Thu Sep 13 10:33:17 2018 LEGENDRE_POLYNOMIAL_TEST: Python version: 3.6.5 Test the LEGENDRE_POLYNOMIAL library. IMTQLX_TEST Python version: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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 3.88578e-16 0 2 0.666667 0.666667 3 1.52656e-16 0 4 0.4 0.4 5 1.31839e-16 0 6 0.285714 0.285714 7 1.63931e-16 0 8 0.222222 0.222222 9 1.43982e-16 0 P_QUADRATURE_RULE_TEST Normal end of execution. PM_POLYNOMIAL_VALUE_TEST: Python version: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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: 3.6.5 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_EPSILON_TEST Python version: 3.6.5 R8_EPSILON produces the R8 roundoff unit. R = R8_EPSILON() = 2.220446e-16 ( 1 + R ) - 1 = 2.220446e-16 ( 1 + (R/2) ) - 1 = 0.000000e+00 R8_EPSILON_TEST Normal end of execution. R8_FACTORIAL_TEST Python version: 3.6.5 R8_FACTORIAL evaluates the factorial function. N Exact Computed 0 1 1 1 1 1 2 2 2 3 6 6 4 24 24 5 120 120 6 720 720 7 5040 5040 8 40320 40320 9 362880 362880 10 3628800 3628800 11 39916800 39916800 12 479001600 479001600 13 6227020800 6227020800 14 87178291200 87178291200 15 1307674368000 1307674368000 16 20922789888000 20922789888000 17 355687428096000 355687428096000 18 6402373705728000 6402373705728000 19 1.21645100408832e+17 1.21645100408832e+17 20 2.43290200817664e+18 2.43290200817664e+18 25 1.551121004333099e+25 1.551121004333099e+25 50 3.041409320171338e+64 3.041409320171338e+64 100 9.332621544394415e+157 9.33262154439441e+157 150 5.713383956445855e+262 5.71338395644585e+262 R8_FACTORIAL_TEST Normal end of execution. R8_SIGN_TEST Python version: 3.6.5 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. R8MAT_PRINT_TEST Python version: 3.6.5 R8MAT_PRINT prints an R8MAT. Here is an R8MAT: Col: 0 1 2 3 4 Row 0 : 11 12 13 14 15 1 : 21 22 23 24 25 2 : 31 32 33 34 35 3 : 41 42 43 44 45 Col: 5 Row 0 : 16 1 : 26 2 : 36 3 : 46 R8MAT_PRINT_TEST: Normal end of execution. R8MAT_PRINT_SOME_TEST Python version: 3.6.5 R8MAT_PRINT_SOME prints some of an R8MAT. Here is an R8MAT: Col: 3 4 5 Row 0 : 14 15 16 1 : 24 25 26 2 : 34 35 36 R8MAT_PRINT_SOME_TEST: Normal end of execution. R8VEC_PRINT_TEST Python version: 3.6.5 R8VEC_PRINT prints an R8VEC. Here is an R8VEC: 0: 123.456 1: 5e-06 2: -1e+06 3: 3.14159 R8VEC_PRINT_TEST: Normal end of execution. R8VEC2_PRINT_TEST Python version: 3.6.5 R8VEC2_PRINT prints a pair of R8VEC's. Print a pair of R8VEC's: 0: 0 0 1: 0.2 0.04 2: 0.4 0.16 3: 0.6 0.36 4: 0.8 0.64 5: 1 1 R8VEC2_PRINT_TEST: Normal end of execution. LEGENDRE_POLYNOMIAL_TEST: Normal end of execution. Thu Sep 13 10:33:17 2018