Tue Oct 19 11:57:18 2021 legendre_polynomial_test(): Python version: 3.6.9 Test legendre_polynomial(). imtqlx_test Python version: 3.6.9 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 imtqlx_test: Normal end of execution. p_exponential_product_test Python version: 3.6.9 p_exponential_product computes an exponential product table for P(n,x): Tij = integral ( -1 <= x <= +1 ) exp(b*x) P(i,x) P(j,x) dx Maximum degree P = 5 Exponential argument coefficient B = 0 Exponential product table: Col: 0 1 2 3 4 Row 0 : 2 -1.60982e-15 -3.46945e-16 -9.71445e-17 2.22045e-16 1 :-1.60982e-15 0.666667 -6.52256e-16 -2.77556e-17 -3.747e-16 2 :-3.46945e-16 -6.52256e-16 0.4 -7.00828e-16 1.94289e-16 3 :-9.71445e-17 -2.77556e-17 -6.93889e-16 0.285714 -4.71845e-16 4 : 2.22045e-16 -3.81639e-16 2.01228e-16 -5.06539e-16 0.222222 5 : -5.6205e-16 4.02456e-16 -2.42861e-16 1.52656e-16 -3.1225e-16 Col: 5 Row 0 : -5.6205e-16 1 : 3.88578e-16 2 :-2.63678e-16 3 : 1.73472e-16 4 : -3.1225e-16 5 : 0.181818 p_exponential_product_test: Normal end of execution. p_exponential_product_test Python version: 3.6.9 p_exponential_product computes an exponential product table for P(n,x): Tij = integral ( -1 <= x <= +1 ) exp(b*x) P(i,x) P(j,x) dx Maximum degree P = 5 Exponential argument coefficient B = 1 Exponential product table: Col: 0 1 2 3 4 Row 0 : 2.3504 0.735759 0.143126 0.0201302 0.00221447 1 : 0.735759 0.878885 0.306382 0.062605 0.00905782 2 : 0.143126 0.306382 0.512112 0.194658 0.0414752 3 : 0.0201302 0.062605 0.194658 0.363558 0.143849 4 : 0.00221447 0.00905782 0.0414752 0.143849 0.28217 5 : 0.000199925 0.00101492 0.00615177 0.0313811 0.114325 Col: 5 Row 0 : 0.000199925 1 : 0.00101492 2 : 0.00615177 3 : 0.0313811 4 : 0.114325 5 : 0.230635 p_exponential_product_test: Normal end of execution. p_integral_test Python version: 3.6.9 p_integral returns the integral of P(n,x) over [-1,+1]. N Integral 0 2 1 0 2 0.6666666666666666 3 0 4 0.4 5 0 6 0.2857142857142857 7 0 8 0.2222222222222222 9 0 10 0.1818181818181818 p_integral_test Normal end of execution. p_polynomial_coefficients_test Python version: 3.6.9 p_polynomial_coefficients determines polynomial coefficients of P(n,x). P(0,x) = 1 P(1,x) = 1 * x P(2,x) = 1.5 * x^2 -0.5 P(3,x) = 2.5 * x^3 -1.5 * x P(4,x) = 4.375 * x^4 -3.75 * x^2 0.375 P(5,x) = 7.875 * x^5 -8.75 * x^3 1.875 * x P(6,x) = 14.4375 * x^6 -19.6875 * x^4 6.5625 * x^2 -0.3125 P(7,x) = 26.8125 * x^7 -43.3125 * x^5 19.6875 * x^3 -2.1875 * x P(8,x) = 50.2734 * x^8 -93.8438 * x^6 54.1406 * x^4 -9.84375 * x^2 0.273438 P(9,x) = 94.9609 * x^9 -201.094 * x^7 140.766 * x^5 -36.0938 * x^3 2.46094 * x P(10,x) = 180.426 * x^10 -427.324 * x^8 351.914 * x^6 -117.305 * x^4 13.5352 * x^2 -0.246094 p_polynomial_coefficients_test Normal end of execution. p_polynomial_prime_test: Python version: 3.6.9 p_polynomial_prime evaluates the derivative of the Legendre polynomial P(N,X). Computed N X P'(N,X) 0 -1 0 1 -1 1 2 -1 -3 3 -1 6 4 -1 -10 5 -1 15 0 -0.8 0 1 -0.8 1 2 -0.8 -2.4 3 -0.8 3.3 4 -0.8 -2.96 5 -0.8 1.203 0 -0.6 0 1 -0.6 1 2 -0.6 -1.8 3 -0.6 1.2 4 -0.6 0.72 5 -0.6 -2.472 0 -0.4 0 1 -0.4 1 2 -0.4 -1.2 3 -0.4 -0.3 4 -0.4 1.88 5 -0.4 -1.317 0 -0.2 0 1 -0.2 1 2 -0.2 -0.6 3 -0.2 -1.2 4 -0.2 1.36 5 -0.2 0.888 0 0 0 1 0 1 2 0 0 3 0 -1.5 4 0 -0 5 0 1.875 0 0.2 0 1 0.2 1 2 0.2 0.6 3 0.2 -1.2 4 0.2 -1.36 5 0.2 0.888 0 0.4 0 1 0.4 1 2 0.4 1.2 3 0.4 -0.3 4 0.4 -1.88 5 0.4 -1.317 0 0.6 0 1 0.6 1 2 0.6 1.8 3 0.6 1.2 4 0.6 -0.72 5 0.6 -2.472 0 0.8 0 1 0.8 1 2 0.8 2.4 3 0.8 3.3 4 0.8 2.96 5 0.8 1.203 0 1 0 1 1 1 2 1 3 3 1 6 4 1 10 5 1 15 p_polynomial_prime_test Normal end of execution. p_polynomial_prime2_test: Python version: 3.6.9 p_polynomial_prime2 evaluates the second derivative of the Legendre polynomial P(N,X). Computed N X P"(N,X) 0 -1 0 1 -1 0 2 -1 3 3 -1 -15 4 -1 45 5 -1 -105 0 -0.8 0 1 -0.8 0 2 -0.8 3 3 -0.8 -12 4 -0.8 26.1 5 -0.8 -38.64 0 -0.6 0 1 -0.6 0 2 -0.6 3 3 -0.6 -9 4 -0.6 11.4 5 -0.6 -2.52 0 -0.4 0 1 -0.4 0 2 -0.4 3 3 -0.4 -6 4 -0.4 0.9 5 -0.4 10.92 0 -0.2 0 1 -0.2 0 2 -0.2 3 3 -0.2 -3 4 -0.2 -5.4 5 -0.2 9.24 0 0 0 1 0 0 2 0 3 3 0 0 4 0 -7.5 5 0 -0 0 0.2 0 1 0.2 0 2 0.2 3 3 0.2 3 4 0.2 -5.4 5 0.2 -9.24 0 0.4 0 1 0.4 0 2 0.4 3 3 0.4 6 4 0.4 0.9 5 0.4 -10.92 0 0.6 0 1 0.6 0 2 0.6 3 3 0.6 9 4 0.6 11.4 5 0.6 2.52 0 0.8 0 1 0.8 0 2 0.8 3 3 0.8 12 4 0.8 26.1 5 0.8 38.64 0 1 0 1 1 0 2 1 3 3 1 15 4 1 45 5 1 105 p_polynomial_prime2_test Normal end of execution. p_polynomial_value_test: Python version: 3.6.9 p_polynomial_value evaluates the Legendre polynomial P(n,x). Tabulated Computed N X P(N,X) P(N,X) Error 0 0.25 1 1 0 1 0.25 0.25 0.25 0 2 0.25 -0.40625 -0.40625 0 3 0.25 -0.335938 -0.335938 0 4 0.25 0.157715 0.157715 0 5 0.25 0.339722 0.339722 0 6 0.25 0.0242767 0.0242767 0 7 0.25 -0.279919 -0.279919 0 8 0.25 -0.152454 -0.152454 -2.77556e-17 9 0.25 0.176824 0.176824 0 10 0.25 0.2212 0.2212 2.77556e-17 3 0 0 -0 0 3 0.1 -0.1475 -0.1475 0 3 0.2 -0.28 -0.28 0 3 0.3 -0.3825 -0.3825 0 3 0.4 -0.44 -0.44 -5.55112e-17 3 0.5 -0.4375 -0.4375 0 3 0.6 -0.36 -0.36 5.55112e-17 3 0.7 -0.1925 -0.1925 1.11022e-16 3 0.8 0.08 0.08 -2.22045e-16 3 0.9 0.4725 0.4725 -1.11022e-16 3 1 1 1 0 p_polynomial_value_test Normal end of execution. p_polynomial_values_test: Python version: 3.6.9 p_polynomial_values stores values of the Legendre polynomials. N X F 0 0.250000 1 1 0.250000 0.25 2 0.250000 -0.40625 3 0.250000 -0.3359375 4 0.250000 0.15771484375 5 0.250000 0.3397216796875 6 0.250000 0.0242767333984375 7 0.250000 -0.2799186706542969 8 0.250000 -0.1524540185928345 9 0.250000 0.1768244206905365 10 0.250000 0.2212002165615559 3 0.000000 0 3 0.100000 -0.1475 3 0.200000 -0.28 3 0.300000 -0.3825 3 0.400000 -0.44 3 0.500000 -0.4375 3 0.600000 -0.36 3 0.700000 -0.1925 3 0.800000 0.08 3 0.900000 0.4725 3 1.000000 1 p_polynomial_values_test: Normal end of execution. p_polynomial_zeros_test: Python version: 3.6.9 p_polynomial_zeros computes the zeros of P(n,x) Check by calling p_polynomial_value there. Computed zeros for P(1,x) 0: 0 Evaluate P(1,z) 0: 0 Computed zeros for P(2,x) 0: -0.57735 1: 0.57735 Evaluate P(2,z) 0: -5.55112e-17 1: -5.55112e-17 Computed zeros for P(3,x) 0: -0.774597 1: -7.20308e-18 2: 0.774597 Evaluate P(3,z) 0: 0 1: 1.08046e-17 2: 4.44089e-16 Computed zeros for P(4,x) 0: -0.861136 1: -0.339981 2: 0.339981 3: 0.861136 Evaluate P(4,z) 0: -1.66533e-16 1: 1.38778e-16 2: -2.77556e-16 3: -1.4988e-15 Computed zeros for P(5,x) 0: -0.90618 1: -0.538469 2: 1.41073e-16 3: 0.538469 4: 0.90618 Evaluate P(5,z) 0: 1.77636e-16 1: -3.10862e-16 2: 2.64513e-16 3: 5.32907e-16 4: 9.76996e-16 p_polynomial_zeros_test Normal end of execution. p_power_product_test: Python version: 3.6.9 p_power_product computes a power product table for P(n,x): Tij = integral ( -1 <= x <= +1 ) x^e P(i,x) P(j,x) dx Maximum degree P = 5 Exponent of X, E = 0 Power product table: Col: 0 1 2 3 4 Row 0 : 2 6.66134e-16 8.04912e-16 7.63278e-16 -6.93889e-17 1 : 6.66134e-16 0.666667 7.21645e-16 2.77556e-16 3.33067e-16 2 : 8.04912e-16 7.21645e-16 0.4 3.46945e-16 -1.66533e-16 3 : 7.63278e-16 2.91434e-16 3.19189e-16 0.285714 1.04083e-16 4 :-6.93889e-17 3.60822e-16 -1.73472e-16 9.71445e-17 0.222222 5 : 4.85723e-17 -5.06539e-16 9.71445e-17 -1.8735e-16 1.94289e-16 Col: 5 Row 0 : 4.85723e-17 1 :-4.85723e-16 2 : 1.00614e-16 3 :-1.83881e-16 4 : 1.8735e-16 5 : 0.181818 p_power_product_test Normal end of execution. p_power_product_test: Python version: 3.6.9 p_power_product computes a power product table for P(n,x): Tij = integral ( -1 <= x <= +1 ) x^e P(i,x) P(j,x) dx Maximum degree P = 5 Exponent of X, E = 1 Power product table: Col: 0 1 2 3 4 Row 0 : 1.249e-16 0.666667 2.22045e-16 4.996e-16 6.80012e-16 1 : 0.666667 1.38778e-16 0.266667 4.71845e-16 6.93889e-17 2 : 2.22045e-16 0.266667 4.57967e-16 0.171429 4.64906e-16 3 : 4.996e-16 4.85723e-16 0.171429 4.44089e-16 0.126984 4 : 6.80012e-16 9.71445e-17 4.51028e-16 0.126984 2.56739e-16 5 :-2.28983e-16 5.55112e-16 -1.52656e-16 2.56739e-16 0.10101 Col: 5 Row 0 :-2.28983e-16 1 : 5.55112e-16 2 :-1.45717e-16 3 : 2.56739e-16 4 : 0.10101 5 : 1.04083e-16 p_power_product_test Normal end of execution. p_quadrature_rule_test: Python version: 3.6.9 p_quadrature_rule computes the quadrature rule associated with P(n,x) X W 0: -0.90618 0.236927 1: -0.538469 0.478629 2: 1.41073e-16 0.568889 3: 0.538469 0.478629 4: 0.90618 0.236927 Use the quadrature rule to estimate: Q = Integral ( -1 <= X < +1 ) X^E dx E Q_Estimate Q_Exact 0 2 2 1 3.89573e-16 0 2 0.666667 0.666667 3 1.82539e-16 0 4 0.4 0.4 5 1.36041e-16 0 6 0.285714 0.285714 7 1.53372e-16 0 8 0.222222 0.222222 9 1.45512e-16 0 p_quadrature_rule_test Normal end of execution. pm_polynomial_value_test: Python version: 3.6.9 pm_polynomial_value evaluates the Legendre polynomial Pm(n,m,x). Tabulated Computed N M X Pm(N,M,X) Pm(N,M,X) Error 1 0 0 0 0 0 2 0 0 -0.5 -0.5 0 3 0 0 0 -0 0 4 0 0 0.375 0.375 0 5 0 0 0 0 0 1 1 0.5 -0.8660254037844386 -0.8660254037844386 0 2 1 0.5 -1.299038105676658 -1.299038105676658 0 3 1 0.5 -0.3247595264191645 -0.3247595264191645 0 4 1 0.5 1.353164693413185 1.353164693413185 -4.44089e-16 3 0 0.2 -0.28 -0.28 0 3 1 0.2 1.175755076535925 1.175755076535925 -4.44089e-16 3 2 0.2 2.88 2.88 0 3 3 0.2 -14.10906091843111 -14.1090609184311 -7.10543e-15 4 2 0.25 -3.955078125 -3.955078125 4.44089e-16 5 2 0.25 -9.99755859375 -9.997558593750002 1.77636e-15 6 3 0.25 82.65311444100485 82.65311444100486 -1.42109e-14 7 3 0.25 20.24442836815152 20.24442836815153 -1.06581e-14 8 4 0.25 -423.7997531890869 -423.7997531890869 -5.68434e-14 9 4 0.25 1638.320624828339 1638.320624828339 0 10 5 0.25 -20256.87389227225 -20256.87389227226 3.63798e-12 pm_polynomial_value_test Normal end of execution. pm_polynomial_values_test: Python version: 3.6.9 pm_polynomial_values stores values of the associated Legendre function. N M X F 1 0 0.000000 0 2 0 0.000000 -0.5 3 0 0.000000 0 4 0 0.000000 0.375 5 0 0.000000 0 1 1 0.500000 -0.8660254037844386 2 1 0.500000 -1.299038105676658 3 1 0.500000 -0.3247595264191645 4 1 0.500000 1.353164693413185 3 0 0.200000 -0.28 3 1 0.200000 1.175755076535925 3 2 0.200000 2.88 3 3 0.200000 -14.10906091843111 4 2 0.250000 -3.955078125 5 2 0.250000 -9.99755859375 6 3 0.250000 82.65311444100485 7 3 0.250000 20.24442836815152 8 4 0.250000 -423.7997531890869 9 4 0.250000 1638.320624828339 10 5 0.250000 -20256.87389227225 pm_polynomial_values_test: Normal end of execution. pmn_polynomial_value_test: Python version: 3.6.9 pmn_polynomial_value evaluates the Legendre polynomial Pmn(n,m,x). Tabulated Computed N M X Pmn(N,M,X) Pmn(N,M,X) Error 0 0 0.5 0.7071067811865475 0.7071067811865476 -1.11022e-16 1 0 0.5 0.6123724356957945 0.6123724356957945 0 1 1 0.5 -0.75 -0.7499999999999999 -1.11022e-16 2 0 0.5 -0.1976423537605237 -0.1976423537605237 2.77556e-17 2 1 0.5 -0.8385254915624211 -0.8385254915624212 1.11022e-16 2 2 0.5 0.7261843774138907 0.7261843774138906 1.11022e-16 3 0 0.5 -0.8184875533567997 -0.8184875533567997 0 3 1 0.5 -0.1753901900050285 -0.1753901900050285 2.77556e-17 3 2 0.5 0.9606516343087123 0.9606516343087123 0 3 3 0.5 -0.6792832849776299 -0.67928328497763 1.11022e-16 4 0 0.5 -0.6131941618102092 -0.6131941618102091 -1.11022e-16 4 1 0.5 0.6418623720763665 0.6418623720763665 0 4 2 0.5 0.4716705890038619 0.4716705890038619 0 4 3 0.5 -1.018924927466445 -1.018924927466445 0 4 4 0.5 0.6239615396237876 0.6239615396237875 1.11022e-16 5 0 0.5 0.2107022704608181 0.2107022704608181 -2.77556e-17 5 1 0.5 0.8256314721961969 0.8256314721961968 1.11022e-16 5 2 0.5 -0.3982651281554632 -0.3982651281554632 -5.55112e-17 5 3 0.5 -0.7040399320721435 -0.7040399320721434 -1.11022e-16 5 4 0.5 1.034723155272289 1.034723155272289 4.44089e-16 5 5 0.5 -0.566741212915553 -0.566741212915553 0 pmn_polynomial_value_test Normal end of execution. pmn_polynomial_values_test: Python version: 3.6.9 pmn_polynomial_values stores values of the normalized associated Legendre function. N M X F 0 0 0.5 0.7071067811865475 1 0 0.5 0.6123724356957945 1 1 0.5 -0.75 2 0 0.5 -0.1976423537605237 2 1 0.5 -0.8385254915624211 2 2 0.5 0.7261843774138907 3 0 0.5 -0.8184875533567997 3 1 0.5 -0.1753901900050285 3 2 0.5 0.9606516343087123 3 3 0.5 -0.6792832849776299 4 0 0.5 -0.6131941618102092 4 1 0.5 0.6418623720763665 4 2 0.5 0.4716705890038619 4 3 0.5 -1.018924927466445 4 4 0.5 0.6239615396237876 5 0 0.5 0.2107022704608181 5 1 0.5 0.8256314721961969 5 2 0.5 -0.3982651281554632 5 3 0.5 -0.7040399320721435 5 4 0.5 1.034723155272289 5 5 0.5 -0.566741212915553 pmn_polynomial_values_test: Normal end of execution. pmns_polynomial_value_test: Python version: 3.6.9 pmns_polynomial_value evaluates the Legendre polynomial Pmns(n,m,x). Tabulated Computed N M X Pmns(N,M,X) Pmns(N,M,X) Error 0 0 0.5 0.2820947917738781 0.2820947917738781 -5.55112e-17 1 0 0.5 0.24430125595146 0.24430125595146 2.77556e-17 1 1 0.5 -0.2992067103010745 -0.2992067103010745 0 2 0 0.5 -0.07884789131313 -0.07884789131313001 1.38778e-17 2 1 0.5 -0.3345232717786446 -0.3345232717786445 -5.55112e-17 2 2 0.5 0.2897056515173922 0.2897056515173921 5.55112e-17 3 0 0.5 -0.326529291016351 -0.326529291016351 0 3 1 0.5 -0.06997056236064664 -0.06997056236064664 0 3 2 0.5 0.3832445536624809 0.3832445536624809 -5.55112e-17 3 3 0.5 -0.2709948227475519 -0.2709948227475519 5.55112e-17 4 0 0.5 -0.24462907724141 -0.24462907724141 -2.77556e-17 4 1 0.5 0.2560660384200185 0.2560660384200185 0 4 2 0.5 0.1881693403754876 0.1881693403754876 2.77556e-17 4 3 0.5 -0.4064922341213279 -0.406492234121328 5.55112e-17 4 4 0.5 0.2489246395003027 0.2489246395003027 -5.55112e-17 5 0 0.5 0.0840580442633982 0.08405804426339822 -1.38778e-17 5 1 0.5 0.3293793022891428 0.3293793022891428 0 5 2 0.5 -0.1588847984307093 -0.1588847984307093 2.77556e-17 5 3 0.5 -0.2808712959945307 -0.2808712959945307 0 5 4 0.5 0.4127948151484925 0.4127948151484925 0 5 5 0.5 -0.2260970318780046 -0.2260970318780046 2.77556e-17 pmns_polynomial_value_test Normal end of execution. pmns_polynomial_values_test: Python version: 3.6.9 pmns_polynomial_values stores values of the associated Legendre function normalized for the surface of a sphere. N M X F 0 0 0.5 0.2820947917738781 1 0 0.5 0.24430125595146 1 1 0.5 -0.2992067103010745 2 0 0.5 -0.07884789131313 2 1 0.5 -0.3345232717786446 2 2 0.5 0.2897056515173922 3 0 0.5 -0.326529291016351 3 1 0.5 -0.06997056236064664 3 2 0.5 0.3832445536624809 3 3 0.5 -0.2709948227475519 4 0 0.5 -0.24462907724141 4 1 0.5 0.2560660384200185 4 2 0.5 0.1881693403754876 4 3 0.5 -0.4064922341213279 4 4 0.5 0.2489246395003027 5 0 0.5 0.0840580442633982 5 1 0.5 0.3293793022891428 5 2 0.5 -0.1588847984307093 5 3 0.5 -0.2808712959945307 5 4 0.5 0.4127948151484925 5 5 0.5 -0.2260970318780046 pmns_polynomial_values_test: Normal end of execution. pn_pair_product_test Python version: 3.6.9 pn_pair_product computes a pair product table for Pn(n,x): Tij = integral ( -1 <= x <= +1 ) Pn(i,x) Pn(j,x) dx The Pn(n,x) polynomials are orthonormal, so T should be the identity matrix. Maximum degree P = 5 Pair product table: Col: 0 1 2 3 4 Row 0 : 1 -1.31839e-16 7.97973e-16 -7.07767e-16 -1.38778e-17 1 :-1.38778e-16 1 -7.49401e-16 7.63278e-16 -1.27676e-15 2 : 7.56339e-16 -7.35523e-16 1 -1.19349e-15 2.66454e-15 3 :-6.38378e-16 7.63278e-16 -1.19349e-15 1 -4.16334e-16 4 : 0 -1.249e-15 2.66454e-15 -4.16334e-16 1 5 :-8.04912e-16 2.01228e-15 -5.82867e-16 3.16414e-15 -9.71445e-16 Col: 5 Row 0 :-7.77156e-16 1 : 1.95677e-15 2 : -4.996e-16 3 : 3.16414e-15 4 :-9.99201e-16 5 : 1 pn_pair_product_test Normal end of execution. pn_polynomial_coefficients_test Python version: 3.6.9 pn_polynomial_coefficients: polynomial coefficients of Pn(n,x). P(0,x) = 0.707107 P(1,x) = 1.22474 * x P(2,x) = 2.37171 * x^2 -0.790569 P(3,x) = 4.67707 * x^3 -2.80624 * x P(4,x) = 9.28078 * x^4 -7.95495 * x^2 0.795495 P(5,x) = 18.4685 * x^5 -20.5206 * x^3 4.39726 * x P(6,x) = 36.8085 * x^6 -50.1935 * x^4 16.7312 * x^2 -0.796722 P(7,x) = 73.4291 * x^7 -118.616 * x^5 53.9164 * x^3 -5.99072 * x P(8,x) = 146.571 * x^8 -273.599 * x^6 157.846 * x^4 -28.6992 * x^2 0.7972 P(9,x) = 292.689 * x^9 -619.813 * x^7 433.869 * x^5 -111.248 * x^3 7.58512 * x P(10,x) = 584.646 * x^10 -1384.69 * x^8 1140.33 * x^6 -380.111 * x^4 43.8589 * x^2 -0.797435 pn_polynomial_coefficients_test Normal end of execution. pn_polynomial_value_test: Python version: 3.6.9 pn_polynomial_value evaluates the normalized Legendre polynomial Pn(n,x). Tabulated Computed N X Pn(N,X) Pn(N,X) Error 0 0.25 0.7071067811865475 0.7071067811865475 0 1 0.25 0.3061862178478972 0.3061862178478972 -5.55112e-17 2 0.25 -0.642337649721702 -0.642337649721702 0 3 0.25 -0.6284815141846855 -0.6284815141846855 0 4 0.25 0.3345637065282053 0.3345637065282053 -5.55112e-17 5 0.25 0.7967179601799685 0.7967179601799685 0 6 0.25 0.06189376866246124 0.06189376866246124 0 7 0.25 -0.766588850921089 -0.766588850921089 0 8 0.25 -0.4444760242953344 -0.4444760242953344 0 9 0.25 0.5450094674858101 0.5450094674858101 0 10 0.25 0.7167706229835538 0.7167706229835538 0 3 0 0 -0 0 3 0.1 -0.2759472322745781 -0.2759472322745781 0 3 0.2 -0.5238320341483518 -0.5238320341483518 0 3 0.3 -0.7155919752205163 -0.7155919752205163 0 3 0.4 -0.823164625090267 -0.823164625090267 0 3 0.5 -0.8184875533567997 -0.8184875533567997 0 3 0.6 -0.6734983296193094 -0.6734983296193094 0 3 0.7 -0.360134523476992 -0.360134523476992 5.55112e-17 3 0.8 0.1496662954709581 0.1496662954709581 5.55112e-17 3 0.9 0.8839665576253438 0.8839665576253438 0 3 1 1.870828693386971 1.870828693386971 4.44089e-16 pn_polynomial_value_test Normal end of execution. pn_polynomial_values_test: Python version: 3.6.9 pn_polynomial_values stores values of the normalized Legendre polynomials. N X F 0 0.25 0.7071067811865475 1 0.25 0.3061862178478972 2 0.25 -0.642337649721702 3 0.25 -0.6284815141846855 4 0.25 0.3345637065282053 5 0.25 0.7967179601799685 6 0.25 0.06189376866246124 7 0.25 -0.766588850921089 8 0.25 -0.4444760242953344 9 0.25 0.5450094674858101 10 0.25 0.7167706229835538 3 0 0 3 0.1 -0.2759472322745781 3 0.2 -0.5238320341483518 3 0.3 -0.7155919752205163 3 0.4 -0.823164625090267 3 0.5 -0.8184875533567997 3 0.6 -0.6734983296193094 3 0.7 -0.360134523476992 3 0.8 0.1496662954709581 3 0.9 0.8839665576253438 3 1 1.870828693386971 pn_polynomial_values_test: Normal end of execution. r8_sign_test Python version: 3.6.9 r8_sign returns the sign of an R8. R8 r8_sign(R8) -1.2500 -1 -0.2500 -1 0.0000 1 0.5000 1 9.0000 1 r8_sign_test Normal end of execution. legendre_polynomial_test: Normal end of execution. Tue Oct 19 11:57:18 2021