15 September 2021 8:27:06.988 AM LAGUERRE_POLYNOMIAL_TEST: FORTRAN90 version. Test LAGUERRE_POLYNOMIAL. 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 1.000000 1.000000000000000 1.000000000000000 0.0 1 1.000000 0.000000000000000 0.000000000000000 0.0 2 1.000000 -0.5000000000000000 -0.5000000000000000 0.0 3 1.000000 -0.6666666666666667 -0.6666666666666666 -.11E-15 4 1.000000 -0.6250000000000000 -0.6250000000000000 0.0 5 1.000000 -0.4666666666666667 -0.4666666666666667 0.0 6 1.000000 -0.2569444444444444 -0.2569444444444445 0.56E-16 7 1.000000 -0.4047619047619048E-01 -0.4047619047619059E-01 0.11E-15 8 1.000000 0.1539930555555556 0.1539930555555554 0.19E-15 9 1.000000 0.3097442680776014 0.3097442680776012 0.17E-15 10 1.000000 0.4189459325396825 0.4189459325396824 0.11E-15 11 1.000000 0.4801341790925124 0.4801341790925122 0.17E-15 12 1.000000 0.4962122235082305 0.4962122235082302 0.28E-15 5 0.500000 -0.4455729166666667 -0.4455729166666667 0.0 5 3.000000 0.8500000000000000 0.8500000000000000 0.0 5 5.000000 -3.166666666666667 -3.166666666666667 -.44E-15 5 10.000000 34.33333333333333 34.33333333333333 0.0 LAGUERRE_POLYNOMIAL_TEST02 L_POLYNOMIAL_COEFFICIENTS determines polynomial coefficients of L(n,x). L( 0) = 1.00000 L( 1) = -1.00000 * x 1.00000 L( 2) = 0.500000 * x^ 2 -2.00000 * x 1.00000 L( 3) = -0.166667 * x^ 3 1.50000 * x^ 2 -3.00000 * x 1.00000 L( 4) = 0.416667E-01 * x^ 4 -0.666667 * x^ 3 3.00000 * x^ 2 -4.00000 * x 1.00000 L( 5) = -0.833333E-02 * x^ 5 0.208333 * x^ 4 -1.66667 * x^ 3 5.00000 * x^ 2 -5.00000 * x 1.00000 L( 6) = 0.138889E-02 * x^ 6 -0.500000E-01 * x^ 5 0.625000 * x^ 4 -3.33333 * x^ 3 7.50000 * x^ 2 -6.00000 * x 1.00000 L( 7) = -0.198413E-03 * x^ 7 0.972222E-02 * x^ 6 -0.175000 * x^ 5 1.45833 * x^ 4 -5.83333 * x^ 3 10.5000 * x^ 2 -7.00000 * x 1.00000 L( 8) = 0.248016E-04 * x^ 8 -0.158730E-02 * x^ 7 0.388889E-01 * x^ 6 -0.466667 * x^ 5 2.91667 * x^ 4 -9.33333 * x^ 3 14.0000 * x^ 2 -8.00000 * x 1.00000 L( 9) = -0.275573E-05 * x^ 9 0.223214E-03 * x^ 8 -0.714286E-02 * x^ 7 0.116667 * x^ 6 -1.05000 * x^ 5 5.25000 * x^ 4 -14.0000 * x^ 3 18.0000 * x^ 2 -9.00000 * x 1.00000 L(10) = 0.275573E-06 * x^10 -0.275573E-04 * x^ 9 0.111607E-02 * x^ 8 -0.238095E-01 * x^ 7 0.291667 * x^ 6 -2.10000 * x^ 5 8.75000 * x^ 4 -20.0000 * x^ 3 22.5000 * x^ 2 -10.0000 * x 1.00000 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): 1: 1.0000000 Evaluate L(1,z): 1: 0.0000000 Computed zeros for L(2,z): 1: 0.58578644 2: 3.4142136 Evaluate L(2,z): 1: -0.16653345E-15 2: -0.16653345E-15 Computed zeros for L(3,z): 1: 0.41577456 2: 2.2942804 3: 6.2899451 Evaluate L(3,z): 1: 0.29605947E-15 2: -0.44408921E-15 3: -0.47369516E-14 Computed zeros for L(4,z): 1: 0.32254769 2: 1.7457611 3: 4.5366203 4: 9.3950709 Evaluate L(4,z): 1: -0.55511151E-16 2: 0.0000000 3: 0.22204460E-15 4: 0.46185278E-13 Computed zeros for L(5,z): 1: 0.26356032 2: 1.4134031 3: 3.5964258 4: 7.0858100 5: 12.640801 Evaluate L(5,z): 1: -0.13322676E-15 2: 0.10658141E-14 3: 0.14210855E-14 4: -0.18474111E-13 5: -0.90949470E-13 LAGUERRE_POLYNOMIAL_TEST04: L_QUADRATURE_RULE computes the quadrature rule associated with L(n,x) X W 1 0.193044 0.409319 2 1.02666 0.421831 3 2.56788 0.147126 4 4.90035 0.206335E-01 5 8.18215 0.107401E-02 6 12.7342 0.158655E-04 7 19.3957 0.317032E-07 Use the quadrature rule to estimate: Q = Integral ( 0 <= X < +00 ) X^E exp(-X) dx E Q_Estimate Q_Exact 0 1.00000 1.00000 1 1.00000 1.00000 2 2.00000 2.00000 3 6.00000 6.00000 4 24.0000 24.0000 5 120.000 120.000 6 720.000 720.000 7 5040.00 5040.00 8 40320.0 40320.0 9 362880. 362880. 10 0.362880E+07 0.362880E+07 11 0.399168E+08 0.399168E+08 12 0.479002E+09 0.479002E+09 13 0.622702E+10 0.622702E+10 LAGUERRE_POLYNOMIAL_TEST05: LM_POLYNOMIAL_VALUES stores values of the Laguerre polynomial Lm(n,m,x) LM_POLYNOMIAL evaluates the polynomial. Tabulated Computed N M X Lm(N,M,X) Lm(N,M,X) Error 1 0 0.000000 1.000000000000000 1.000000000000000 0.0 2 0 0.000000 1.000000000000000 1.000000000000000 0.0 3 0 0.000000 1.000000000000000 1.000000000000000 0.0 4 0 0.000000 1.000000000000000 1.000000000000000 0.0 5 0 0.000000 1.000000000000000 1.000000000000000 0.0 1 1 0.500000 1.500000000000000 1.500000000000000 0.0 2 1 0.500000 1.625000000000000 1.625000000000000 0.0 3 1 0.500000 1.479166666666667 1.479166666666667 0.22E-15 4 1 0.500000 1.148437500000000 1.148437500000000 0.0 3 0 0.200000 0.4586666666666667 0.4586666666666665 0.22E-15 3 1 0.200000 2.878666666666667 2.878666666666666 0.89E-15 3 2 0.200000 8.098666666666666 8.098666666666665 0.18E-14 3 3 0.200000 17.11866666666667 17.11866666666667 0.36E-14 4 2 0.250000 10.45328776041667 10.45328776041666 0.53E-14 5 2 0.250000 13.29019368489583 13.29019368489583 0.0 6 3 0.250000 56.22453647189670 56.22453647189671 -.71E-14 7 3 0.250000 74.84729341779436 74.84729341779438 -.14E-13 8 4 0.250000 323.8912982762806 323.8912982762805 0.11E-12 9 4 0.250000 442.6100000097533 442.6100000097532 0.11E-12 10 5 0.250000 1936.876572288250 1936.876572288250 0.23E-12 LAGUERRE_POLYNOMIAL_TEST06 LM_POLYNOMIAL_COEFFICIENTS determines polynomial coefficients of Lm(n,m,x). Lm( 0, 0) = 1.00000 Lm( 1, 0) = -1.00000 * x 1.00000 Lm( 2, 0) = 0.500000 * x^ 2 -2.00000 * x 1.00000 Lm( 3, 0) = -0.166667 * x^ 3 1.50000 * x^ 2 -3.00000 * x 1.00000 Lm( 4, 0) = 0.416667E-01 * x^ 4 -0.666667 * x^ 3 3.00000 * x^ 2 -4.00000 * x 1.00000 Lm( 5, 0) = -0.833333E-02 * x^ 5 0.208333 * x^ 4 -1.66667 * x^ 3 5.00000 * x^ 2 -5.00000 * x 1.00000 Lm( 0, 1) = 1.00000 Lm( 1, 1) = -1.00000 * x 2.00000 Lm( 2, 1) = 0.500000 * x^ 2 -3.00000 * x 3.00000 Lm( 3, 1) = -0.166667 * x^ 3 2.00000 * x^ 2 -6.00000 * x 4.00000 Lm( 4, 1) = 0.416667E-01 * x^ 4 -0.833333 * x^ 3 5.00000 * x^ 2 -10.0000 * x 5.00000 Lm( 5, 1) = -0.833333E-02 * x^ 5 0.250000 * x^ 4 -2.50000 * x^ 3 10.0000 * x^ 2 -15.0000 * x 6.00000 Lm( 0, 2) = 1.00000 Lm( 1, 2) = -1.00000 * x 3.00000 Lm( 2, 2) = 0.500000 * x^ 2 -4.00000 * x 6.00000 Lm( 3, 2) = -0.166667 * x^ 3 2.50000 * x^ 2 -10.0000 * x 10.0000 Lm( 4, 2) = 0.416667E-01 * x^ 4 -1.00000 * x^ 3 7.50000 * x^ 2 -20.0000 * x 15.0000 Lm( 5, 2) = -0.833333E-02 * x^ 5 0.291667 * x^ 4 -3.50000 * x^ 3 17.5000 * x^ 2 -35.0000 * x 21.0000 Lm( 0, 3) = 1.00000 Lm( 1, 3) = -1.00000 * x 4.00000 Lm( 2, 3) = 0.500000 * x^ 2 -5.00000 * x 10.0000 Lm( 3, 3) = -0.166667 * x^ 3 3.00000 * x^ 2 -15.0000 * x 20.0000 Lm( 4, 3) = 0.416667E-01 * x^ 4 -1.16667 * x^ 3 10.5000 * x^ 2 -35.0000 * x 35.0000 Lm( 5, 3) = -0.833333E-02 * x^ 5 0.333333 * x^ 4 -4.66667 * x^ 3 28.0000 * x^ 2 -70.0000 * x 56.0000 Lm( 0, 4) = 1.00000 Lm( 1, 4) = -1.00000 * x 5.00000 Lm( 2, 4) = 0.500000 * x^ 2 -6.00000 * x 15.0000 Lm( 3, 4) = -0.166667 * x^ 3 3.50000 * x^ 2 -21.0000 * x 35.0000 Lm( 4, 4) = 0.416667E-01 * x^ 4 -1.33333 * x^ 3 14.0000 * x^ 2 -56.0000 * x 70.0000 Lm( 5, 4) = -0.833333E-02 * x^ 5 0.375000 * x^ 4 -6.00000 * x^ 3 42.0000 * x^ 2 -126.000 * x 126.000 LAGUERRE_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 = 0.00000 Exponential product table: Col 1 2 3 4 5 Row 1: 1.00000 0.129083E-15 0.336141E-15 0.430790E-15 0.335157E-15 2: 0.129083E-15 1.00000 0.457784E-15 0.654917E-16 0.213012E-15 3: 0.336141E-15 0.457784E-15 1.00000 0.988118E-15 0.201052E-15 4: 0.430790E-15 0.516137E-16 0.988118E-15 1.00000 0.582217E-16 5: 0.335157E-15 0.185256E-15 0.171575E-15 0.581132E-16 1.00000 6: 0.294847E-15 0.279948E-15 -0.492878E-15 0.579398E-15 0.117094E-14 Col 6 Row 1: 0.294847E-15 2: 0.301198E-15 3: -0.465123E-15 4: 0.634475E-15 5: 0.117094E-14 6: 1.00000 LAGUERRE_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 = 0.500000 Exponential product table: Col 1 2 3 4 5 Row 1: 2.00000 -2.00000 1.99998 -1.99973 1.99749 2: -2.00000 9.99996 -17.9991 25.9883 -33.8934 3: 1.99998 -17.9991 65.9815 -145.762 255.900 4: -1.99973 25.9883 -145.762 487.031 -1160.53 5: 1.99749 -33.8934 255.900 -1160.53 3629.63 6: -1.98292 41.2967 -388.545 2203.41 -8474.02 Col 6 Row 1: -1.98292 2: 41.2967 3: -388.545 4: 2203.41 5: -8474.02 6: 23413.3 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 1 2 3 4 5 Row 1: 1.00000 -0.571092E-15 -0.274032E-16 0.175966E-15 0.235922E-15 2: -0.571092E-15 1.00000 -0.586987E-15 -0.618429E-15 -0.225514E-15 3: -0.274032E-16 -0.586987E-15 1.00000 -0.159595E-15 -0.197065E-14 4: 0.175966E-15 -0.619296E-15 -0.104083E-15 1.00000 0.555112E-15 5: 0.235922E-15 -0.222045E-15 -0.197065E-14 0.582867E-15 1.00000 6: 0.225731E-15 -0.532560E-15 -0.122818E-14 0.277556E-16 0.222045E-15 Col 6 Row 1: 0.225731E-15 2: -0.503070E-15 3: -0.123512E-14 4: 0.277556E-16 5: 0.166533E-15 6: 1.00000 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 1 2 3 4 5 Row 1: 1.00000 -1.00000 -0.333080E-15 -0.246602E-15 0.331766E-16 2: -1.00000 3.00000 -2.00000 0.112237E-14 0.582867E-15 3: -0.333080E-15 -2.00000 5.00000 -3.00000 0.777156E-15 4: -0.246602E-15 0.954098E-15 -3.00000 7.00000 -4.00000 5: 0.331766E-16 0.582867E-15 0.333067E-15 -4.00000 9.00000 6: 0.494830E-15 0.763278E-16 -0.316414E-14 0.621725E-14 -5.00000 Col 6 Row 1: 0.494830E-15 2: 0.298372E-15 3: -0.338618E-14 4: 0.621725E-14 5: -5.00000 6: 11.0000 LAGUERRE_POLYNOMIAL_TEST: Normal end of execution. 15 September 2021 8:27:06.988 AM