# sandia_rules

sandia_rules, a C code which generates a variety of quadrature rules of various orders.

This library is used, in turn, by several other libraries, including SPARSE_GRID_MIXED, SPARSE_GRID_MIXED_GROWTH, and SGMGA. This means that a program that calls any one 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

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:

• name_COMPUTE returns points X and weights W;
• name_COMPUTE_POINTS returns points X;
• name_COMPUTE_WEIGHTS returns weights W;
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.

• name_COMPUTE_NP ( ORDER, NP, P, X, W ) unpacks parameters, calls name_COMPUTE, returns points X and weights W;
• name_COMPUTE_POINTS_NP ( ORDER, NP, P, X ) unpacks parameters, calls name_COMPUTE_POINTS, returns points X;
• name_COMPUTE_WEIGHTS_NP ( ORDER, NP, P, W ) unpacks parameters, calls name_COMPUTE_WEIGHTS, returns weights W;

### Languages:

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

### Related Data and Programs:

HERMITE_RULE, a C code which can compute and print a Gauss-Hermite quadrature rule.

LAGUERRE_RULE, a C code which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is a C code which can compute and print a Gauss-Legendre quadrature rule.

PRODUCT_RULE, a C code which constructs a product rule from 1D factor rules.

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

QUADRATURE_RULES_LEGENDRE, a dataset directory which contains triples of files defining standard Gauss-Legendre quadrature rules.

### Reference:

1. Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
2. William Cody,
An Overview of Software Development for Special Functions,
in Numerical Analysis Dundee, 1975,
edited by GA Watson,
Lecture Notes in Mathematics 506,
Springer, 1976.
3. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight,
Journal of Computational and Applied Mathematics,
Volume 71, 1996, pages 299-309.
5. John Hart, Ward Cheney, Charles Lawson, Hans Maehly, Charles Mesztenyi, John Rice, Henry Thatcher, Christoph Witzgall,
Computer Approximations,
Wiley, 1968,
LC: QA297.C64.
6. Arthur Stroud, Don Secrest,