17 January 2017 08:43:08 PM LAGUERRE_POLYNOMIAL_PRB: C version. Test the LAGUERRE_POLYNOMIAL library. LAGUERRE_POLYNOMIAL_TEST01: L_POLYNOMIAL_VALUES stores values of the Laguerre polynomials. L_POLYNOMIAL evaluates the polynomial. Tabulated Computed N X L(N,X) L(N,X) Error 0 1 1 1 0 1 1 0 0 0 2 1 -0.5 -0.5 0 3 1 -0.6666666666666667 -0.6666666666666666 -1.11022e-16 4 1 -0.625 -0.625 0 5 1 -0.4666666666666667 -0.4666666666666667 0 6 1 -0.2569444444444444 -0.2569444444444445 5.55112e-17 7 1 -0.04047619047619048 -0.04047619047619059 1.11022e-16 8 1 0.1539930555555556 0.1539930555555554 1.94289e-16 9 1 0.3097442680776014 0.3097442680776012 1.66533e-16 10 1 0.4189459325396825 0.4189459325396824 1.11022e-16 11 1 0.4801341790925124 0.4801341790925122 1.66533e-16 12 1 0.4962122235082305 0.4962122235082302 2.77556e-16 5 0.5 -0.4455729166666667 -0.4455729166666667 0 5 3 0.85 0.85 0 5 5 -3.166666666666667 -3.166666666666667 -4.44089e-16 5 10 34.33333333333333 34.33333333333333 0 LAGUERRE_POLYNOMIAL_TEST02 L_POLYNOMIAL_COEFFICIENTS determines Laguerre polynomial coefficients. L(0,x) = 1 L(1,x) = -1 * x 1 L(2,x) = 0.5 * x^2 -2 * x 1 L(3,x) = -0.166667 * x^3 1.5 * x^2 -3 * x 1 L(4,x) = 0.0416667 * x^4 -0.666667 * x^3 3 * x^2 -4 * x 1 L(5,x) = -0.00833333 * x^5 0.208333 * x^4 -1.66667 * x^3 5 * x^2 -5 * x 1 L(6,x) = 0.00138889 * x^6 -0.05 * x^5 0.625 * x^4 -3.33333 * x^3 7.5 * x^2 -6 * x 1 L(7,x) = -0.000198413 * x^7 0.00972222 * x^6 -0.175 * x^5 1.45833 * x^4 -5.83333 * x^3 10.5 * x^2 -7 * x 1 L(8,x) = 2.48016e-05 * x^8 -0.0015873 * x^7 0.0388889 * x^6 -0.466667 * x^5 2.91667 * x^4 -9.33333 * x^3 14 * x^2 -8 * x 1 L(9,x) = -2.75573e-06 * x^9 0.000223214 * x^8 -0.00714286 * x^7 0.116667 * x^6 -1.05 * x^5 5.25 * x^4 -14 * x^3 18 * x^2 -9 * x 1 L(10,x) = 2.75573e-07 * x^10 -2.75573e-05 * x^9 0.00111607 * x^8 -0.0238095 * x^7 0.291667 * x^6 -2.1 * x^5 8.75 * x^4 -20 * x^3 22.5 * x^2 -10 * x 1 LAGUERRE_POLYNOMIAL_TEST03: L_POLYNOMIAL_ZEROS computes the zeros of L(n,x) Check by calling L_POLYNOMIAL there. Computed zeros for L(1,z): 0: 1.000000 Evaluate L(1,z): 0: 0.000000 Computed zeros for L(2,z): 0: 0.585786 1: 3.414214 Evaluate L(2,z): 0: -0.000000 1: -0.000000 Computed zeros for L(3,z): 0: 0.415775 1: 2.294280 2: 6.289945 Evaluate L(3,z): 0: 0.000000 1: -0.000000 2: -0.000000 Computed zeros for L(4,z): 0: 0.322548 1: 1.745761 2: 4.536620 3: 9.395071 Evaluate L(4,z): 0: -0.000000 1: 0.000000 2: 0.000000 3: 0.000000 Computed zeros for L(5,z): 0: 0.263560 1: 1.413403 2: 3.596426 3: 7.085810 4: 12.640801 Evaluate L(5,z): 0: -0.000000 1: 0.000000 2: 0.000000 3: -0.000000 4: -0.000000 LAGUERRE_POLYNOMIAL_TEST04: L_QUADRATURE_RULE computes the quadrature rule associated with L(n,x) X W 0: 0.193044 0.409319 1: 1.026665 0.421831 2: 2.567877 0.147126 3: 4.900353 0.020634 4: 8.182153 0.001074 5: 12.734180 0.000016 6: 19.395728 0.000000 Use the quadrature rule to estimate: Q = Integral ( 0 <= X < +00 ) X^E exp(-X) dx E Q_Estimate Q_Exact 0 1 1 1 1 1 2 2 2 3 6 6 4 24 24 5 120 120 6 720 720 7 5040 5040 8 40320 40320 9 362880 362880 10 3.6288e+06 3.6288e+06 11 3.99168e+07 3.99168e+07 12 4.79002e+08 4.79002e+08 13 6.22702e+09 6.22702e+09 LAGUERRE_POLYNOMIAL_TEST05: LM_POLYNOMIAL_VALUES stores values of the Laguerre polynomials. LM_POLYNOMIAL evaluates the polynomial. Tabulated Computed N M X Lm(N,M,X) Lm(N,M,X) Error 1 0 0 1 1 0 2 0 0 1 1 0 3 0 0 1 1 0 4 0 0 1 1 0 5 0 0 1 1 0 1 1 0.5 1.5 1.5 0 2 1 0.5 1.625 1.625 0 3 1 0.5 1.479166666666667 1.479166666666667 2.22045e-16 4 1 0.5 1.1484375 1.1484375 0 3 0 0.2 0.4586666666666667 0.4586666666666665 2.22045e-16 3 1 0.2 2.878666666666667 2.878666666666666 8.88178e-16 3 2 0.2 8.098666666666666 8.098666666666665 1.77636e-15 3 3 0.2 17.11866666666667 17.11866666666667 3.55271e-15 4 2 0.25 10.45328776041667 10.45328776041666 5.32907e-15 5 2 0.25 13.29019368489583 13.29019368489583 0 6 3 0.25 56.2245364718967 56.22453647189671 -7.10543e-15 7 3 0.25 74.84729341779436 74.84729341779438 -1.42109e-14 8 4 0.25 323.8912982762806 323.8912982762805 1.13687e-13 9 4 0.25 442.6100000097533 442.6100000097532 1.13687e-13 10 5 0.25 1936.87657228825 1936.87657228825 2.27374e-13 LAGUERRE_POLYNOMIAL_TEST06 LM_POLYNOMIAL_COEFFICIENTS determines Laguerre polynomial coefficients. Lm(0,0,x) = 1 Lm(1,0,x) = -1 * x 1 Lm(2,0,x) = 0.5 * x^2 -2 * x 1 Lm(3,0,x) = -0.166667 * x^3 1.5 * x^2 -3 * x 1 Lm(4,0,x) = 0.0416667 * x^4 -0.666667 * x^3 3 * x^2 -4 * x 1 Lm(5,0,x) = -0.00833333 * x^5 0.208333 * x^4 -1.66667 * x^3 5 * x^2 -5 * x 1 Lm(0,1,x) = 1 Lm(1,1,x) = -1 * x 2 Lm(2,1,x) = 0.5 * x^2 -3 * x 3 Lm(3,1,x) = -0.166667 * x^3 2 * x^2 -6 * x 4 Lm(4,1,x) = 0.0416667 * x^4 -0.833333 * x^3 5 * x^2 -10 * x 5 Lm(5,1,x) = -0.00833333 * x^5 0.25 * x^4 -2.5 * x^3 10 * x^2 -15 * x 6 Lm(0,2,x) = 1 Lm(1,2,x) = -1 * x 3 Lm(2,2,x) = 0.5 * x^2 -4 * x 6 Lm(3,2,x) = -0.166667 * x^3 2.5 * x^2 -10 * x 10 Lm(4,2,x) = 0.0416667 * x^4 -1 * x^3 7.5 * x^2 -20 * x 15 Lm(5,2,x) = -0.00833333 * x^5 0.291667 * x^4 -3.5 * x^3 17.5 * x^2 -35 * x 21 Lm(0,3,x) = 1 Lm(1,3,x) = -1 * x 4 Lm(2,3,x) = 0.5 * x^2 -5 * x 10 Lm(3,3,x) = -0.166667 * x^3 3 * x^2 -15 * x 20 Lm(4,3,x) = 0.0416667 * x^4 -1.16667 * x^3 10.5 * x^2 -35 * x 35 Lm(5,3,x) = -0.00833333 * x^5 0.333333 * x^4 -4.66667 * x^3 28 * x^2 -70 * x 56 Lm(0,4,x) = 1 Lm(1,4,x) = -1 * x 5 Lm(2,4,x) = 0.5 * x^2 -6 * x 15 Lm(3,4,x) = -0.166667 * x^3 3.5 * x^2 -21 * x 35 Lm(4,4,x) = 0.0416667 * x^4 -1.33333 * x^3 14 * x^2 -56 * x 70 Lm(5,4,x) = -0.00833333 * x^5 0.375 * x^4 -6 * x^3 42 * x^2 -126 * x 126 LAGUERREE_POLYNOMIAL_TEST07 Compute an exponential product table for L(n,x): Tij = integral ( 0 <= x < +oo ) exp(b*x) Ln(i,x) Ln(j,x) exp(-x) dx Maximum degree P = 5 Exponential argument coefficient B = 0 Exponential product table: Col: 0 1 2 3 4 Row 0: 1.000000 0.000000 0.000000 0.000000 0.000000 1: 0.000000 1.000000 0.000000 0.000000 0.000000 2: 0.000000 0.000000 1.000000 0.000000 0.000000 3: 0.000000 0.000000 0.000000 1.000000 0.000000 4: 0.000000 0.000000 0.000000 0.000000 1.000000 5: 0.000000 0.000000 -0.000000 0.000000 0.000000 Col: 5 Row 0: 0.000000 1: 0.000000 2: -0.000000 3: 0.000000 4: 0.000000 5: 1.000000 LAGUERREE_POLYNOMIAL_TEST07 Compute an exponential product table for L(n,x): Tij = integral ( 0 <= x < +oo ) exp(b*x) Ln(i,x) Ln(j,x) exp(-x) dx Maximum degree P = 5 Exponential argument coefficient B = 1 Exponential product table: Col: 0 1 2 3 4 Row 0: 31.468942 -457.615068 4062.416650 -24470.343336 104929.038563 1: -457.615068 9071.532378 -90575.926743 578725.464216 -2565446.262182 2: 4062.416650 -90575.926743 965704.447121 -6411972.171243 29100687.935897 3: -24470.343336 578725.464216 -6411972.171243 43655452.833002 -201415061.218088 4: 104929.038563 -2565446.262182 29100687.935897 -201415061.218088 939857714.042539 5: -325626.516067 8119745.096081 -93465087.150013 653856197.241933 -3074411195.696336 Col: 5 Row 0: -325626.516067 1: 8119745.096081 2: -93465087.150013 3: 653856197.241933 4: -3074411195.696335 5: 10109937564.943697 LAGUERRE_POLYNOMIAL_TEST08 Compute a power product table for L(n,x). Tij = integral ( 0 <= x < +oo ) x^e L(i,x) L(j,x) exp(-x) dx Maximum degree P = 5 Exponent of X, E = 0 Power product table: Col: 0 1 2 3 4 Row 0: 1.000000 -0.000000 -0.000000 0.000000 0.000000 1: -0.000000 1.000000 -0.000000 -0.000000 -0.000000 2: -0.000000 -0.000000 1.000000 -0.000000 -0.000000 3: 0.000000 -0.000000 -0.000000 1.000000 0.000000 4: 0.000000 -0.000000 -0.000000 0.000000 1.000000 5: 0.000000 -0.000000 -0.000000 0.000000 0.000000 Col: 5 Row 0: 0.000000 1: -0.000000 2: -0.000000 3: 0.000000 4: 0.000000 5: 1.000000 LAGUERRE_POLYNOMIAL_TEST08 Compute a power product table for L(n,x). Tij = integral ( 0 <= x < +oo ) x^e L(i,x) L(j,x) exp(-x) dx Maximum degree P = 5 Exponent of X, E = 1 Power product table: Col: 0 1 2 3 4 Row 0: 1.000000 -1.000000 -0.000000 -0.000000 0.000000 1: -1.000000 3.000000 -2.000000 0.000000 0.000000 2: -0.000000 -2.000000 5.000000 -3.000000 0.000000 3: -0.000000 0.000000 -3.000000 7.000000 -4.000000 4: 0.000000 0.000000 0.000000 -4.000000 9.000000 5: 0.000000 0.000000 -0.000000 0.000000 -5.000000 Col: 5 Row 0: 0.000000 1: 0.000000 2: -0.000000 3: 0.000000 4: -5.000000 5: 11.000000 LAGUERRE_POLYNOMIAL_PRB: Normal end of execution. 17 January 2017 08:43:08 PM