19 October 2014 11:21:52 AM HERMITE_POLYNOMIAL_PRB: 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 Evaluate H(1,z): 0: 0 Computed zeros for H(2,z): 0: -0.707107 1: 0.707107 Evaluate H(2,z): 0: -4.44089e-16 1: -4.44089e-16 Computed zeros for H(3,z): 0: -1.22474 1: -9.86284e-17 2: 1.22474 Evaluate H(3,z): 0: -8.88178e-15 1: 1.18354e-15 2: 8.88178e-15 Computed zeros for H(4,z): 0: -1.65068 1: -0.524648 2: 0.524648 3: 1.65068 Evaluate H(4,z): 0: -1.06581e-13 1: -8.88178e-16 2: 2.66454e-15 3: -4.26326e-14 Computed zeros for H(5,z): 0: -2.02018 1: -0.958572 2: 2.40258e-16 3: 0.958572 4: 2.02018 Evaluate H(5,z): 0: 0 1: -2.13163e-14 2: 2.8831e-14 3: -4.26326e-14 4: 0 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 Evaluate He(1,z): 0: 0 Computed zeros for He(2,z): 0: -1 1: 1 Evaluate He(2,z): 0: 0 1: 0 Computed zeros for He(3,z): 0: -1.73205 1: -1.39482e-16 2: 1.73205 Evaluate He(3,z): 0: -3.10862e-15 1: 4.18445e-16 2: 3.10862e-15 Computed zeros for He(4,z): 0: -2.33441 1: -0.741964 2: 0.741964 3: 2.33441 Evaluate He(4,z): 0: -1.95399e-14 1: -4.44089e-16 2: 4.44089e-16 3: -8.88178e-15 Computed zeros for He(5,z): 0: -2.85697 1: -1.35563 2: 3.39776e-16 3: 1.35563 4: 2.85697 Evaluate He(5,z): 0: 1.42109e-14 1: -3.55271e-15 2: 5.09664e-15 3: -1.15463e-14 4: -1.42109e-14 HERMITE_POLYNOMIAL_TEST06: H_QUADRATURE_RULE computes the quadrature rule associated with H(n,x) X W 0: -2.65196 0.000971781 1: -1.67355 0.0545156 2: -0.816288 0.425607 3: -1.05979e-16 0.810265 4: 0.816288 0.425607 5: 1.67355 0.0545156 6: 2.65196 0.000971781 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 1.7486e-15 0 6 3.32335 3.32335 7 6.43929e-15 0 8 11.6317 11.6317 9 2.93099e-14 0 10 52.3428 52.3428 11 1.42109e-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: -3.75044 0.00137431 1: -2.36676 0.0770967 2: -1.15441 0.6019 3: -1.49876e-16 1.14589 4: 1.15441 0.6019 5: 2.36676 0.0770967 6: 3.75044 0.00137431 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 2.19269e-15 0 4 7.51988 7.51988 5 1.08802e-14 0 6 37.5994 37.5994 7 7.81597e-14 0 8 263.196 263.196 9 6.25278e-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 4.34968e-16 7.60568e-16 -5.61617e-17 -3.98986e-17 1: 4.34968e-16 1 4.62521e-16 1.2954e-15 -3.88578e-16 2: 7.60568e-16 4.27609e-16 1 -5.20417e-17 7.25114e-16 3: -1.01156e-16 1.26722e-15 -2.42861e-17 1 9.71445e-16 4: -1.21431e-17 -4.16334e-16 7.5287e-16 9.64506e-16 1 5: -1.07987e-16 -6.97359e-16 5.34295e-16 1.66533e-16 -1.11022e-16 Col: 5 Row 0: -9.41087e-17 1: -6.67869e-16 2: 5.48173e-16 3: 1.249e-16 4: -1.11022e-16 5: 1 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: 1.28403 0.907943 0.453972 0.185333 0.0655251 1: 0.907943 1.92604 1.60503 0.917352 0.416999 2: 0.453972 1.60503 2.72855 2.42443 1.50583 3: 0.185333 0.917352 2.42443 3.71832 3.41422 4: 0.0655251 0.416999 1.50583 3.41422 4.92527 5: 0.0207208 0.161169 0.739903 2.24593 4.6102 Col: 5 Row 0: 0.0207208 1: 0.161169 2: 0.739903 3: 2.24593 4: 4.6102 5: 6.37677 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 -6.47052e-16 -8.7777e-16 -3.46945e-17 4.78784e-16 1: -6.47052e-16 1 -9.15934e-16 -5.68989e-16 -1.80411e-16 2: -8.22259e-16 -9.08995e-16 1 -1.30451e-15 1.66533e-16 3: -3.46945e-17 -5.96745e-16 -1.30451e-15 1 -2.10942e-15 4: 4.4062e-16 -1.52656e-16 1.66533e-16 -2.05391e-15 1 5: -1.73472e-17 7.49401e-16 -5.27356e-16 2.60902e-15 -1.27676e-15 Col: 5 Row 0: -9.02056e-17 1: 7.21645e-16 2: -5.55112e-16 3: 2.55351e-15 4: -1.22125e-15 5: 1 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: 2.22045e-16 0.707107 3.71231e-16 2.87964e-16 6.93889e-17 1: 0.707107 7.80626e-16 1 8.74301e-16 1.66533e-16 2: 3.4521e-16 1 1.42941e-15 1.22474 9.99201e-16 3: 2.60209e-16 7.63278e-16 1.22474 2.16493e-15 1.41421 4: 1.04083e-16 1.94289e-16 9.99201e-16 1.41421 1.66533e-15 5: -2.15106e-16 -2.77556e-17 -8.32667e-16 4.44089e-16 1.58114 Col: 5 Row 0: -1.8735e-16 1: -2.77556e-17 2: -8.32667e-16 3: 4.44089e-16 4: 1.58114 5: 2.33147e-15 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 3.93335e-16 8.43889e-16 -3.15503e-17 -7.069e-17 1: 4.48846e-16 1 5.18249e-16 1.47712e-15 -3.5822e-16 2: 9.13278e-16 5.73977e-16 1 1.49186e-16 9.29812e-16 3: -6.26669e-17 1.47755e-15 2.04697e-16 1 9.50628e-16 4: -6.72205e-17 -3.29597e-16 9.33281e-16 8.95117e-16 1 5: -1.74773e-16 -7.94503e-16 2.91434e-16 1.38778e-17 -1.38778e-16 Col: 5 Row 0: -1.74773e-16 1: -8.22259e-16 2: 2.498e-16 3: 0 4: -1.66533e-16 5: 1 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: 1.64872 1.64872 1.16582 0.673087 0.336543 1: 1.64872 3.29744 3.49747 2.69235 1.6827 2: 1.16582 3.49747 5.77052 6.18726 4.99725 3: 0.673087 2.69235 6.18726 9.34255 10.0284 4: 0.336543 1.6827 4.99725 10.0284 14.3501 5: 0.150499 0.902934 3.29819 8.34976 15.3556 Col: 5 Row 0: 0.150499 1: 0.902934 2: 3.29819 3: 8.34976 4: 15.3556 5: 21.0802 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 -6.47052e-16 -7.59809e-16 -2.77556e-17 4.3715e-16 1: -6.19296e-16 1 -7.70217e-16 -4.57967e-16 -1.38778e-17 2: -7.32053e-16 -8.8124e-16 1 -9.15934e-16 5.55112e-16 3: -2.77556e-17 -4.71845e-16 -9.15934e-16 1 -1.55431e-15 4: 4.4062e-16 2.77556e-17 5.27356e-16 -1.55431e-15 1 5: 6.93889e-18 9.4369e-16 -2.77556e-16 3.08087e-15 -5.55112e-16 Col: 5 Row 0: 5.89806e-17 1: 9.71445e-16 2: -2.77556e-16 3: 3.05311e-15 4: -5.82867e-16 5: 1 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: 3.17454e-16 1 4.57967e-16 4.23273e-16 -1.17961e-16 1: 1 8.91648e-16 1.41421 5.82867e-16 5.27356e-16 2: 4.26742e-16 1.41421 1.33227e-15 1.73205 9.4369e-16 3: 4.78784e-16 6.10623e-16 1.73205 1.88738e-15 2 4: -9.02056e-17 6.10623e-16 9.4369e-16 2 2.44249e-15 5: -2.15106e-16 8.32667e-17 -1.11022e-16 1.11022e-15 2.23607 Col: 5 Row 0: -2.22045e-16 1: 2.77556e-17 2: -5.55112e-17 3: 1.11022e-15 4: 2.23607 5: 2.88658e-15 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 4.90466e-16 8.26487e-16 -3.85976e-17 3.31766e-17 1: 5.32113e-16 1 6.22115e-16 1.46281e-15 -4.56232e-16 2: 8.29957e-16 6.29054e-16 1 9.02056e-17 1.00961e-15 3: -1.09504e-17 1.47625e-15 6.245e-17 1 1.04777e-15 4: 2.27682e-17 -4.7011e-16 1.03736e-15 1.04777e-15 1 5: -1.52656e-16 -6.31439e-16 4.19803e-16 2.35922e-16 -5.55112e-17 Col: 5 Row 0: -1.52656e-16 1: -6.31439e-16 2: 4.33681e-16 3: 2.08167e-16 4: -8.32667e-17 5: 1 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: 1.28403 0.907943 0.453972 0.185333 0.0655251 1: 0.907943 1.92604 1.60503 0.917352 0.416999 2: 0.453972 1.60503 2.72855 2.42443 1.50583 3: 0.185333 0.917352 2.42443 3.71832 3.41422 4: 0.0655251 0.416999 1.50583 3.41422 4.92527 5: 0.0207208 0.161169 0.739903 2.24593 4.6102 Col: 5 Row 0: 0.0207208 1: 0.161169 2: 0.739903 3: 2.24593 4: 4.6102 5: 6.37677 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 -5.06539e-16 -9.02056e-16 -2.77556e-17 4.57967e-16 1: -5.34295e-16 1 -9.50628e-16 -6.245e-16 2.77556e-17 2: -8.46545e-16 -9.50628e-16 1 -1.08247e-15 3.88578e-16 3: -2.08167e-17 -6.52256e-16 -1.08247e-15 1 -1.72085e-15 4: 4.47559e-16 0 3.88578e-16 -1.60982e-15 1 5: -1.38778e-17 1.02696e-15 -3.60822e-16 2.9976e-15 -8.32667e-16 Col: 5 Row 0: 5.20417e-17 1: 1.02696e-15 2: -3.60822e-16 3: 2.9976e-15 4: -8.32667e-16 5: 1 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: 1.11022e-16 0.707107 3.13985e-16 2.32453e-16 -2.08167e-17 1: 0.707107 6.10623e-16 1 7.77156e-16 1.66533e-16 2: 3.13985e-16 1 1.27676e-15 1.22474 6.10623e-16 3: 2.08167e-16 6.52256e-16 1.22474 1.4988e-15 1.41421 4: -3.46945e-17 2.22045e-16 6.10623e-16 1.41421 1.77636e-15 5: -6.245e-17 -2.22045e-16 0 5.55112e-16 1.58114 Col: 5 Row 0: -9.02056e-17 1: -2.22045e-16 2: -5.55112e-17 3: 5.55112e-16 4: 1.58114 5: 2.66454e-15 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_PRB: Normal end of execution. 19 October 2014 11:21:52 AM