Fri Apr 5 10:55:57 2024 gegenbauer_polynomial_test(): python version: 3.10.12 numpy version: 1.26.4 Test gegenbauer_polynomial(). gegenbauer_alpha_check_test(): gegenbauer_alpha_check() checks that ALPHA is legal; ALPHA Check? 0.4467 True 1.4855 True 3.8878 True 3.9865 True -4.6580 False -3.5772 False 2.0696 True -1.8566 False 3.5798 True 3.0740 True gegenbauer_ek_compute_test(): gegenbauer_ek_compute() computes Gauss-Gegenbauer rules Abscissas and weights for a generalized Gauss Gegenbauer rule with ALPHA = 0.5 Integration interval is [-1,+1] 0 1.570796326794897 0 0 0.7853981633974484 -0.4999999999999999 1 0.7853981633974484 0.4999999999999999 0 0.3926990816987244 -0.7071067811865476 1 0.7853981633974487 4.837676228855989e-17 2 0.3926990816987246 0.7071067811865474 0 0.217078713422706 -0.8090169943749473 1 0.5683194499747422 -0.3090169943749473 2 0.5683194499747424 0.3090169943749473 3 0.217078713422706 0.8090169943749475 0 0.1308996938995748 -0.8660254037844385 1 0.3926990816987244 -0.4999999999999998 2 0.5235987755982987 1.884809048867118e-17 3 0.3926990816987244 0.5 4 0.1308996938995748 0.8660254037844383 0 0.0844886908915886 -0.9009688679024193 1 0.2743330560697776 -0.6234898018587333 2 0.4265764164360817 -0.2225209339563144 3 0.4265764164360816 0.2225209339563145 4 0.2743330560697779 0.6234898018587334 5 0.08448869089158867 0.900968867902419 0 0.05750944903191339 -0.9238795325112866 1 0.1963495408493622 -0.7071067811865474 2 0.3351896326668113 -0.3826834323650896 3 0.3926990816987241 2.455208374201258e-17 4 0.3351896326668114 0.3826834323650897 5 0.1963495408493624 0.7071067811865471 6 0.05750944903191314 0.9238795325112868 0 0.04083294770910707 -0.9396926207859084 1 0.144225600795673 -0.7660444431189782 2 0.2617993877991496 -0.4999999999999999 3 0.3385402270935186 -0.1736481776669303 4 0.3385402270935185 0.1736481776669304 5 0.261799387799149 0.5 6 0.1442256007956726 0.7660444431189781 7 0.04083294770910745 0.9396926207859085 0 0.02999954037160817 -0.9510565162951534 1 0.1085393567113531 -0.8090169943749473 2 0.2056199086476265 -0.587785252292473 3 0.2841597249873712 -0.3090169943749474 4 0.3141592653589791 -8.481943400504541e-17 5 0.2841597249873714 0.3090169943749473 6 0.2056199086476269 0.5877852522924731 7 0.1085393567113536 0.8090169943749473 8 0.02999954037160817 0.9510565162951534 0 0.02266894250185888 -0.9594929736144975 1 0.08347854093418892 -0.8412535328311812 2 0.1631221774548166 -0.654860733945285 3 0.2363135602034876 -0.4154150130018863 4 0.2798149423030963 -0.1423148382732852 5 0.2798149423030966 0.1423148382732851 6 0.2363135602034878 0.4154150130018863 7 0.1631221774548166 0.6548607339452849 8 0.08347854093418941 0.8412535328311812 9 0.02266894250185881 0.9594929736144975 gegenbauer_integral_test(): gegenbauer_integral() evaluates Integral ( -1 < x < +1 ) x^n * (1-x*x)^alpha dx N Value 0 1.748038369528081 1 0 2 0.4994395341508806 3 0 4 0.2724215640822985 5 0 6 0.1816143760548655 7 0 8 0.133821119198322 9 0 10 0.1047295715465128 gegenbauer_polynomial_value_test(): gegenbauer_polynomial()_value evaluates the Gegenbauer polynomial. M ALPHA X GPV GEGENBAUER 0 0.50 0.20 1.000000 1.000000 1 0.50 0.20 0.200000 0.200000 2 0.50 0.20 -0.440000 -0.440000 3 0.50 0.20 -0.280000 -0.280000 4 0.50 0.20 0.232000 0.232000 5 0.50 0.20 0.307520 0.307520 6 0.50 0.20 -0.080576 -0.080576 7 0.50 0.20 -0.293517 -0.293517 8 0.50 0.20 -0.039565 -0.039565 9 0.50 0.20 0.245971 0.245957 10 0.50 0.20 0.129072 0.129072 2 0.00 0.40 0.000000 0.000000 2 1.00 0.40 -0.360000 -0.360000 2 2.00 0.40 -0.080000 -0.080000 2 3.00 0.40 0.840000 0.840000 2 4.00 0.40 2.400000 2.400000 2 5.00 0.40 4.600000 4.600000 2 6.00 0.40 7.440000 7.440000 2 7.00 0.40 10.920000 10.920000 2 8.00 0.40 15.040000 15.040000 2 9.00 0.40 19.800000 19.800000 2 10.00 0.40 25.200000 25.200000 5 3.00 -0.50 -9.000000 9.000000 5 3.00 -0.40 -0.161280 -0.161280 5 3.00 -0.30 -6.672960 -6.672960 5 3.00 -0.20 -8.375040 -8.375040 5 3.00 -0.10 -5.526720 -5.526720 5 3.00 0.00 0.000000 0.000000 5 3.00 0.10 5.526720 5.526720 5 3.00 0.20 8.375040 8.375040 5 3.00 0.30 6.672960 6.672960 5 3.00 0.40 0.161280 0.161280 5 3.00 0.50 -9.000000 -9.000000 5 3.00 0.60 -15.425280 -15.425280 5 3.00 0.70 -9.696960 -9.696960 5 3.00 0.80 22.440960 22.440960 5 3.00 0.90 100.889280 100.889280 5 3.00 1.00 252.000000 252.000000 gegenbauer_ss_compute_test(): gegenbauer_ss_compute() computes Gauss-Gegenbauer rules; Abscissas and weights for a generalized Gauss Gegenbauer rule with ALPHA = 0.5 # W X 0 1.570796326794897 0 0 0.7853981633974484 -0.5 1 0.7853981633974484 0.5 0 0.3926990816987239 -0.7071067811865475 1 0.7853981633974484 0 2 0.3926990816987239 0.7071067811865475 0 0.217078713422706 -0.8090169943749475 1 0.5683194499747424 -0.3090169943749475 2 0.5683194499747424 0.3090169943749474 3 0.217078713422706 0.8090169943749475 0 0.130899693899574 -0.8660254037844387 1 0.3926990816987242 -0.5 2 0.5235987755982989 0 3 0.3926990816987242 0.5 4 0.130899693899575 0.8660254037844387 0 0.08448869089158841 -0.9009688679024191 1 0.2743330560697777 -0.6234898018587335 2 0.4265764164360819 -0.2225209339563144 3 0.4265764164360819 0.2225209339563144 4 0.2743330560697777 0.6234898018587335 5 0.08448869089158884 0.900968867902419 0 0.05750944903191331 -0.9238795325112867 1 0.1963495408493619 -0.7071067811865475 2 0.3351896326668111 -0.3826834323650898 3 0.3926990816987242 0 4 0.3351896326668108 0.3826834323650898 5 0.1963495408493624 0.7071067811865476 6 0.05750944903191331 0.9238795325112867 0 0.04083294770910693 -0.9396926207859084 1 0.1442256007956728 -0.766044443118978 2 0.2617993877991495 -0.5 3 0.3385402270935191 -0.1736481776669303 4 0.3385402270935191 0.1736481776669303 5 0.2617993877991495 0.5 6 0.1442256007956728 0.766044443118978 7 0.04083294770910754 0.9396926207859084 0 0.02999954037160841 -0.9510565162951536 1 0.108539356711353 -0.8090169943749475 2 0.2056199086476264 -0.5877852522924731 3 0.2841597249873712 -0.3090169943749475 4 0.3141592653589794 0 5 0.2841597249873712 0.3090169943749475 6 0.2056199086476264 0.5877852522924731 7 0.108539356711353 0.8090169943749475 8 0.02999954037160841 0.9510565162951536 0 0.02266894250185901 -0.9594929736144974 1 0.08347854093418892 -0.8412535328311812 2 0.1631221774548165 -0.6548607339452851 3 0.2363135602034873 -0.4154150130018864 4 0.2798149423030965 -0.1423148382732851 5 0.2798149423030966 0.1423148382732851 6 0.2363135602034873 0.4154150130018864 7 0.1631221774548165 0.6548607339452851 8 0.08347854093418892 0.8412535328311812 9 0.02266894250185901 0.9594929736144974 gegenbauer_to_monomial_matrix_test(): gegenbauer_to_monomial_matrix() evaluates the matrix which converts Gegenbauer polyjomial coefficients to monomial coefficients. Gegenbauer to Monomial matrix G: [[ 1. 0. -0.5 0. 0.375] [ 0. 1. 0. -1.5 0. ] [ 0. 0. 1.5 0. -3.75 ] [ 0. 0. 0. 2.5 0. ] [ 0. 0. 0. 0. 4.375]] Monomial form of Gegenbauer polynomial #0 p(x) = 1 Monomial form of Gegenbauer polynomial #1 p(x) = 1 * x Monomial form of Gegenbauer polynomial #2 p(x) = 1.5 * x^2 - 0.5 Monomial form of Gegenbauer polynomial #3 p(x) = 2.5 * x^3 - 1.5 * x Monomial form of Gegenbauer polynomial #4 p(x) = 4.375 * x^4 - 3.75 * x^2 + 0.375 imtqlx_test(): imtqlx() takes a symmetric tridiagonal matrix A and computes its eigenvalues LAM. It also accepts a vector Z and computes Q'*Z, where Q is the matrix that diagonalizes A. Computed eigenvalues: 0: 0.267949 1: 1 2: 2 3: 3 4: 3.73205 Exact eigenvalues: 0: 0.267949 1: 1 2: 2 3: 3 4: 3.73205 Vector Z: 0: 1 1: 1 2: 1 3: 1 4: 1 Vector Q*Z: 0: -2.1547 1: -1.8855e-16 2: 0.57735 3: 1.66533e-16 4: -0.154701 r8_hyper_2f1_test(): r8_hyper_2f1() evaluates the hypergeometric 2F1 function. A B C X 2F1 2F1 DIFF (tabulated) (computed) -2.5 3.3 6.7 0.25 0.723561 0.723561 2.22045e-16 -0.5 1.1 6.7 0.25 0.979111 0.979111 1.11022e-16 0.5 1.1 6.7 0.25 1.02166 1.02166 0 2.5 3.3 6.7 0.25 1.40516 1.40516 4.44089e-16 -2.5 3.3 6.7 0.55 0.469614 0.469614 5.55112e-17 -0.5 1.1 6.7 0.55 0.952962 0.952962 3.33067e-16 0.5 1.1 6.7 0.55 1.05128 1.05128 8.88178e-16 2.5 3.3 6.7 0.55 2.39991 2.39991 1.77636e-15 -2.5 3.3 6.7 0.85 0.291061 0.291061 1.66533e-16 -0.5 1.1 6.7 0.85 0.92537 0.92537 5.55112e-16 0.5 1.1 6.7 0.85 1.08655 1.08655 2.22045e-16 2.5 3.3 6.7 0.85 5.73816 5.73816 8.61533e-14 3.3 6.7 -5.5 0.25 15090.7 15090.7 1.09139e-11 1.1 6.7 -0.5 0.25 -104.312 -104.312 2.84217e-14 1.1 6.7 0.5 0.25 21.1751 21.1751 1.06581e-14 3.3 6.7 4.5 0.25 4.19469 4.19469 8.88178e-16 3.3 6.7 -5.5 0.55 1.01708e+10 1.01708e+10 1.14441e-05 1.1 6.7 -0.5 0.55 -24708.6 -24708.6 1.81899e-11 1.1 6.7 0.5 0.55 1372.23 1372.23 2.27374e-12 3.3 6.7 4.5 0.55 58.0927 58.0927 2.84217e-14 3.3 6.7 -5.5 0.85 5.86821e+18 5.86821e+18 12288 1.1 6.7 -0.5 0.85 -4.4635e+08 -4.4635e+08 4.76837e-07 1.1 6.7 0.5 0.85 5.38351e+06 5.38351e+06 7.45058e-09 3.3 6.7 4.5 0.85 20396.9 20396.9 2.18279e-11 r8_psi_test(): r8_psi() evaluates the PSI function. X PSI(X) r8_psi(X) 0.1 -10.42375494041108 -10.42375494041108 0.2 -5.289039896592188 -5.289039896592188 0.3 -3.502524222200133 -3.502524222200133 0.4 -2.561384544585116 -2.561384544585116 0.5 -1.963510026021423 -1.963510026021424 0.6 -1.54061921389319 -1.540619213893191 0.7 -1.220023553697935 -1.220023553697935 0.8 -0.9650085667061385 -0.9650085667061382 0.9 -0.7549269499470515 -0.7549269499470511 1 -0.5772156649015329 -0.5772156649015329 1.1 -0.4237549404110768 -0.4237549404110768 1.2 -0.2890398965921883 -0.2890398965921884 1.3 -0.1691908888667997 -0.1691908888667995 1.4 -0.06138454458511615 -0.06138454458511624 1.5 0.03648997397857652 0.03648997397857652 1.6 0.1260474527734763 0.1260474527734763 1.7 0.208547874873494 0.208547874873494 1.8 0.2849914332938615 0.2849914332938615 1.9 0.3561841611640597 0.3561841611640596 2 0.4227843350984671 0.4227843350984672 gegenbauer_polynomial_test(): Normal end of execution. Fri Apr 5 10:55:58 2024