Tue Oct 19 11:33:56 2021 exactness_test(): Python version: 3.6.9 Test exactness(). chebyshev1_exactness_test Python version: 3.6.9 Test Gauss-Chebyshev1 rules for the Chebyshev1 integral. Density function rho(x) = 1/sqrt(1-x^2). Region: -1 <= x <= +1. exactness: 2*N-1. chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 1 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 1.0000000000000000 chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 2 Degree Relative Error 0 0.0000000000000003 1 0.0000000000000002 2 0.0000000000000003 3 0.0000000000000003 4 0.3333333333333331 chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 3 Degree Relative Error 0 0.0000000000000003 1 0.0000000000000000 2 0.0000000000000006 3 0.0000000000000000 4 0.0000000000000006 5 0.0000000000000000 6 0.0999999999999992 chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 4 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000000 3 0.0000000000000001 4 0.0000000000000000 5 0.0000000000000000 6 0.0000000000000000 7 0.0000000000000000 8 0.0285714285714287 chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 5 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000001 3 0.0000000000000001 4 0.0000000000000002 5 0.0000000000000000 6 0.0000000000000002 7 0.0000000000000000 8 0.0000000000000005 9 0.0000000000000000 10 0.0079365079365082 chebyshev1_exactness_test Normal end of execution. chebyshev1_integral_test Python version: 3.6.9 chebyshev1_integral returns Chebyshev1 integrals of monomials: M(k) = integral ( -1 <= x <= 1 ) x^k / sqrt ( 1 - x^2 ) dx K M(K) 0 3.14159 1 0 2 1.5708 3 0 4 1.1781 5 0 6 0.981748 7 0 8 0.859029 9 0 10 0.773126 chebyshev1_integral_test Normal end of execution. chebyshev2_exactness_test Python version: 3.6.9 Test Gauss-Chebyshev2 rules for the Chebyshev2 integral. Density function rho(x) = sqrt(1-x^2). Region: -1 <= x <= +1. exactness: 2*N-1. chebyshev2_exactness: Quadrature rule for Chebyshev2 integral. Order N = 1 Degree Relative Error 0 0.0000000000000003 1 0.0000000000000000 2 1.0000000000000000 chebyshev2_exactness: Quadrature rule for Chebyshev2 integral. Order N = 2 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000002 2 0.0000000000000000 3 0.0000000000000000 4 0.5000000000000000 chebyshev2_exactness: Quadrature rule for Chebyshev2 integral. Order N = 3 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000002 2 0.0000000000000000 3 0.0000000000000001 4 0.0000000000000000 5 0.0000000000000000 6 0.1999999999999999 chebyshev2_exactness: Quadrature rule for Chebyshev2 integral. Order N = 4 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000001 3 0.0000000000000000 4 0.0000000000000001 5 0.0000000000000000 6 0.0000000000000002 7 0.0000000000000000 8 0.0714285714285715 chebyshev2_exactness: Quadrature rule for Chebyshev2 integral. Order N = 5 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000000 5 0.0000000000000000 6 0.0000000000000005 7 0.0000000000000000 8 0.0000000000000005 9 0.0000000000000000 10 0.0238095238095231 chebyshev2_exactness_test Normal end of execution. chebyshev2_integral_test Python version: 3.6.9 chebyshev2_integral returns Chebyshev2 integrals of monomials: M(k) = integral ( -1 <= x <= 1 ) x^k * sqrt ( 1 - x^2 ) dx K M(K) 0 1.5708 1 0 2 0.392699 3 0 4 0.19635 5 0 6 0.122718 7 0 8 0.0859029 9 0 10 0.0644272 chebyshev2_integral_test Normal end of execution. chebyshev3_exactness_test Python version: 3.6.9 Test Gauss-Chebyshev3 rules for the Chebyshev1 integral. Density function rho(x) = 1/sqrt(1-x^2). Region: -1 <= x <= +1. exactness: 2*N-3. chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 1 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 1.0000000000000000 chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 2 Degree Relative Error 0 0.0000000000000003 1 0.0000000000000000 2 1.0000000000000007 chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 3 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.3333333333333333 chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 4 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000000 5 0.0000000000000000 6 0.1000000000000000 chebyshev1_exactness: Quadrature rule for Chebyshev1 integral. Order N = 5 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000000 3 0.0000000000000001 4 0.0000000000000002 5 0.0000000000000001 6 0.0000000000000001 7 0.0000000000000001 8 0.0285714285714283 chebyshev3_exactness_test Normal end of execution. clenshaw_curtis_exactness_test Python version: 3.6.9 Test Clenshaw-Curtis rules on Legendre integrals. Density function rho(x) = 1. Region: -1 <= x <= +1. exactness: N for N odd, N-1 for N even. legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 1 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 1.0000000000000000 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 2 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 2.0000000000000004 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 3 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.6666666666666665 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 4 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000002 3 0.0000000000000000 4 0.1666666666666668 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 5 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000002 3 0.0000000000000000 4 0.0000000000000001 5 0.0000000000000000 6 0.0666666666666664 clenshaw_curtis_exactness_test Normal end of execution. fejer1_exactness_test Python version: 3.6.9 Test Fejer Type 1 rules on Legendre integrals. Density function rho(x) = 1. Region: -1 <= x <= +1. exactness: N for N odd, N-1 for N even. legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 1 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 1.0000000000000000 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 2 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.4999999999999997 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 3 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000002 3 0.0000000000000000 4 0.2500000000000002 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 4 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000002 3 0.0000000000000000 4 0.0416666666666664 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 5 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000002 3 0.0000000000000000 4 0.0000000000000003 5 0.0000000000000000 6 0.0208333333333333 fejer1_exactness_test Normal end of execution. fejer2_exactness_test; Python version: 3.6.9 Test Fejer Type 2 rules on Legendre integrals. Density function rho(x) = 1. Region: -1 <= x <= +1. exactness: N for N odd, N-1 for N even. legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 1 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 1.0000000000000000 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 2 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.2499999999999999 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 3 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000002 3 0.0000000000000000 4 0.1666666666666666 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 4 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000002 3 0.0000000000000000 4 0.0624999999999999 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 5 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000002 3 0.0000000000000000 4 0.0000000000000001 5 0.0000000000000000 6 0.0374999999999995 fejer2_exactness_test Normal end of execution. gegenbauer_exactness_test Python version: 3.6.9 Test Gauss-Gegenbauer rules on Gegenbauer integrals. Density function rho(x) = 1. Using Lambda = 1.75 Region: -1 <= x <= +1. exactness: 2*N-1. gegenbauer_exactness: Quadrature rule for Gegenbauer integral. Lambda = 1.75 Rule of order N = 1 Degree Relative Error 0 0.0000000000000005 1 0.0000000000000000 2 1.0000000000000000 gegenbauer_exactness: Quadrature rule for Gegenbauer integral. Lambda = 1.75 Rule of order N = 2 Degree Relative Error 0 0.0000000000000007 1 0.0000000000000000 2 0.0000000000000012 3 0.0000000000000000 4 0.5454545454545461 gegenbauer_exactness: Quadrature rule for Gegenbauer integral. Lambda = 1.75 Rule of order N = 3 Degree Relative Error 0 0.0000000000000002 1 0.0000000000000000 2 0.0000000000000005 3 0.0000000000000000 4 0.0000000000000012 5 0.0000000000000000 6 0.2400000000000015 gegenbauer_exactness: Quadrature rule for Gegenbauer integral. Lambda = 1.75 Rule of order N = 4 Degree Relative Error 0 0.0000000000000005 1 0.0000000000000000 2 0.0000000000000015 3 0.0000000000000001 4 0.0000000000000024 5 0.0000000000000001 6 0.0000000000000030 7 0.0000000000000001 8 0.0938345864661686 gegenbauer_exactness: Quadrature rule for Gegenbauer integral. Lambda = 1.75 Rule of order N = 5 Degree Relative Error 0 0.0000000000000005 1 0.0000000000000002 2 0.0000000000000010 3 0.0000000000000000 4 0.0000000000000008 5 0.0000000000000000 6 0.0000000000000010 7 0.0000000000000000 8 0.0000000000000014 9 0.0000000000000000 10 0.0339980385746993 gegenbauer_exactness_test Normal end of execution. gegenbauer_integral_test Python version: 3.6.9 gegenbauer_integral returns Gegenbauer integrals of monomials: M(k) = integral ( -1 <= x <= 1 ) (1-x^2)^(lambda-1/2) dx Here, we use lambda = 1.75 K M(K) 0 1.2486 1 0 2 0.227018 3 0 4 0.0908072 5 0 6 0.0477933 7 0 8 0.0290915 9 0 10 0.0193944 gegenbauer_integral_test Normal end of execution. hermite_1_exactness_test Python version: 3.6.9 Test Gauss-Hermite rules on Hermite integrals. Density function rho(x) = 1. Region: -oo < x < +oo. exactness: 2N-1. hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 1 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 1.0000000000000000 hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 2 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000003 3 0.0000000000000000 4 0.6666666666666666 hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 3 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000003 3 0.0000000000000000 4 0.0000000000000005 5 0.0000000000000000 6 0.4000000000000005 hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 4 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000003 5 0.0000000000000001 6 0.0000000000000007 7 0.0000000000000002 8 0.2285714285714292 hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 5 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000001 3 0.0000000000000001 4 0.0000000000000000 5 0.0000000000000000 6 0.0000000000000003 7 0.0000000000000001 8 0.0000000000000002 9 0.0000000000000006 10 0.1269841269841271 hermite_1_exactness_test Normal end of execution. hermite_exactness_test Python version: 3.6.9 Test Gauss-Hermite rules on Hermite integrals. Density function rho(x) = exp(-x^2). Region: -oo < x < +oo. exactness: 2N-1. hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 1 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 1.0000000000000000 hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 2 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000004 3 0.0000000000000000 4 0.6666666666666666 hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 3 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000003 5 0.0000000000000000 6 0.4000000000000004 hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 4 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000002 5 0.0000000000000001 6 0.0000000000000005 7 0.0000000000000002 8 0.2285714285714292 hermite_exactness: Quadrature rule for Hermite integral. Rule of order N = 5 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000001 3 0.0000000000000001 4 0.0000000000000000 5 0.0000000000000000 6 0.0000000000000003 7 0.0000000000000001 8 0.0000000000000002 9 0.0000000000000006 10 0.1269841269841271 hermite_exactness_test Normal end of execution. hermite_integral_test Python version: 3.6.9 hermite_integral returns Hermite integrals of monomials: M(k) = integral ( -oo <= x <= +oo ) x^k exp(-x^2) dx K M(K) 0 1.77245 1 0 2 0.886227 3 0 4 1.32934 5 0 6 3.32335 7 0 8 11.6317 9 0 10 52.3428 hermite_integral_test Normal end of execution. laguerre_1_exactness_test Python version: 3.6.9 Test quadrature rules on Laguerre integrals. Density function rho(x) = 1. Region: 0 <= x < +oo. exactness: 2N-1. Quadrature rule for Laguerre integral. Rule of order N = 1 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.5000000000000000 Quadrature rule for Laguerre integral. Rule of order N = 2 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.1666666666666668 Quadrature rule for Laguerre integral. Rule of order N = 3 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000000 5 0.0000000000000000 6 0.0500000000000000 Quadrature rule for Laguerre integral. Rule of order N = 4 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000001 2 0.0000000000000001 3 0.0000000000000003 4 0.0000000000000003 5 0.0000000000000002 6 0.0000000000000002 7 0.0000000000000002 8 0.0142857142857146 Quadrature rule for Laguerre integral. Rule of order N = 5 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000001 2 0.0000000000000002 3 0.0000000000000001 4 0.0000000000000001 5 0.0000000000000002 6 0.0000000000000002 7 0.0000000000000002 8 0.0000000000000002 9 0.0000000000000002 10 0.0039682539682544 laguerre_1_exactness_test Normal end of execution. laguerre_exactness_test Python version: 3.6.9 Test Gauss-Laguerre rules on Laguerre integrals. Density function rho(x) = exp(-x). Region: 0 <= x < +oo. exactness: 2N-1. Quadrature rule for Laguerre integral. Rule of order N = 1 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.5000000000000000 Quadrature rule for Laguerre integral. Rule of order N = 2 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.1666666666666668 Quadrature rule for Laguerre integral. Rule of order N = 3 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000001 5 0.0000000000000001 6 0.0499999999999998 Quadrature rule for Laguerre integral. Rule of order N = 4 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000001 3 0.0000000000000003 4 0.0000000000000003 5 0.0000000000000002 6 0.0000000000000002 7 0.0000000000000002 8 0.0142857142857146 Quadrature rule for Laguerre integral. Rule of order N = 5 Degree Relative Error 0 0.0000000000000001 1 0.0000000000000001 2 0.0000000000000001 3 0.0000000000000000 4 0.0000000000000001 5 0.0000000000000002 6 0.0000000000000002 7 0.0000000000000002 8 0.0000000000000002 9 0.0000000000000003 10 0.0039682539682544 laguerre_exactness_test Normal end of execution. laguerre_integral_test Python version: 3.6.9 laguerre_integral returns Laguerre integrals of monomials: M(k) = integral ( 0 <= x < +oo ) x^k exp(-x) dx K M(K) 0 1 1 1 2 2 3 6 4 24 5 120 6 720 7 5040 8 40320 9 362880 10 3.6288e+06 laguerre_integral_test Normal end of execution. legendre_exactness_test Python version: 3.6.9 Test Gauss-Legendre rules on Legendre integrals. Density function rho(x) = 1. Region: -1 <= x <= +1. exactness: 2*N-1. legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 1 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 1.0000000000000000 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 2 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.4444444444444446 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 3 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000002 3 0.0000000000000000 4 0.0000000000000001 5 0.0000000000000000 6 0.1599999999999997 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 4 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000000 5 0.0000000000000000 6 0.0000000000000002 7 0.0000000000000000 8 0.0522448979591837 legendre_exactness: Quadrature rule for Legendre integral. Rule of order N = 5 Degree Relative Error 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000000 4 0.0000000000000001 5 0.0000000000000000 6 0.0000000000000000 7 0.0000000000000000 8 0.0000000000000001 9 0.0000000000000000 10 0.0161249685059213 legendre_exactness_test Normal end of execution. legendre_integral_test Python version: 3.6.9 legendre_integral returns Legendre integrals of monomials: M(k) = integral ( -1 <= x <= 1 ) x^k dx K M(K) 0 2 1 0 2 0.666667 3 0 4 0.4 5 0 6 0.285714 7 0 8 0.222222 9 0 10 0.181818 legendre_integral_test Normal end of execution. exactness_test: Normal end of execution. Tue Oct 19 11:33:56 2021