QUADRULE
Quadrature Rules

QUADRULE is a C library which sets up a variety of quadrature rules, used to approximate the integral of a function over various domains.

QUADRULE returns the abscissas and weights for a variety of one dimensional quadrature rules for approximating the integral of a function. The best rule is generally Gauss-Legendre quadrature, but other rules offer special features, including the ability to handle certain weight functions, to approximate an integral on an infinite integration region, or to estimate the approximation error.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

QUADRULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Programs:

ALPERT_RULE, a C library which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

CCN_RULE, a C program which defines one of a set of nested Clenshaw Curtis quadrature rules.

CLENSHAW_CURTIS_RULE, a C program which defines a Clenshaw Curtis quadrature rule.

KRONROD, a C library which can compute a Gauss and Gauss-Kronrod pair of quadrature rules of arbitrary order, by Robert Piessens, Maria Branders.

LAGUERRE_RULE, a C program which can compute and print a Gauss-Laguerre quadrature rule for estimating the integral of a function with density exp(-x) over the interval [0,+oo).

LEGENDRE_RULE, a C program which computes a 1D Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, a C program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

LINE_FELIPPA_RULE, a C library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

LINE_NCO_RULE, a C library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

PATTERSON_RULE, a C program which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

POWER_RULE, a C program which constructs a power rule, that is, a product quadrature rule from identical 1D factor rules.

PRODUCT_RULE, a C program which constructs a product quadrature rule from distinct 1D factor rules.

QUADMOM, a C library which computes a Gaussian quadrature rule for a weight function rho(x) based on the Golub-Welsch procedure that only requires knowledge of the moments of rho(x).

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_TET, a dataset directory which contains triples of files defining various quadrature rules on tetrahedrons.

QUADRATURE_RULES_TRI, a dataset directory which contains quadrature rules to be applied to triangular regions.

SANDIA_RULES, a C library which produces 1D quadrature rules of Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite, generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.

SGMGA, a C library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.

SPARSE_GRID_HW, a C library which creates sparse grids based on Gauss-Legendre, Gauss-Hermite, Gauss-Patterson, or a nested variation of Gauss-Hermite rules, by Florian Heiss and Viktor Winschel.

STROUD, a C library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_INT, a C library which defines test integrands for 1D quadrature rules.

Source Code:

• quadrule.c, the source code;
• quadrule.h, the include file;

Examples and Tests:

• quadrule_test.c, the calling program;
• quadrule_test.txt, the output file.

List of Routines:

• CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
• CGQF computes knots and weights of a Gauss quadrature formula.
• CHEBYSHEV_SET sets abscissas and weights for Chebyshev quadrature.
• CHEBYSHEV1_COMPUTE computes a Gauss-Chebyshev type 1 quadrature rule.
• CHEBYSHEV1_INTEGRAL evaluates a monomial Chebyshev type 1 integral.
• CHEBYSHEV1_SET sets a Chebyshev Type 1 quadrature rule.
• CHEBYSHEV2_COMPUTE computes a Gauss-Chebyshev type 2 quadrature rule.
• CHEBYSHEV2_INTEGRAL evaluates a monomial Chebyshev type 2 integral.
• CHEBYSHEV2_SET sets a Chebyshev Type 2 quadrature rule.
• CHEBYSHEV3_COMPUTE computes a Gauss-Chebyshev type 3 quadrature rule.
• CHEBYSHEV3_INTEGRAL evaluates a monomial Chebyshev type 3 integral.
• CHEBYSHEV3_SET sets a Chebyshev Type 3 quadrature rule.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• CLENSHAW_CURTIS_COMPUTE computes a Clenshaw Curtis quadrature rule.
• CLENSHAW_CURTIS_SET sets a Clenshaw-Curtis quadrature rule.
• FEJER1_COMPUTE computes a Fejer type 1 quadrature rule.
• FEJER1_SET sets abscissas and weights for Fejer type 1 quadrature.
• FEJER2_COMPUTE computes a Fejer type 2 quadrature rule.
• FEJER2_SET sets abscissas and weights for Fejer type 2 quadrature.
• GEGENBAUER_INTEGRAL: the integral of a monomial with Gegenbauer weight.
• GEGENBAUER_EK_COMPUTE computes a Gauss-Gegenbauer quadrature rule.
• GEGENBAUER_SS_COMPUTE computes a Gauss-Gegenbauer quadrature rule.
• GEGENBAUER_SS_RECUR: value and derivative of a Gegenbauer polynomial.
• GEGENBAUER_SS_ROOT improves an approximate root of a Gegenbauer polynomial.
• GEN_HERMITE_DR_COMPUTE: generalized Gauss-Hermite rule by Davis, Rabinowitz.
• GEN_HERMITE_EK_COMPUTE: generalized Gauss-Hermite by Elhay-Kautsky.
• GEN_HERMITE_INTEGRAL evaluates a monomial generalized Hermite integral.
• GEN_LAGUERRE_EK_COMPUTE: generalized Gauss-Laguerre quadrature rule.
• GEN_LAGUERRE_INTEGRAL evaluates a monomial generalized Laguerre integral.
• GEN_LAGUERRE_SS_COMPUTE computes a generalized Gauss-Laguerre quadrature rule.
• GEN_LAGUERRE_SS_RECUR evaluates a generalized Laguerre polynomial.
• GEN_LAGUERRE_SS_ROOT improves a root of a generalized Laguerre polynomial.
• HERMITE_EK_COMPUTE computes a Gauss-Hermite quadrature rule.
• HERMITE_GK16_SET sets a Hermite Genz-Keister 16 rule.
• HERMITE_GK18_SET sets a Hermite Genz-Keister 18 rule.
• HERMITE_GK22_SET sets a Hermite Genz-Keister 22 rule.
• HERMITE_GK24_SET sets a Hermite Genz-Keister 24 rule.
• HERMITE_INTEGRAL evaluates a monomial Hermite integral.
• HERMITE_PROBABILIST_SET: probabilist Hermite quadrature.
• HERMITE_SET sets abscissas and weights for Hermite quadrature.
• HERMITE_1_SET sets abscissas and weights for Hermite quadrature.
• HERMITE_SS_COMPUTE computes a Gauss-Hermite quadrature rule.
• HERMITE_SS_RECUR finds the value and derivative of a Hermite polynomial.
• HERMITE_SS_ROOT improves an approximate root of a Hermite polynomial.
• I4_FACTORIAL2 computes the double factorial function N!!
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_POWER returns the value of I^J.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• JACOBI_EK_COMPUTE: Elhay-Kautsky method for Gauss-Jacobi quadrature rule.
• JACOBI_INTEGRAL evaluates the integral of a monomial with Jacobi weight.
• JACOBI_SS_COMPUTE computes a Gauss-Jacobi quadrature rule.
• JACOBI_SS_RECUR finds the value and derivative of a Jacobi polynomial.
• JACOBI_SS_ROOT improves an approximate root of a Jacobi polynomial.
• KRONROD_SET sets abscissas and weights for Gauss-Kronrod quadrature.
• LAGUERRE_EK_COMPUTE: Laguerre quadrature rule by the Elhay-Kautsky method.
• LAGUERRE_INTEGRAL evaluates a monomial Laguerre integral.
• LAGUERRE_SS_COMPUTE computes a Gauss-Laguerre quadrature rule.
• LAGUERRE_SS_RECUR evaluates a Laguerre polynomial.
• LAGUERRE_SS_ROOT improves a root of a Laguerre polynomial.
• LAGUERRE_SET sets abscissas and weights for Laguerre quadrature.
• LAGUERRE_1_SET sets abscissas and weights for Laguerre quadrature.
• LAGUERRE_SUM carries out Laguerre quadrature over [A, +oo ).
• LEGENDRE_DR_COMPUTE: Gauss-Legendre quadrature by Davis-Rabinowitz method.
• LEGENDRE_EK_COMPUTE: Legendre quadrature rule by the Elhay-Kautsky method.
• LEGENDRE_INTEGRAL evaluates a monomial Legendre integral.
• LEGENDRE_RECUR finds the value and derivative of a Legendre polynomial.
• LEGENDRE_SET sets abscissas and weights for Gauss-Legendre quadrature.
• LOBATTO_COMPUTE computes a Lobatto quadrature rule.
• LOBATTO_SET sets abscissas and weights for Lobatto quadrature.
• NC_COMPUTE_WEIGHTS computes weights for a Newton-Cotes quadrature rule.
• NCC_COMPUTE computes a Newton-Cotes Closed quadrature rule.
• NCC_SET sets abscissas and weights for closed Newton-Cotes quadrature.
• NCO_COMPUTE computes a Newton-Cotes Open quadrature rule.
• NCO_SET sets abscissas and weights for open Newton-Cotes quadrature.
• NCOH_COMPUTE computes a Newton-Cotes "open half" quadrature rule.
• NCOH_SET sets abscissas and weights for Newton-Cotes "open half" rules.
• PARCHK checks parameters ALPHA and BETA for classical weight functions.
• PATTERSON_SET sets abscissas and weights for Gauss-Patterson quadrature.
• PSI_VALUES returns some values of the Psi or Digamma function.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 round off unit.
• R8_FACTORIAL computes the factorial of N, also denoted "N!".
• R8_FACTORIAL2 computes the double factorial function N!!
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HUGE returns a "huge" R8.
• R8_HYPER_2F1 evaluates the hypergeometric function 2F1(A,B,C,X).
• R8_MAX returns the maximum of two R8's.
• R8_PSI evaluates the function Psi(X).
• R8_SIGN returns the sign of an R8.
• R8VEC_COPY copies an R8VEC.
• R8VEC_DOT_PRODUCT computes the dot product of a pair of R8VEC's.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• R8VEC_PRINT prints an R8VEC.
• R8VEC_REVERSE reverses the elements of an R8VEC.
• RADAU_COMPUTE computes a Radau quadrature rule.
• RADAU_SET sets abscissas and weights for Radau quadrature.
• SCQF scales a quadrature formula to a nonstandard interval.
• SGQF computes knots and weights of a Gauss Quadrature formula.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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