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 FORTRAN77 version and a FORTRAN90 version and a MATHEMATICA version and a MATLAB version.

Related Programs:

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

DUNAVANT, a C++ library which defines Dunavant rules for quadrature on a triangle.

FEKETE, a C++ library which defines a Fekete rule for quadrature or interpolation over a triangle.

FELIPPA, a C++ library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

INTLIB, a FORTRAN90 library which numerically estimates integrals.

KEAST, a C++ library which defines a number of quadrature rules for a tetrahedron.

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

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

NCC_TETRAHEDRON, a C++ library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

NCC_TRIANGLE, a C++ library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TETRAHEDRON, a C++ library which defines Newton-Cotes open quadrature rules on a tetrahedron.

NCO_TRIANGLE, a C++ library which defines Newton-Cotes open quadrature rules on a triangle.

NINT_EXACTNESS, a C++ program which demonstrates how to measure the polynomial exactness of a multidimensional quadrature rule.

NINTLIB, a C++ library which contains a variety of routines for numerical estimation of integrals in multiple dimensions.

PRODUCT_RULE, a C++ program which can create a multidimensional quadrature rule as a product of one dimensional rules.

QUADPACK, a FORTRAN90 library which contains a variety of routines for numerical estimation of integrals in 1D.

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.

QUADRATURE_TEST, a MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.

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.

TEST_INT_2D, a C library which defines test integrands for 2D quadrature rules.

TEST_INT_LAGUERRE, a C library which defines test integrands for the interval [a,+oo) and weight exp(-x);

TEST_NINT, a C++ library which defines a number of functions that may be used as test integrands for quadrature rules in multiple dimensions.

TEST_TRI_INT, a C++ library which can be used to test algorithms for quadrature over a triangle.

TRIANGLE_EXACTNESS, a C++ program which investigates the polynomial exactness of a quadrature rule for the triangle.

WANDZURA, a C++ library which defines Wandzura rules for quadrature on a triangle.

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Claudio Canuto, Yousuff Hussaini, Alfio Quarteroni, Thomas Zang,
   Spectral Methods in Fluid Dynamics,
   Springer, 1993,
   ISNB13: 978-3540522058,
   LC: QA377.S676.
3. Charles Clenshaw, Alan Curtis,
   A Method for Numerical Integration on an Automatic Computer,
   Numerische Mathematik,
   Volume 2, Number 1, December 1960, pages 197-205.
4. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
5. Sylvan Elhay, Jaroslav Kautsky,
   Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
   ACM Transactions on Mathematical Software,
   Volume 13, Number 4, December 1987, pages 399-415.
6. Hermann Engels,
   Numerical Quadrature and Cubature,
   Academic Press, 1980,
   ISBN: 012238850X,
   LC: QA299.3E5.
7. Gwynne Evans,
   Practical Numerical Integration,
   Wiley, 1993,
   ISBN: 047193898X,
   LC: QA299.3E93.
8. Simeon Fatunla,
   Numerical Methods for Initial Value Problems in Ordinary Differential Equations,
   Academic Press, 1988,
   ISBN: 0122499301,
   LC: QA372.F35.
9. Walter Gautschi,
   Numerical Quadrature in the Presence of a Singularity,
   SIAM Journal on Numerical Analysis,
   Volume 4, Number 3, September 1967, pages 357-362.
10. Alan Genz, Bradley Keister,
    Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight,
    Journal of Computational and Applied Mathematics,
    Volume 71, 1996, pages 299-309.
11. Florian Heiss, Viktor Winschel,
    Likelihood approximation by numerical integration on sparse grids,
    Journal of Econometrics,
    Volume 144, 2008, pages 62-80.
12. Francis Hildebrand,
    Introduction to Numerical Analysis,
    Dover, 1987,
    ISBN13: 978-0486653631,
    LC: QA300.H5.
13. Zdenek Kopal,
    Numerical Analysis,
    John Wiley, 1955,
    LC: QA297.K6.
14. Vladimir Krylov,
    Approximate Calculation of Integrals,
    Dover, 2006,
    ISBN: 0486445798,
    LC: QA311.K713.
15. Prem Kythe, Michael Schaeferkotter,
    Handbook of Computational Methods for Integration,
    Chapman and Hall, 2004,
    ISBN: 1-58488-428-2,
    LC: QA299.3.K98.
16. Leon Lapidus, John Seinfeld,
    Numerical Solution of Ordinary Differential Equations,
    Mathematics in Science and Engineering, Volume 74,
    Academic Press, 1971,
    ISBN: 0124366503,
    LC: QA3.M32.v74
17. Thomas Patterson,
    The Optimal Addition of Points to Quadrature Formulae,
    Mathematics of Computation,
    Volume 22, Number 104, October 1968, pages 847-856.
18. Robert Piessens, Elise deDoncker-Kapenga, Christian Ueberhuber, David Kahaner,
    QUADPACK: A Subroutine Package for Automatic Integration,
    Springer, 1983,
    ISBN: 3540125531,
    LC: QA299.3.Q36.
19. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.
20. Lloyd Trefethen,
    Is Gauss Quadrature Better Than Clenshaw-Curtis?,
    SIAM Review,
    Volume 50, Number 1, March 2008, pages 67-87.
21. Joerg Waldvogel,
    Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
    BIT Numerical Mathematics,
    Volume 43, Number 1, 2003, pages 1-18.
22. Stephen Wolfram,
    The Mathematica Book,
    Fourth Edition,
    Cambridge University Press, 1999,
    ISBN: 0-521-64314-7,
    LC: QA76.95.W65.
23. Daniel Zwillinger, editor,
    CRC Standard Mathematical Tables and Formulae,
    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.

Source Code:

• quadrule.c, the source code;
• quadrule.h, the include file;
• quadrule.sh, commands to compile the source code;

Examples and Tests:

• quadrule_prb.c, the calling program;
• quadrule_prb.sh, commands to compile, link and run the calling program;
• quadrule_prb_output.txt, the output file.

List of Routines:

• BASHFORTH_SET sets abscissas and weights for Adams-Bashforth quadrature.
• BDF_SET sets weights for backward differentiation ODE weights.
• BDFC_SET sets weights for backward differentiation corrector quadrature.
• BDFP_SET sets weights for backward differentiation predictor quadrature.
• BDF_SUM carries out explicit backward difference quadrature on [0,1].
• CH_CAP capitalizes a single character.
• CHEB_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.
• CHEBYSHEV2_COMPUTE computes a Gauss-Chebyshev type 2 quadrature rule.
• CHEBYSHEV2_INTEGRAL evaluates a monomial Chebyshev type 2 integral.
• CHEBYSHEV3_COMPUTE computes a Gauss-Chebyshev type 3 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_COMPUTE computes a Gauss-Gegenbauer quadrature rule.
• GEGENBAUER_INTEGRAL evaluates the integral of a monomial with Gegenbauer weight.
• GEGENBAUER_RECUR finds the value and derivative of a Gegenbauer polynomial.
• GEGENBAUER_ROOT improves an approximate root of a Gegenbauer polynomial.
• GEN_HERMITE_COMPUTE computes a generalized Gauss-Hermite rule.
• GEN_HERMITE_INTEGRAL evaluates a monomial generalized Hermite integral.
• GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre quadrature rule.
• GEN_LAGUERRE_INTEGRAL evaluates a monomial generalized Laguerre integral.
• GEN_LAGUERRE_RECUR evaluates a generalized Laguerre polynomial.
• GEN_LAGUERRE_ROOT improves a root of a generalized Laguerre polynomial.
• HERMITE_EK_COMPUTE computes a Gauss-Hermite quadrature rule.
• HERMITE_GENZ_KEISTER_SET sets a Hermite Genz-Keister rule.
• HERMITE_INTEGRAL evaluates a monomial Hermite integral.
• HERMITE_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_MIN returns the smaller of two I4's.
• I4_POWER returns the value of I^J.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• JACOBI_COMPUTE computes a Gauss-Jacobi quadrature rule.
• JACOBI_INTEGRAL evaluates the integral of a monomial with Jacobi weight.
• JACOBI_RECUR finds the value and derivative of a Jacobi polynomial.
• JACOBI_ROOT improves an approximate root of a Jacobi polynomial.
• KRONROD_SET sets abscissas and weights for Gauss-Kronrod quadrature.
• LAGUERRE_COMPUTE computes a Gauss-Laguerre quadrature rule.
• LAGUERRE_INTEGRAL evaluates a monomial Laguerre integral.
• LAGUERRE_RECUR evaluates a Laguerre polynomial.
• LAGUERRE_ROOT improves a root of a Laguerre polynomial.
• LAGUERRE_SET sets abscissas and weights for Laguerre quadrature.
• LAGUERRE_SUM carries out Laguerre quadrature over [ A, +oo ).
• LEGENDRE_COMPUTE_DR: Gauss-Legendre quadrature by Davis-Rabinowitz 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.
• LEGENDRE_SET_COS: Gauss-Legendre rules for COS(X)*F(X) on [-PI/2,PI/2].
• LEGENDRE_SET_COS2: Gauss-Legendre rules for COS(X)*F(X) on [0,PI/2].
• LEGENDRE_SET_LOG sets a Gauss-Legendre rule for - LOG(X) * F(X) on [0,1].
• LEGENDRE_SET_SQRTX_01 sets Gauss-Legendre rules for SQRT(X)*F(X) on [0,1].
• LEGENDRE_SET_SQRTX2_01: Gauss-Legendre rules for F(X)/SQRT(X) on [0,1].
• LEGENDRE_SET_X0_01 sets a Gauss-Legendre rule for F(X) on [0,1].
• LEGENDRE_SET_X1 sets a Gauss-Legendre rule for ( 1 + X ) * F(X) on [-1,1].
• LEGENDRE_SET_X1_01 sets a Gauss-Legendre rule for X * F(X) on [0,1].
• LEGENDRE_SET_X2 sets Gauss-Legendre rules for ( 1 + X )^2*F(X) on [-1,1].
• LEGENDRE_SET_X2_01 sets a Gauss-Legendre rule for X*X * F(X) on [0,1].
• LOBATTO_COMPUTE computes a Lobatto quadrature rule.
• LOBATTO_SET sets abscissas and weights for Lobatto quadrature.
• MOULTON_SET sets weights for Adams-Moulton quadrature.
• NC_COMPUTE computes a Newton-Cotes quadrature rule.
• NCC_COMPUTE computes a Newton-Cotes Closed quadrature rule.
• NCC_COMPUTE_POINTS: points of a Newton-Cotes Closed quadrature rule.
• NCC_COMPUTE_WEIGHTS: weights of 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_COMPUTE_POINTS: points of a Newton-Cotes Open quadrature rule.
• NCO_COMPUTE_WEIGHTS: weights of 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.
• PATTERSON_SET sets abscissas and weights for Gauss-Patterson quadrature.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff 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_REVERSE reverses the elements of an R8VEC.
• RADAU_COMPUTE computes a Radau quadrature rule.
• RADAU_SET sets abscissas and weights for Radau quadrature.
• RULE_ADJUST maps a quadrature rule from [A,B] to [C,D].
• S_EQI reports whether two strings are equal, ignoring case.
• SUM_SUB carries out a composite quadrature rule.
• SUMMER carries out a quadrature rule over a single interval.
• SUMMER_GK carries out Gauss-Kronrod quadrature over a single interval.
• SUM_SUB_GK carries out a composite Gauss-Kronrod rule.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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