Quadrature Rules of Gaussian Type

SANDIA_RULES is a FORTRAN90 library which generates a variety of quadrature rules of various orders and types.

This library is used, in turn, by several other libraries, including SPARSE_GRID_MIXED and SGMGA. This means that a program that calls any of those libraries must have access to a compiled copy of SANDIA_RULES as well.

Name Usual domain Weight function
Gauss-Legendre [-1,+1] 1
Clenshaw-Curtis [-1,+1] 1
Fejer Type 2 [-1,+1] 1
Gauss-Chebyshev 1 [-1,+1] 1/sqrt(1-x2)
Gauss-Chebyshev 2 [-1,+1] sqrt(1-x2)
Gauss-Gegenbauer [-1,+1] (1-x2)alpha
Gauss-Jacobi [-1,+1] (1-x)alpha (1+x)beta
Gauss-Laguerre [0,+oo) e-x
Generalized Gauss-Laguerre [0,+oo) xalpha e-x
Gauss-Hermite (-oo,+oo) e-x*x
Generalized Gauss-Hermite (-oo,+oo) |x|alpha e-x*x
Hermite Genz-Keister (-oo,+oo) e-x*x
Newton-Cotes-Closed [-1,+1] 1
Newton-Cotes-Open [-1,+1] 1
Newton-Cotes-Open-Half [-1,+1] 1

For example, a Gauss-Gegenbauer quadrature rule is used to approximate:

        Integral ( -1 <= x <= +1 ) f(x) (1-x^2)^alpha dx
where alpha is a real parameter chosen by the user.

The approximation to the integral is formed by computing a weighted sum of function values at specific points:

        Sum ( 1 <= i <= n ) w(i) * f(x(i))
The quantities x are the abscissas of the quadrature rule, the values w are the weights of the quadrature rule, and the number of terms n in the sum is the order of the quadrature rule.

As a matter of convenience, most of the quadrature rules are available through three related functions:

In some cases, it is possible to compute points or weights separately; in other cases, the point and weight functions actually call the underlying function for the entire rule, and then discard the unrequested information.

Some of these quadrature rules expect a parameter ALPHA, and perhaps also a parameter BETA, in order to fully define the rule. Therefore, the argument lists of these functions vary. They always include the input quantity ORDER, but may have one or two additional inputs. In order to offer a uniform interface, there is also a family of functions with a standard set of input arguments, ORDER, NP, and P. Here NP is parameter counter, and P is the parameter value vector P. Using this interface, it is possible to call all the quadrature functions with the same argument list. The uniform interface functions can be identified by the suffix _NP that appears in their names. Generally, these functions "unpack" the parameter vector where needed, and then call the corresponding basic function. Of course, for many rules NP is zero and P may be a null pointer.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


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

Related Data and Programs:

QUADRULE, a FORTRAN90 library which defines quadrature rules for 1-dimensional domains.

R8LIB, a FORTRAN90 library which contains many utility routines, using "R8" or "double precision real" arithmetic.

SGMGA, a FORTRAN90 library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.


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Source Code:

Examples and Tests:

List of Routines:

You can go up one level to the FORTRAN90 source codes.

Last revised on 17 October 2011.