,j) end do write ( *, '(i5,a,5a14)' ) i, ':', ( ctemp(j), j = 1, inc ) end do end do return end subroutine r8mat_svd_linpack ( m, n, a, u, s, v ) c*********************************************************************72 c cc R8MAT_SVD_LINPACK gets the SVD of a matrix using a call to LINPACK. c c Discussion: c c The singular value decomposition of a real MxN matrix A has the form: c c A = U * S * V' c c where c c U is MxM orthogonal, c S is MxN, and entirely zero except for the diagonal; c V is NxN orthogonal. c c Moreover, the nonzero entries of S are positive, and appear c in order, from largest magnitude to smallest. c c This routine calls the LINPACK routine DSVDC to compute the c factorization. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 19 June 2012 c c Author: c c John Burkardt c c Parameters: c c Input, integer M, N, the number of rows and columns c in the matrix A. c c Input, double precision A(M,N), the matrix whose singular value c decomposition we are investigating. c c Output, double precision U(M,M), S(M,N), V(N,N), the factors c that form the singular value decomposition of A. c implicit none integer m integer n double precision a(m,n) double precision a_copy(m,n) double precision e(max(m+1,n)) integer i integer info integer j integer lda integer ldu integer ldv integer job integer lwork double precision s(m,n) double precision sdiag(max(m+1,n)) double precision u(m,m) double precision v(n,n) double precision work(m) c c Compute the eigenvalues and eigenvectors. c job = 11 lda = m ldu = m ldv = n c c The input matrix is destroyed by the routine. Since we need to keep c it around, we only pass a copy to the routine. c do j = 1, n do i = 1, m a_copy(i,j) = a(i,j) end do end do call dsvdc ( a_copy, lda, m, n, sdiag, e, u, ldu, v, ldv, work, & job, info ) if ( info .ne. 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_SVD_LINPACK - Failure!' write ( *, '(a)' ) ' The SVD could not be calculated.' write ( *, '(a)' ) ' LINPACK routine DSVDC returned a nonzero' write ( *, '(a,i8)' ) ' value of the error flag, INFO = ', info return end if c c Make the MxN matrix S from the diagonal values in SDIAG. c do j = 1, n do i = 1, m s(i,j) = 0.0D+00 end do end do do i = 1, min ( m, n ) s(i,i) = sdiag(i) end do c c Note that we do NOT need to transpose the V that comes out of LINPACK! c return end subroutine r8mat_svd_low_rank ( m, n, r, u, s, v, a ) c*********************************************************************72 c cc R8MAT_SVD_LOW_RANK forms a rank R approximation using the SVD. c c Discussion: c c The singular value decomposition of a real MxN matrix A has the form: c c A = U * S * V' c c where c c U is MxM orthogonal, c S is MxN, and entirely zero except for the diagonal; c V is NxN orthogonal. c c For R <= min ( M, N ), we can form a rank R approximation to A: c c Ar = Ur * Sr * Vr' c c where c c Ur is MxR orthogonal, c Sr is RxR, and entirely zero except for the diagonal; c Vr is NxR orthogonal. c c Strictly speaking, we use RM = min ( R, M ), RN = min ( R, N ): c c Ur is M x RM, c Sr is RM x RN, c Vr is N x RN c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 09 May 2013 c c Author: c c John Burkardt c c Parameters: c c Input, integer M, N, the number of rows and columns in the matrix A. c c Input, integer R, the desired rank of the approximation. c c Input, double precision U(M,M), S(M,N), V(N,N), the factors c that form the singular value decomposition of A. c c Output, double precision A(M,N), the rank R SVD approximation to A. c implicit none integer m integer n double precision a(m,n) integer r integer rm integer rn double precision s(m,n) double precision sv(m,n) double precision u(m,m) double precision v(n,n) rm = min ( r, m ) rn = min ( r, n ) c c SV = Sr * Vr' c call r8mat_mmt ( rm, rn, n, s(1:rm,1:rn), v(1:n,1:rn), sv ) c c A = Ur * SV = Ur * Sr * Vr' c call r8mat_mm ( m, rm, n, u(1:m,1:rm), sv, a ) return end subroutine r8row_reverse ( m, n, a ) c********************************************************************72 c cc R8ROW_REVERSE reverses the order of the rows of an R8MAT. c c Discussion: c c To reverse the rows is to start with something like c c 11 12 13 14 15 c 21 22 23 24 25 c 31 32 33 34 35 c 41 42 43 44 45 c 51 52 53 54 55 c c and return c c 51 52 53 54 55 c 41 42 43 44 45 c 31 32 33 34 35 c 21 22 23 24 25 c 11 12 13 14 15 c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 06 May 2013 c c Author: c c John Burkardt c c Parameters: c c Input, integer M, N, the number of rows and columns. c c Input/output, double precision A(M,N), the matrix. c implicit none integer m integer n double precision a(m,n) integer i integer ihi integer j double precision t ihi = m / 2 do i = 1, ihi do j = 1, n t = a(i,j) a(i,j) = a(m+1-i,j) a(m+1-i,j) = t end do end do return end subroutine r8vec_cum0 ( n, a, a_cum ) c*********************************************************************72 c cc R8VEC_CUM0 computes the cumulutive sums of an R8VEC. c c Discussion: c c An R8VEC is a vector of R8 values. c c Input: c c A = (/ 1.0, 2.0, 3.0, 4.0 /) c c Output: c c A_CUM = (/ 0.0, 1.0, 3.0, 6.0, 10.0 /) c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 07 May 2012 c c Author: c c John Burkardt c c Parameters: c c Input, integer N, the number of entries in the vector. c c Input, double precision A(N), the vector to be summed. c c Output, double precision A_CUM(0:N), the cumulative sums. c implicit none integer n double precision a(n) double precision a_cum(0:n) integer i a_cum(0) = 0.0D+00 do i = 1, n a_cum(i) = a_cum(i-1) + a(i) end do return end subroutine r8vec_print ( n, a, title ) c*********************************************************************72 c cc R8VEC_PRINT prints an R8VEC. c c Discussion: c c An R8VEC is a vector of R8's. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 12 January 2007 c c Author: c c John Burkardt c c Parameters: c c Input, integer N, the number of components of the vector. c c Input, double precision A(N), the vector to be printed. c c Input, character * ( * ) TITLE, a title. c implicit none integer n double precision a(n) integer i character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(2x,i8,a,1x,g16.8)' ) i, ':', a(i) end do return end function r8vec_sum ( n, v1 ) c*********************************************************************72 c cc R8VEC_SUM sums the entries of an R8VEC. c c Discussion: c c An R8VEC is a vector of R8's. c c In FORTRAN90, the system routine SUM should be called c directly. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 22 July 2008 c c Author: c c John Burkardt c c Parameters: c c Input, integer N, the dimension of the vectors. c c Input, double precision V1(N), the vector. c c Output, double precision R8VEC_SUM, the sum of the entries. c implicit none integer n integer i double precision r8vec_sum double precision v1(n) double precision value value = 0.0D+00 do i = 1, n value = value + v1(i) end do r8vec_sum = value return end function s_len_trim ( s ) c*********************************************************************72 c cc S_LEN_TRIM returns the length of a string to the last nonblank. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 05 March 2004 c c Author: c c John Burkardt c c Parameters: c c Input, character*(*) S, a string. c c Output, integer S_LEN_TRIM, the length of the string to the last nonblank. c implicit none integer i character*(*) s integer s_len_trim do i = len ( s ), 1, -1 if ( s(i:i) .ne. ' ' ) then s_len_trim = i return end if end do s_len_trim = 0 return end subroutine s_to_r8 ( s, dval, ierror, length ) c*********************************************************************72 c cc S_TO_R8 reads an R8 from a string. c c Discussion: c c The routine will read as many characters as possible until it reaches c the end of the string, or encounters a character which cannot be c part of the number. c c Legal input is: c c 1 blanks, c 2 '+' or '-' sign, c 2.5 blanks c 3 integer part, c 4 decimal point, c 5 fraction part, c 6 'E' or 'e' or 'D' or 'd', exponent marker, c 7 exponent sign, c 8 exponent integer part, c 9 exponent decimal point, c 10 exponent fraction part, c 11 blanks, c 12 final comma or semicolon, c c with most quantities optional. c c Example: c c S DVAL c c '1' 1.0 c ' 1 ' 1.0 c '1A' 1.0 c '12,34,56' 12.0 c ' 34 7' 34.0 c '-1E2ABCD' -100.0 c '-1X2ABCD' -1.0 c ' 2E-1' 0.2 c '23.45' 23.45 c '-4.2E+2' -420.0 c '17d2' 1700.0 c '-14e-2' -0.14 c 'e2' 100.0 c '-12.73e-9.23' -12.73 * 10.0**(-9.23) c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 28 April 2008 c c Author: c c John Burkardt c c Parameters: c c Input, character * ( * ) S, the string containing the c data to be read. Reading will begin at position 1 and c terminate at the end of the string, or when no more c characters can be read to form a legal real. Blanks, c commas, or other nonnumeric data will, in particular, c cause the conversion to halt. c c Output, double precision DVAL, the value read from the string. c c Output, integer IERROR, error flag. c 0, no errors occurred. c 1, 2, 6 or 7, the input number was garbled. The c value of IERROR is the last type of input successfully c read. For instance, 1 means initial blanks, 2 means c a plus or minus sign, and so on. c c Output, integer LENGTH, the number of characters read c to form the number, including any terminating c characters such as a trailing comma or blanks. c implicit none logical ch_eqi character c double precision dval integer ierror integer ihave integer isgn integer iterm integer jbot integer jsgn integer jtop integer length integer nchar integer ndig double precision rbot double precision rexp double precision rtop character * ( * ) s integer s_len_trim nchar = s_len_trim ( s ) ierror = 0 dval = 0.0D+00 length = -1 isgn = 1 rtop = 0 rbot = 1 jsgn = 1 jtop = 0 jbot = 1 ihave = 1 iterm = 0 10 continue length = length + 1 if ( nchar .lt. length + 1 ) then go to 20 end if c = s(length+1:length+1) c c Blank character. c if ( c .eq. ' ' ) then if ( ihave .eq. 2 ) then else if ( ihave .eq. 6 .or. ihave .eq. 7 ) then iterm = 1 else if ( 1 .lt. ihave ) then ihave = 11 end if c c Comma. c else if ( c .eq. ',' .or. c .eq. ';' ) then if ( ihave .ne. 1 ) then iterm = 1 ihave = 12 length = length + 1 end if c c Minus sign. c else if ( c .eq. '-' ) then if ( ihave .eq. 1 ) then ihave = 2 isgn = -1 else if ( ihave .eq. 6 ) then ihave = 7 jsgn = -1 else iterm = 1 end if c c Plus sign. c else if ( c .eq. '+' ) then if ( ihave .eq. 1 ) then ihave = 2 else if ( ihave .eq. 6 ) then ihave = 7 else iterm = 1 end if c c Decimal point. c else if ( c .eq. '.' ) then if ( ihave .lt. 4 ) then ihave = 4 else if ( 6 .le. ihave .and. ihave .le. 8 ) then ihave = 9 else iterm = 1 end if c c Scientific notation exponent marker. c else if ( ch_eqi ( c, 'E' ) .or. ch_eqi ( c, 'D' ) ) then if ( ihave .lt. 6 ) then ihave = 6 else iterm = 1 end if c c Digit. c else if ( ihave .lt. 11 .and. lle ( '0', c ) & .and. lle ( c, '9' ) ) then if ( ihave .le. 2 ) then ihave = 3 else if ( ihave .eq. 4 ) then ihave = 5 else if ( ihave .eq. 6 .or. ihave .eq. 7 ) then ihave = 8 else if ( ihave .eq. 9 ) then ihave = 10 end if call ch_to_digit ( c, ndig ) if ( ihave .eq. 3 ) then rtop = 10.0D+00 * rtop + dble ( ndig ) else if ( ihave .eq. 5 ) then rtop = 10.0D+00 * rtop + dble ( ndig ) rbot = 10.0D+00 * rbot else if ( ihave .eq. 8 ) then jtop = 10 * jtop + ndig else if ( ihave .eq. 10 ) then jtop = 10 * jtop + ndig jbot = 10 * jbot end if c c Anything else is regarded as a terminator. c else iterm = 1 end if c c If we haven't seen a terminator, and we haven't examined the c entire string, go get the next character. c if ( iterm .eq. 1 ) then go to 20 end if go to 10 20 continue c c If we haven't seen a terminator, and we have examined the c entire string, then we're done, and LENGTH is equal to NCHAR. c if ( iterm .ne. 1 .and. length+1 .eq. nchar ) then length = nchar end if c c Number seems to have terminated. Have we got a legal number? c Not if we terminated in states 1, 2, 6 or 7. c if ( ihave .eq. 1 .or. ihave .eq. 2 .or. & ihave .eq. 6 .or. ihave .eq. 7 ) then ierror = ihave write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'S_TO_R8 - Serious error!' write ( *, '(a)' ) ' Illegal or nonnumeric input:' write ( *, '(a,a)' ) ' ', s return end if c c Number seems OK. Form it. c if ( jtop .eq. 0 ) then rexp = 1.0D+00 else if ( jbot .eq. 1 ) then rexp = 10.0D+00 ** ( jsgn * jtop ) else rexp = 10.0D+00 ** ( dble ( jsgn * jtop ) / dble ( jbot ) ) end if end if dval = dble ( isgn ) * rexp * rtop / rbot return end subroutine s_to_r8vec ( s, n, rvec, ierror ) c*********************************************************************72 c cc S_TO_R8VEC reads an R8VEC from a string. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 28 April 2008 c c Author: c c John Burkardt c c Parameters: c c Input, character * ( * ) S, the string to be read. c c Input, integer N, the number of values expected. c c Output, double precision RVEC(N), the values read from the string. c c Output, integer IERROR, error flag. c 0, no errors occurred. c -K, could not read data for entries -K through N. c implicit none integer n integer i integer ierror integer ilo integer lchar double precision rvec(n) character * ( * ) s i = 0 ierror = 0 ilo = 1 10 continue if ( i .lt. n ) then i = i + 1 call s_to_r8 ( s(ilo:), rvec(i), ierror, lchar ) if ( ierror .ne. 0 ) then ierror = -i go to 20 end if ilo = ilo + lchar go to 10 end if 20 continue return end subroutine s_word_count ( s, nword ) c*********************************************************************72 c cc S_WORD_COUNT counts the number of "words" in a string. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 28 April 2008 c c Author: c c John Burkardt c c Parameters: c c Input, character * ( * ) S, the string to be examined. c c Output, integer NWORD, the number of "words" in the string. c Words are presumed to be separated by one or more blanks. c implicit none logical blank integer i integer lens integer nword character * ( * ) s nword = 0 lens = len ( s ) if ( lens .le. 0 ) then return end if blank = .true. do i = 1, lens if ( s(i:i) .eq. ' ' ) then blank = .true. else if ( blank ) then nword = nword + 1 blank = .false. end if end do return end subroutine timestamp ( ) c*********************************************************************72 c cc TIMESTAMP prints out the current YMDHMS date as a timestamp. c c Licensing: c c This code is distributed under the GNU LGPL license. c c Modified: c c 12 January 2007 c c Author: c c John Burkardt c c Parameters: c c None c implicit none character * ( 8 ) ampm integer d character * ( 8 ) date integer h integer m integer mm character * ( 9 ) month(12) integer n integer s character * ( 10 ) time integer y save month data month / & 'January ', 'February ', 'March ', 'April ', & 'May ', 'June ', 'July ', 'August ', & 'September', 'October ', 'November ', 'December ' / call date_and_time ( date, time ) read ( date, '(i4,i2,i2)' ) y, m, d read ( time, '(i2,i2,i2,1x,i3)' ) h, n, s, mm if ( h .lt. 12 ) then ampm = 'AM' else if ( h .eq. 12 ) then if ( n .eq. 0 .and. s .eq. 0 ) then ampm = 'Noon' else ampm = 'PM' end if else h = h - 12 if ( h .lt. 12 ) then ampm = 'PM' else if ( h .eq. 12 ) then if ( n .eq. 0 .and. s .eq. 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, month(m), y, h, ':', n, ':', s, '.', mm, ampm return end