QUADRULE_FAST
Efficient Quadrature Rule Implementation

QUADRULE_FAST is a MATLAB program, using double precision arithmetic, which implements fast and efficient forms of several popular quadrature rules.

The quadrature rules are defined on the interval [-1,1], and assume there is no additional weighting factor in the data.

The fast implementations are exhibited and discussed in the the papers by Trefethen and Waldvogel.

Related Data and Programs:

CC_DISPLAY is a MATLAB program which can display various 2D Clenshaw Curtis grids.

CLENSHAW_CURTIS is a MATLAB program which can compute Clenshaw Curtis rules for 1 dimensional or multidimensional problems.

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

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

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

NINTLIB is a MATLAB library containing a variety of routines for numerical estimation of integrals in multiple dimensions.

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 containing a variety of routines for numerical estimation of integrals in 1D.

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 a MATLAB library containing quadrature rules.

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

TEST_INT is a FORTRAN90 library containing 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.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
2. Charles Clenshaw, Alan Curtis,
   A Method for Numerical Integration on an Automatic Computer,
   Numerische Mathematik,
   Volume 2, Number 1, December 1960, pages 197-205.
3. Lloyd Trefethen,
   Is Gauss Quadrature Better Than Clenshaw-Curtis?,
   SIAM Review,
   to appear.
4. Joerg Waldvogel,
   Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
   BIT Numerical Mathematics,
   Volume 43, Number 1, 2003, pages 1-18.

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

Source Code:

• clenshaw_curtis_integrate.m applies a Clenshaw Curtis quadrature rule.
• clenshaw_curtis_integrate_fast.m applies a Clenshaw Curtis quadrature rule, using an efficient MATLAB implementation.
• clenshaw_curtis_rule_compute.m computes a Clenshaw Curtis quadrature rule.
• clenshaw_curtis_rule_set.m sets a Clenshaw Curtis quadrature rule by looking it up in a table.
• fejer1_integrate_fast.m applies a Fejer type 1 quadrature rule, using an efficient MATLAB implementation.
• fejer1_rule_compute.m computes a Fejer type 1 quadrature rule.
• fejer1_rule_set.m sets a Fejer type 1 quadrature rule by looking it up in a table.
• fejer2_integrate_fast.m applies a Fejer type 2 quadrature rule, using an efficient MATLAB implementation.
• fejer2_rule_compute.m computes a Fejer type 2 quadrature rule.
• fejer2_rule_set.m sets a Fejer type 2 quadrature rule by looking it up in a table.
• gauss_legendre_integrate_fast.m applies a Gauss-Legendre quadrature rule, using an efficient MATLAB implementation.
• gauss_legendre_rule_compute.m computes a Gauss Legendre quadrature rule.
• gauss_legendre_rule_set.m sets a Gauss Legendre quadrature rule by looking it up in a table.
• timestamp.m prints the current YMDHMS date as a time stamp.

Examples and Tests:

• quadrule_fast_test.m, a sample calling program.
• quadrule_fast_test.out, the output from a run of the sample program.
• quadrule_fast_test01.m, compares CLENSHAW_CURTIS_RULE_COMPUTE and CLENSHAW_CURTIS_RULE_SET.
• quadrule_fast_test02.m, tests CLENSHAW_CURTIS_RULE_COMPUTE.
• quadrule_fast_test03.m, tests CLENSHAW_CURTIS_INTEGRATE and CLENSHAW_CURTIS_INTEGRATE_FAST.
• quadrule_fast_test04.m, compares GAUSS_LEGENDRE_RULE_COMPUTE and GAUSS_LEGENDRE_RULE_SET.
• quadrule_fast_test05.m, tests GAUSS_LEGENDRE_RULE_COMPUTE.
• quadrule_fast_test06.m, tests GAUSS_LEGENDRE_INTEGRATE_FAST.
• quadrule_fast_test07.m, compares FEJER1_RULE_COMPUTE and FEJER1_RULE_SET.
• quadrule_fast_test08.m, tests FEJER1_RULE_COMPUTE.
• quadrule_fast_test09.m, tests FEJER1_INTEGRATE_FAST.
• quadrule_fast_test10.m, compares FEJER2_RULE_COMPUTE and FEJER2_RULE_SET.
• quadrule_fast_test11.m, tests FEJER2_RULE_COMPUTE.
• quadrule_fast_test12.m, tests FEJER2_INTEGRATE_FAST.
• f1.m, test integrand function #1.
• f2.m, test integrand function #2.
• f3.m, test integrand function #3.

You can go up one level to the MATLAB source codes.