clenshaw_curtis_grid

clenshaw_curtis_grid, a C++ code 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:
LEVELNH
012
13=2+11
25=4+11/2
39=8+11/4
417=16+11/8
533=32+11/16

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.
```

Languages:

clenshaw_curtis_grid is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CLENSHAW_CURTIS_RULE a C++ code which defines a Clenshaw Curtis quadrature rule.

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

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

NINTLIB a C++ code containing a variety of routines for numerical estimation of integrals in multiple dimensions.

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

STROUD a C++ code which contains rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_INT_2D a C++ code which defines test integrands for 2D quadrature rules.

TEST_NINT a C++ code which contains functions that may be used as test integrands for quadrature rules in multiple dimensions.

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. W Morven Gentleman,
Communications of the ACM,
Volume 15, Number 5, May 1972, pages 353-355.
4. Lloyd Trefethen,
Is Gauss Quadrature Better than Clenshaw-Curtis?,
SIAM Review,
Volume 50, Number 1, 2008, pages 67-87.
5. Joerg Waldvogel,
Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
BIT Numerical Mathematics,
Volume 43, Number 1, 2003, pages 1-18.

Source Code:

Last revised on 20 February 2020.