28 February 2022 02:19:14 PM HERMITE_POLYNOMIAL_TEST: C version. Test the HERMITE_POLYNOMIAL library. HERMITE_POLYNOMIAL_TEST01: H_POLYNOMIAL_VALUES stores values of the physicist's Hermite polynomials. H_POLYNOMIAL_VALUE evaluates the polynomial. Tabulated Computed N X H(N,X) H(N,X) Error 0 5 1 1 0 1 5 10 10 0 2 5 98 98 0 3 5 940 940 0 4 5 8812 8812 0 5 5 80600 80600 0 6 5 717880 717880 0 7 5 6211600 6211600 0 8 5 52065680 52065680 0 9 5 421271200 421271200 0 10 5 3275529760 3275529760 0 11 5 24329873600 24329873600 0 12 5 171237081280 171237081280 0 5 0 0 0 0 5 0.5 41 41 0 5 1 -8 -8 0 5 3 3816 3816 0 5 10 3041200 3041200 0 HERMITE_POLYNOMIAL_TEST02: HE_POLYNOMIAL_VALUES stores values of the probabilist's Hermite polynomials. HE_POLYNOMIAL_VALUE evaluates the polynomial. Tabulated Computed N X He(N,X) He(N,X) Error 0 5 1 1 0 1 5 5 5 0 2 5 24 24 0 3 5 110 110 0 4 5 478 478 0 5 5 1950 1950 0 6 5 7360 7360 0 7 5 25100 25100 0 8 5 73980 73980 0 9 5 169100 169100 0 10 5 179680 179680 0 11 5 -792600 -792600 0 12 5 -5939480 -5939480 0 5 0 0 0 0 5 0.5 6.28125 6.28125 0 5 1 6 6 0 5 3 18 18 0 5 10 90150 90150 0 HERMITE_POLYNOMIAL_TEST03: HF_FUNCTION_VALUES stores values of the Hermite function Hf(n,x). HF_FUNCTION_VALUE evaluates the function. Tabulated Computed N X Hf(N,X) Hf(N,X) Error 0 0 0.7511255444649425 0.7511255444649425 0 1 0 0 0 0 2 0 -0.5311259660135985 -0.5311259660135984 -1.11022e-16 3 0 0 -0 0 4 0 0.4599685791773266 0.4599685791773266 0 5 0 0 0 0 0 1 0.4555806720113325 0.4555806720113325 0 1 1 0.6442883651134752 0.6442883651134752 0 2 1 0.3221441825567376 0.3221441825567377 -5.55112e-17 3 1 -0.2630296236233334 -0.2630296236233334 5.55112e-17 4 1 -0.464975076292511 -0.464975076292511 0 5 1 -0.05881521185179581 -0.05881521185179584 3.46945e-17 6 1 0.3905052515434106 0.3905052515434106 0 7 1 0.2631861423064045 0.2631861423064046 -5.55112e-17 8 1 -0.2336911435996523 -0.2336911435996523 0 9 1 -0.358297336147284 -0.3582973361472841 1.11022e-16 10 1 0.06146344487883041 0.06146344487883037 4.16334e-17 11 1 0.3678312067984882 0.3678312067984882 -5.55112e-17 12 1 0.09131969309166278 0.09131969309166282 -4.16334e-17 5 0.5 0.4385750950032321 0.4385750950032322 -5.55112e-17 5 2 -0.02624689527931006 -0.02624689527930978 -2.84495e-16 5 3 0.5138426125477819 0.5138426125477823 -4.44089e-16 5 4 0.09355563118061758 0.09355563118061762 -4.16334e-17 HERMITE_POLYNOMIAL_TEST04: H_POLYNOMIAL_ZEROS computes the zeros of H(n,x) Check by calling H_POLYNOMIAL there. Computed zeros for H(1,z): 0: 0.000000 Evaluate H(1,z): 0: 0.000000 Computed zeros for H(2,z): 0: -0.707107 1: 0.707107 Evaluate H(2,z): 0: -0.000000 1: -0.000000 Computed zeros for H(3,z): 0: -1.224745 1: -0.000000 2: 1.224745 Evaluate H(3,z): 0: -0.000000 1: 0.000000 2: 0.000000 Computed zeros for H(4,z): 0: -1.650680 1: -0.524648 2: 0.524648 3: 1.650680 Evaluate H(4,z): 0: -0.000000 1: -0.000000 2: 0.000000 3: -0.000000 Computed zeros for H(5,z): 0: -2.020183 1: -0.958572 2: 0.000000 3: 0.958572 4: 2.020183 Evaluate H(5,z): 0: 0.000000 1: -0.000000 2: 0.000000 3: -0.000000 4: 0.000000 HERMITE_POLYNOMIAL_TEST05: HE_POLYNOMIAL_ZEROS computes the zeros of He(n,x) Check by calling HE_POLYNOMIAL there. Computed zeros for He(1,z): 0: 0.000000 Evaluate He(1,z): 0: 0.000000 Computed zeros for He(2,z): 0: -1.000000 1: 1.000000 Evaluate He(2,z): 0: 0.000000 1: 0.000000 Computed zeros for He(3,z): 0: -1.732051 1: -0.000000 2: 1.732051 Evaluate He(3,z): 0: -0.000000 1: 0.000000 2: 0.000000 Computed zeros for He(4,z): 0: -2.334414 1: -0.741964 2: 0.741964 3: 2.334414 Evaluate He(4,z): 0: -0.000000 1: -0.000000 2: 0.000000 3: -0.000000 Computed zeros for He(5,z): 0: -2.856970 1: -1.355626 2: 0.000000 3: 1.355626 4: 2.856970 Evaluate He(5,z): 0: 0.000000 1: -0.000000 2: 0.000000 3: -0.000000 4: -0.000000 HERMITE_POLYNOMIAL_TEST06: H_QUADRATURE_RULE computes the quadrature rule associated with H(n,x) X W 0: -2.651961 0.000972 1: -1.673552 0.054516 2: -0.816288 0.425607 3: -0.000000 0.810265 4: 0.816288 0.425607 5: 1.673552 0.054516 6: 2.651961 0.000972 Use the quadrature rule to estimate: Q = Integral ( -oo < X < +00 ) X^E exp(-X^2) dx E Q_Estimate Q_Exact 0 1.77245 1.77245 1 3.47378e-16 0 2 0.886227 0.886227 3 5.44703e-16 0 4 1.32934 1.32934 5 1.63758e-15 0 6 3.32335 3.32335 7 5.77316e-15 0 8 11.6317 11.6317 9 2.66454e-14 0 10 52.3428 52.3428 11 1.35003e-13 0 12 287.885 287.885 13 7.38964e-13 0 HERMITE_POLYNOMIAL_TEST07: HE_QUADRATURE_RULE computes the quadrature rule associated with He(n,x) X W 0: -3.750440 0.001374 1: -2.366759 0.077097 2: -1.154405 0.601900 3: -0.000000 1.145887 4: 1.154405 0.601900 5: 2.366759 0.077097 6: 3.750440 0.001374 Use the quadrature rule to estimate: Q = Integral ( -oo < X < +00 ) X^E exp(-X^2) dx E Q_Estimate Q_Exact 0 2.50663 2.50663 1 6.95624e-16 0 2 2.50663 2.50663 3 2.41474e-15 0 4 7.51988 7.51988 5 1.17684e-14 0 6 37.5994 37.5994 7 7.10543e-14 0 8 263.196 263.196 9 5.96856e-13 0 10 2368.76 2368.76 11 7.27596e-12 0 12 26056.4 26056.4 13 8.73115e-11 0 HERMITE_POLYNOMIAL_TEST08 Compute a normalized physicist''s Hermite exponential product table. Tij = integral ( -oo < X < +oo ) exp(B*X) Hn(I,X) Hn(J,X) exp(-X*X) dx where Hn(I,X) = normalized physicist''s Hermite polynomial of degree I. 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 HERMITE_POLYNOMIAL_TEST08 Compute a normalized physicist''s Hermite exponential product table. Tij = integral ( -oo < X < +oo ) exp(B*X) Hn(I,X) Hn(J,X) exp(-X*X) dx where Hn(I,X) = normalized physicist''s Hermite polynomial of degree I. Maximum degree P = 5 Exponential argument coefficient B = 1 Exponential product table: Col: 0 1 2 3 4 Row 0: 1.284025 0.907943 0.453972 0.185333 0.065525 1: 0.907943 1.926038 1.605032 0.917352 0.416999 2: 0.453972 1.605032 2.728554 2.424430 1.505829 3: 0.185333 0.917352 2.424430 3.718321 3.414223 4: 0.065525 0.416999 1.505829 3.414223 4.925266 5: 0.020721 0.161169 0.739903 2.245935 4.610201 Col: 5 Row 0: 0.020721 1: 0.161169 2: 0.739903 3: 2.245935 4: 4.610201 5: 6.376773 HERMITE_POLYNOMIAL_TEST09 Compute a normalized physicist''s Hermite power product table. Tij = integral ( -oo < X < +oo ) X^E Hn(I,X) Hn(J,X) exp(-X*X) dx where Hn(I,X) = normalized physicist''s Hermite polynomial of degree I. 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 HERMITE_POLYNOMIAL_TEST09 Compute a normalized physicist''s Hermite power product table. Tij = integral ( -oo < X < +oo ) X^E Hn(I,X) Hn(J,X) exp(-X*X) dx where Hn(I,X) = normalized physicist''s Hermite polynomial of degree I. Maximum degree P = 5 Exponent of X, E = 1 Power product table: Col: 0 1 2 3 4 Row 0: 0.000000 0.707107 0.000000 0.000000 0.000000 1: 0.707107 0.000000 1.000000 0.000000 0.000000 2: 0.000000 1.000000 0.000000 1.224745 0.000000 3: 0.000000 0.000000 1.224745 0.000000 1.414214 4: 0.000000 0.000000 0.000000 1.414214 0.000000 5: -0.000000 -0.000000 -0.000000 0.000000 1.581139 Col: 5 Row 0: -0.000000 1: -0.000000 2: -0.000000 3: 0.000000 4: 1.581139 5: 0.000000 HERMITE_POLYNOMIAL_TEST10 Compute a normalized probabilist''s Hermite exponential product table. Tij = integral ( -oo < X < +oo ) exp(B*X) Hen(I,X) Hen(J,X) exp(-0.5*X*X) dx where Hen(I,X) = normalized probabilist''s Hermite polynomial of degree I. 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 HERMITE_POLYNOMIAL_TEST10 Compute a normalized probabilist''s Hermite exponential product table. Tij = integral ( -oo < X < +oo ) exp(B*X) Hen(I,X) Hen(J,X) exp(-0.5*X*X) dx where Hen(I,X) = normalized probabilist''s Hermite polynomial of degree I. Maximum degree P = 5 Exponential argument coefficient B = 1 Exponential product table: Col: 0 1 2 3 4 Row 0: 1.648721 1.648721 1.165822 0.673087 0.336543 1: 1.648721 3.297443 3.497466 2.692348 1.682701 2: 1.165822 3.497466 5.770521 6.187257 4.997248 3: 0.673087 2.692348 6.187257 9.342554 10.028450 4: 0.336543 1.682701 4.997248 10.028450 14.350125 5: 0.150499 0.902934 3.298186 8.349759 15.355564 Col: 5 Row 0: 0.150499 1: 0.902934 2: 3.298186 3: 8.349759 4: 15.355564 5: 21.080228 HERMITE_POLYNOMIAL_TEST11 Compute a normalized probabilist''s Hermite power product table. Tij = integral ( -oo < X < +oo ) X^E Hen(I,X) Hen(J,X) exp(-X*X) dx where Hen(I,X) = normalized probabilist''s Hermite polynomial of degree I. 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 HERMITE_POLYNOMIAL_TEST11 Compute a normalized probabilist''s Hermite power product table. Tij = integral ( -oo < X < +oo ) X^E Hen(I,X) Hen(J,X) exp(-X*X) dx where Hen(I,X) = normalized probabilist''s Hermite polynomial of degree I. Maximum degree P = 5 Exponent of X, E = 1 Power product table: Col: 0 1 2 3 4 Row 0: 0.000000 1.000000 0.000000 0.000000 -0.000000 1: 1.000000 0.000000 1.414214 0.000000 0.000000 2: 0.000000 1.414214 0.000000 1.732051 0.000000 3: 0.000000 0.000000 1.732051 0.000000 2.000000 4: -0.000000 0.000000 0.000000 2.000000 0.000000 5: -0.000000 0.000000 -0.000000 0.000000 2.236068 Col: 5 Row 0: -0.000000 1: 0.000000 2: -0.000000 3: 0.000000 4: 2.236068 5: 0.000000 HERMITE_POLYNOMIAL_TEST12 Compute a Hermite function exponential product table. Tij = integral ( -oo < X < +oo ) exp(B*X) Hf(I,X) Hf(J,X) dx where Hf(I,X) = Hermite function of "degree" I. 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 HERMITE_POLYNOMIAL_TEST12 Compute a Hermite function exponential product table. Tij = integral ( -oo < X < +oo ) exp(B*X) Hf(I,X) Hf(J,X) dx where Hf(I,X) = Hermite function of "degree" I. Maximum degree P = 5 Exponential argument coefficient B = 1 Exponential product table: Col: 0 1 2 3 4 Row 0: 1.284025 0.907943 0.453972 0.185333 0.065525 1: 0.907943 1.926038 1.605032 0.917352 0.416999 2: 0.453972 1.605032 2.728554 2.424430 1.505829 3: 0.185333 0.917352 2.424430 3.718321 3.414223 4: 0.065525 0.416999 1.505829 3.414223 4.925266 5: 0.020721 0.161169 0.739903 2.245935 4.610201 Col: 5 Row 0: 0.020721 1: 0.161169 2: 0.739903 3: 2.245935 4: 4.610201 5: 6.376773 HERMITE_POLYNOMIAL_TEST13 Compute a Hermite function product table. Tij = integral ( -oo < X < +oo ) X^E Hf(I,X) Hf(J,X) exp(-X*X) dx where Hf(I,X) = Hermite function of "degree" I. 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 HERMITE_POLYNOMIAL_TEST13 Compute a Hermite function product table. Tij = integral ( -oo < X < +oo ) X^E Hf(I,X) Hf(J,X) exp(-X*X) dx where Hf(I,X) = Hermite function of "degree" I. Maximum degree P = 5 Exponent of X, E = 1 Power product table: Col: 0 1 2 3 4 Row 0: 0.000000 0.707107 0.000000 0.000000 -0.000000 1: 0.707107 0.000000 1.000000 0.000000 0.000000 2: 0.000000 1.000000 0.000000 1.224745 0.000000 3: 0.000000 0.000000 1.224745 0.000000 1.414214 4: -0.000000 0.000000 0.000000 1.414214 0.000000 5: -0.000000 -0.000000 0.000000 0.000000 1.581139 Col: 5 Row 0: -0.000000 1: -0.000000 2: -0.000000 3: 0.000000 4: 1.581139 5: 0.000000 HERMITE_POLYNOMIAL_TEST14 H_POLYNOMIAL_COEFFICIENTS determines physicist's Hermite polynomial coefficients. H(0,x) = 1 H(1,x) = 2 * x H(2,x) = 4 * x^2 -2 H(3,x) = 8 * x^3 -12 * x H(4,x) = 16 * x^4 -48 * x^2 12 H(5,x) = 32 * x^5 -160 * x^3 120 * x H(6,x) = 64 * x^6 -480 * x^4 720 * x^2 -120 H(7,x) = 128 * x^7 -1344 * x^5 3360 * x^3 -1680 * x H(8,x) = 256 * x^8 -3584 * x^6 13440 * x^4 -13440 * x^2 1680 H(9,x) = 512 * x^9 -9216 * x^7 48384 * x^5 -80640 * x^3 30240 * x H(10,x) = 1024 * x^10 -23040 * x^8 161280 * x^6 -403200 * x^4 302400 * x^2 -30240 HERMITE_POLYNOMIAL_TEST15 HE_POLYNOMIAL_COEFFICIENTS determines probabilist's Hermite polynomial coefficients. He(0) = 1 He(1) = 1 * x He(2) = 1 * x^2 -1 He(3) = 1 * x^3 -3 * x He(4) = 1 * x^4 -6 * x^2 3 He(5) = 1 * x^5 -10 * x^3 15 * x He(6) = 1 * x^6 -15 * x^4 45 * x^2 -15 He(7) = 1 * x^7 -21 * x^5 105 * x^3 -105 * x He(8) = 1 * x^8 -28 * x^6 210 * x^4 -420 * x^2 105 He(9) = 1 * x^9 -36 * x^7 378 * x^5 -1260 * x^3 945 * x He(10) = 1 * x^10 -45 * x^8 630 * x^6 -3150 * x^4 4725 * x^2 -945 HERMITE_POLYNOMIAL_TEST: Normal end of execution. 28 February 2022 02:19:14 PM