CLENSHAW_CURTIS is a MATLAB library which sets up a Clenshaw Curtis quadrature rule in multiple dimensions.
Routines are available to look up or compute the weights and abscissas of the 1D rule.
It is easy to generate a Clenshaw Curtis grid of any order N. Of special interest are nested grids, particularly those for which the nesting involves repeated divisions of the interval length by 2. In such a case, the data computed from the previous step can be reused, and the new data allows for an inexpensive estimate of the rate of error decrease. In the 1D case, we can keep track of such a nested sequence of rules by the "level", which, except for the zeroth level, has a simple relationship to the order N:
| LEVEL | N | H |
|---|---|---|
| 0 | 1 | 2 |
| 1 | 3=2+1 | 1 |
| 2 | 5=4+1 | 1/2 |
| 3 | 9=8+1 | 1/4 |
| 4 | 17=16+1 | 1/8 |
| 5 | 33=32+1 | 1/16 |
The code includes a routine to set the abscissas of a multiple dimension product rule.
The code includes routines which pack into a single array all the abscissas associated with several Clenshaw Curtis grids of abscissas, with the grids selected in one of two ways.
Two selection rules are available.
For the first "MINMAX" selection rule, we define the quantity Q to be the sum of the 1D orders:
Q = sum ( 1 <= I <= DIM_NUM ) ORDER(I)
and then select all grids for which
Q_MIN <= Q <= Q_MAX.
For the second "CONSTRAINED" selection rule, we define the quantity Q to be the weighted sum of the 1D orders:
Q = sum ( 1 <= I <= DIM_NUM ) ALPHA(I) * ORDER(I)
and we require that the orders be bounded:
ORDER_MIN(I) <= ORDER(I) <= ORDER_MAX(I).
and then select all grids for which
Q <= Q_MAX.
CC_DISPLAY is a MATLAB program which can display various 2D Clenshaw Curtis grids.
CLENSHAW_CURTIS is available in a C++ version and a FORTRAN90 version and a MATLAB version.
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.
QUADRULE_FAST is a MATLAB library of routines defining efficient versions of a few 1D quadrature rules.
SPARSE_GRID_CLOSED is a FORTRAN90 library of routines for generating points on sparse grids associated with closed quadrature rules, including the Clenshaw-Curtis rule.
SPARSE_GRID_DISPLAY is a library of MATLAB routines which can display a sparse grid in 2D or 3D.
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_NINT is a MATLAB library containing a number of functions that may be used as test integrands for quadrature rules in multiple dimensions.
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.
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.
There are several files created by the example programs.
You can go up one level to the MATLAB source codes.