QUADRULE
Quadrature Rules

QUADRULE is a library of MATLAB routines, using double precision arithmetic, which set 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.

Related Data and Programs:

CLENSHAW_CURTIS is a MATLAB library which can set up a Clenshaw Curtis quadrature grid in multiple dimensions.

DUNAVANT is a MATLAB library of routines for defining Dunavant rules for quadrature on a triangle.

FEKETE is a MATLAB library of routines for defining a Fekete rule for quadrature or interpolation over a triangle.

INTLIB is a FORTRAN90 library of routines for numerical estimation of integrals in 1D.

KEAST is a MATLAB library which defines a number of quadrature rules for a tetrahedron.

NCC_TETRAHEDRON is a MATLAB library defining Newton-Cotes closed quadrature rules on a tetrahedron.

NCC_TRIANGLE is a MATLAB library defining Newton-Cotes closed quadrature rules on a triangle.

NCO_TETRAHEDRON is a MATLAB library defining Newton-Cotes open quadrature rules on a tetrahedron.

NCO_TRIANGLE is a MATLAB library defining Newton-Cotes open quadrature rules on a triangle.

NINT_EXACTNESS is a MATLAB program which demonstrates how to measure the polynomial exactness of a multidimensional quadrature rule.

NINT_EXACTBESS_TRI is an executable MATLAB program which investigates the polynomial exactness of a quadrature rule for the triangle.

NINTLIB is a MATLAB library contains of routines for numerical estimation of integrals in ND.

PRODUCT_RULE is an executable MATLAB program which can create a multidimensional quadrature rule as a product of one dimensional rules.

QUADPACK is a FORTRAN90 library of routines for numerical estimation of integrals in 1D.

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

QUADRATURE_RULES_TET is a dataset directory of triples of files defining various quadrature rules on tetrahedrons.

QUADRATURE_RULES_TRI is a collection of quadrature rules to be applied to triangular regions.

QUADRATURE_TEST an executable 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.

QUADRULE is also available in a C++ version and a FORTRAN90 version.

QUADRULE_FAST is a MATLAB library of routines defining efficient versions of a few 1D quadrature rules.

STROUD is a MATLAB library of routines defining quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_INT is a FORTRAN90 library of routines defining a number of functions that may be used as test integrands for quadrature rules in 1D.

TEST_TRI_INT is a MATLAB library of functions that can be used to test algorithms for quadrature over a triangle.

TESTPACK is a MATLAB library of functions which define a set of integrands used to test multidimensional quadrature.

TOMS351 is a FORTRAN77 library which estimates an integral using Romberg integration.

TOMS379 is a FORTRAN77 library which estimates an integral.

TOMS418 is a FORTRAN77 library which estimates the integral of a function with a sine or cosine factor.

TOMS424 is a FORTRAN77 library which estimates the integral of a function using Clenshaw-Curtis quadrature.

TOMS468 is a FORTRAN77 library for the "automatic" integration of a function.

WANDZURA is a MATLAB library of routines for defining 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. Francis Hildebrand,
    Introduction to Numerical Analysis,
    Dover, 1987,
    ISBN13: 978-0486653631,
    LC: QA300.H5.
11. Zdenek Kopal,
    Numerical Analysis,
    John Wiley, 1955,
    LC: QA297.K6.
12. Vladimir Krylov,
    Approximate Calculation of Integrals,
    Dover, 2006,
    ISBN: 0486445798,
    LC: QA311.K713.
13. Prem Kythe, Michael Schaeferkotter,
    Handbook of Computational Methods for Integration,
    Chapman and Hall, 2004,
    ISBN: 1-58488-428-2,
    LC: QA299.3.K98.
14. 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
15. Thomas Patterson,
    The Optimal Addition of Points to Quadrature Formulae,
    Mathematics of Computation,
    Volume 22, Number 104, October 1968, pages 847-856.
16. Robert Piessens, Elise deDoncker-Kapenga, Christian Ueberhuber, David Kahaner,
    QUADPACK: A Subroutine Package for Automatic Integration,
    Springer, 1983,
    ISBN: 3540125531,
    LC: QA299.3.Q36.
17. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.
18. LLoyd Trefethen,
    Is Gauss Quadrature Better Than Clenshaw-Curtis?,
    SIAM Review,
    to appear.
19. Joerg Waldvogel,
    Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
    BIT Numerical Mathematics,
    Volume 43, Number 1, 2003, pages 1-18.
20. Stephen Wolfram,
    The Mathematica Book,
    Fourth Edition,
    Cambridge University Press, 1999,
    ISBN: 0-521-64314-7,
    LC: QA76.95.W65.
21. Daniel Zwillinger, editor,
    CRC Standard Mathematical Tables and Formulae,
    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.

Tar File:

A GZIP'ed TAR file of the contents of this directory is available. This is only done as a convenience for users who want ALL the files, and don't want to download them individually. This is not a convenience for me, so don't be surprised if the tar file is somewhat out of date.

• quadrule_m_src.tar.csh the commands used to create the GZIP'ed TAR file.
• quadrule_m_src.tar.gz the GZIP'ed TAR file.

Source Code:

• bashforth_set.m, sets abscissas and weights for Adams-Bashforth quadrature.
• bdf_set.m, sets weights for backward differentiation ODE weights.
• bdfc_set.m, sets weights for backward differentiation corrector quadrature.
• bdfp_set.m, sets weights for backward differentiation predictor quadrature.
• bdf_sum.m, carries out an explicit backward difference quadrature rule for [0,1].
• cheb_set.m, sets abscissas and weights for Chebyshev quadrature.
• chebyshev1_compute.m computes a Gauss-Chebyshev type 1 quadrature rule.
• chebyshev1_integral.m evaluates a monomial Chebyshev type 1 integral.
• chebyshev2_compute.m computes a Gauss-Chebyshev type 2 quadrature rule.
• chebyshev2_integral.m evaluates a monomial Chebyshev type 2 integral.
• chebyshev3_compute.m, computes a Gauss-Chebyshev type 3 quadrature rule.
• clenshaw_curtis_compute.m, computes a Clenshaw-Curtis quadrature rule.
• clenshaw_curtis_set.m, sets up Clenshaw-Curtis quadrature.
• fejer1_compute.m, computes a Fejer type 1 quadrature rule.
• fejer1_set.m, sets a Fejer type 1 quadrature rule.
• fejer2_compute.m, computes a Fejer type 2 quadrature rule.
• fejer2_set.m, sets a Fejer type 2 quadrature rule.
• gegenbauer_compute.m computes a Gauss-Gegenbauer quadrature rule.
• gegenbauer_integral.m evaluates the integral of a monomial with Gegenbauer weight.
• gegenbauer_recur.m finds the value and derivative of a Gegenbauer polynomial.
• gegenbauer_root.m improves an approximate root of a Gegenbauer polynomial.
• gen_hermite_compute.m, computes a generalized Gauss-Hermite quadrature rule.
• gen_hermite_integral.m, evaluates a monomial generalized Hermite integral.
• gen_laguerre_compute.m, computes a generalized Gauss-Laguerre quadrature rule.
• gen_laguerre_integral.m, evaluates a monomial generalized Laguerre integral.
• gen_laguerre_recur.m, finds the value and derivative of a generalized Laguerre polynomial.
• gen_laguerre_root.m, improves an approximate root of a generalized Laguerre polynomial.
• hermite_compute.m, computes a Gauss-Hermite quadrature rule.
• hermite_integral.m, returns the value of a Hermite integral.
• hermite_recur.m, finds the value and derivative of a Hermite polynomial.
• hermite_root.m, improves an approximate root of a Hermite polynomial.
• hermite_set.m, sets abscissas and weights for Hermite quadrature.
• jacobi_compute.m, computes a Gauss-Jacobi quadrature rule.
• jacobi_integral.m, evaluates the integral of a monomial with Jacobi weight.
• jacobi_recur.m, finds the value and derivative of a Jacobi polynomial.
• jacobi_root.m, improves an approximate root of a Jacobi polynomial.
• kronrod_set.m, sets abscissas and weights for Gauss-Kronrod quadrature.
• laguerre_compute.m, computes a Gauss-Laguerre quadrature rule.
• laguerre_integral.m, evaluates a monomial Laguerre integral.
• laguerre_recur.m, finds the value and derivative of a Laguerre polynomial.
• laguerre_root.m, improves an approximate root of a Laguerre polynomial.
• laguerre_set.m, sets abscissas and weights for Laguerre quadrature.
• laguerre_sum.m, carries out Laguerre quadrature over [ A, +Infinity ).
• legendre_compute.m, computes a Gauss-Legendre quadrature rule.
• legendre_integral.m, evaluates a monomial Legendre integral.
• legendre_recur.m, finds the value and derivative of a Legendre polynomial.
• legendre_set.m, sets abscissas and weights for Gauss-Legendre quadrature.
• legendre_set_cos.m, sets a Gauss-Legendre rule for COS(X) * F(X) on [-PI/2,PI/2].
• legendre_set_cos2.m, sets a Gauss-Legendre rule for COS(X) * F(X) on [0,PI/2].
• legendre_set_log.m, sets a Gauss-Legendre rule for - LOG(X) * F(X) on [0,1].
• legendre_set_sqrtx_01.m, sets a Gauss-Legendre rule for SQRT(X) * F(X) on [0,1].
• legendre_set_sqrtx2_01.m, sets a Gauss-Legendre rule for F(X) / SQRT(X) on [0,1].
• legendre_set_x0_01.m, sets a Gauss-Legendre rule for F(X) on [0,1].
• legendre_set_x1.m, sets a Gauss-Legendre rule for ( 1 + X ) * F(X) on [-1,1].
• legendre_set_x1_01.m, sets a Gauss-Legendre rule for X * F(X) on [0,1].
• legendre_set_x2.m, sets a Gauss-Legendre rule for ( 1 + X )**2 * F(X) on [-1,1].
• legendre_set_x2_01.m, sets a Gauss-Legendre rule for X**2 * F(X) on [0,1].
• lobatto_compute.m, computes abscissas and weights for Lobatto quadrature.
• lobatto_set.m, sets abscissas and weights for Lobatto quadrature.
• log_gamma.m, calculates the natural logarithm of GAMMA(X).
• moulton_set.m, sets weights for Adams-Moulton quadrature.
• nc_compute.m, computes a Newton-Cotes quadrature rule.
• ncc_compute.m, computes a Newton-Cotes closed quadrature rule.
• ncc_set.m, sets abscissas and weights for closed Newton-Cotes quadrature.
• nco_compute.m, computes a Newton-Cotes open quadrature rule.
• nco_set.m, sets abscissas and weights for open Newton-Cotes quadrature.
• ncoh_compute.m, computes a Newton-Cotes "open half" quadrature rule.
• ncoh_set.m, sets abscissas and weights for "open half" Newton-Cotes quadrature.
• patterson_set.m, sets abscissas and weights for Patterson quadrature.
• r8_epsilon.m, returns the R8 roundoff unit.
• r8_factorial.m, evaluates the factorial function.
• r8_factorial2.m, evaluates the double factorial function.
• r8_gamma.m, computes the gamma function.
• r8_hyper_2f1.m, evaluates the hypergeometric function F(A,B,C,X).
• r8_psi.m, evaluates the Psi function.
• r8vec_reverse.m, reverses the elements of an R8VEC.
• radau_compute.m, computes abscissas and weights for Radau quadrature.
• radau_set.m, sets abscissas and weights for Radau quadrature.
• rule_adjust.m, maps a quadrature rule from [A,B] to [C,D].
• summer.m, carries out a quadrature rule over a single interval.
• summer_gk.m, carries out Gauss-Kronrod quadrature over a single interval.
• sum_sub.m, carries out a composite quadrature rule.
• sum_sub_gk.m, carries out a composite Gauss-Kronrod rule.
• timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• quadrule_test.m, calls all the tests;
• quadrule_test_output.txt, the output from a rule of all the tests;
• quadrule_test01.m, tests BASHFORTH_SET and SUMMER;
• quadrule_test02.m, tests BDFC_SET and BDF_SUM;
• quadrule_test03.m, tests BDFP_SET and BDF_SUM;
• quadrule_test04.m, tests CHEB_SET and SUM_SUB.
• quadrule_test05.m, tests CHEBYSHEV1_COMPUTE and SUMMER.
• quadrule_test06.m, tests CHEBYSHEV2_COMPUTE and SUMMER.
• quadrule_test07.m, tests CHEBYSHEV3_COMPUTE and SUMMER.
• quadrule_test0725.m, tests CLENSHAW_CURTIS_COMPUTE.
• quadrule_test075.m, tests CLENSHAW_CURTIS_SET and SUMMER.
• quadrule_test076.m, compares FEJER1_SET and FEJER1_COMPUTE.
• quadrule_test078.m, compares FEJER2_SET and FEJER2_COMPUTE.
• quadrule_test08.m, tests HERMITE_COMPUTE and SUMMER.
• quadrule_test085.m, tests HERMITE_COMPUTE and HERMITE_INTEGRAL.
• quadrule_test087.m, tests HERMITE_SET.
• quadrule_test09.m, tests HERMITE_SET and SUMMER.
• quadrule_test10.m, tests JACOBI_COMPUTE and SUM_SUB.
• quadrule_test105.m, tests JACOBI_COMPUTE.
• quadrule_test108.m, tests JACOBI_COMPUTE.
• quadrule_test11.m, tests KRONROD_SET, LEGENDRE_SET and SUMMER_GK.
• quadrule_test12.m, tests KRONROD_SET, LEGENDRE_SET and SUM_SUB_GK.
• quadrule_test13.m, tests LAGUERRE_COMPUTE and LAGUERRE_SUM.
• quadrule_test14.m, tests LAGUERRE_COMPUTE and LAGUERRE_SUM.
• quadrule_test15.m, tests LAGUERRE_COMPUTE and LAGUERRE_SUM.
• quadrule_test16.m, tests LAGUERRE_COMPUTE and LAGUERRE_SUM.
• quadrule_test165.m, tests LAGUERRE_COMPUTE.
• quadrule_test17.m, tests LAGUERRE_SET and LAGUERRE_SUM.
• quadrule_test18.m, tests LEGENDRE_COMPUTE and LEGENDRE_SET.
• quadrule_test185.m, tests LEGENDRE_COMPUTE.
• quadrule_test19.m, tests LEGENDRE_COMPUTE and SUM_SUB.
• quadrule_test20.m, tests LEGENDRE_COMPUTE and SUM_SUB.
• quadrule_test21.m, tests LEGENDRE_SET and SUM_SUB.
• quadrule_test22.m, tests LEGENDRE_SET, LEGENDRE_SET_X0_01 and RULE_ADJUST.
• quadrule_test23.m, tests LEGENDRE_SET_COS and SUM_SUB.
• quadrule_test24.m, tests LEGENDRE_SET_SQRTX_01 and SUM_SUB.
• quadrule_test25.m, tests LEGENDRE_SET_SQRTX2_01 and SUM_SUB.
• quadrule_test26.m, tests LEGENDRE_SET_COS2 and SUM_SUB.
• quadrule_test27.m, tests LEGENDRE_SET_LOG and SUM_SUB.
• quadrule_test28.m, tests LEGENDRE_SET_X0_01 and SUM_SUB.
• quadrule_test29.m, tests LEGENDRE_SET_X1 and SUM_SUB.
• quadrule_test30.m, tests LEGENDRE_SET_X1, LEGENDRE_SET_X1_01 and RULE_ADJUST.
• quadrule_test31.m, tests LEGENDRE_SET_X1_01 and SUM_SUB.
• quadrule_test32.m, tests LEGENDRE_SET_X2 and SUM_SUB
• quadrule_test33.m, tests LEGENDRE_SET_X2, LEGENDRE_SET_X2_01 and RULE_ADJUST.
• quadrule_test34.m, tests LEGENDRE_SET_X2_01 and SUM_SUB
• quadrule_test345.m, tests LOBATTO_COMPUTE and LOBATTO_SET.
• quadrule_test35.m, tests LOBATTO_SET and SUM_SUB.
• quadrule_test36.m, tests MOULTON_SET and SUMMER.
• quadrule_test37.m, tests NCC_SET and SUM_SUB.
• quadrule_test38.m, tests NCC_COMPUTE and SUM_SUB.
• quadrule_test39.m, tests NCO_SET and SUM_SUB.
• quadrule_test40.m, tests NCO_COMPUTE and SUM_SUB.
• quadrule_test401.m, tests NCOH_SET and SUM_SUB.
• quadrule_test402.m, tests NCOH_COMPUTE and SUM_SUB.
• quadrule_test403.m, tests PATTERSON_SET and SUM_SUB.
• quadrule_test404.m, tests RADAU_COMPUTE and RADAU_SET.
• quadrule_test41.m, tests RADAU_SET and SUM_SUB.
• f1sd1.m, evaluates the function 1.0D+00/ sqrt ( 1.1 - x**2 ).
• fxsd1.m, evaluates the function x / sqrt ( 1.1 - x**2 ).
• fx2sd1.m, evaluates the function x**2 / sqrt ( 1.1 - x**2 ).
• func.m, evaluates a function of X, as chosen by the user.
• func_set.m, sets the function to be returned by FUNC.
• fname.m, returns the name of the function that will be evaluated in FUNC.

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