TOMS655
Weights for Interpolatory Quadrature

TOMS655 is a MATLAB library which computes weights for interpolatory quadrature schemes.

Thus, the typical use of this library is for the user to specify a quadrature interval, a weight function, and a sequence of abscissas (which may be repeated), and to request the corresponding weight vector so that an interpolatory quadrature rule is produced.

Note that when an abscissa is repeated, this indicates that, at this point, not only the function value but one or more derivatives are to be used in the quadrature formula.

The library is also suitable for the simpler task of computing both the abscissas and weights for a variety of classical Gaussian quadrature rules, including

Name	Interval	Weight function
Legendre	(a,b)	1.0
Chebyshev Type 1	(a,b)	((b-x)*(x-a))^(-0.5)
Gegenbauer	(a,b)	((b-x)*(x-a))^alpha
Jacobi	(a,b)	(b-x)^alpha*(x-a)^beta
Laguerre and Generalized Laguerre	(a,+oo)	(x-a)^alpha*exp(-b*(x-a))
Hermite and Generalized Hermite	(-oo,+oo)	|x-a|^alpha*exp(-b*(x-a)^2)
Exponential	(a,b)	|x-(a+b)/2.0|^alpha
Rational	(a,+oo)	(x-a)^alpha*(x+b)^beta

The original, true, correct version of ACM TOMS Algorithm 655 is available through ACM: http://www.acm.org/pubs/calgo or NETLIB: http://www.netlib.org/toms/index.html.

Licensing:

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

Languages:

TOMS655 is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CHEBYSHEV1_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

GEGENBAUER_RULE, a MATLAB program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a MATLAB program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a MATLAB program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a MATLAB program which computes a Gauss-Hermite quadrature rule.

JACOBI_RULE, a MATLAB program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a MATLAB program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a MATLAB program which computes a Gauss-Legendre quadrature rule.

QUADRULE, a MATLAB library which contains information about quadrature rules, both as tabulated values, and as computational procedures.

Reference:

1. Sylvan Elhay, Jaroslav Kautsky,
   Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
   ACM Transactions on Mathematical Software,
   Volume 13, Number 4, December 1987, pages 399-415.
2. Jaroslav Kautsky, Sylvan Elhay,
   Calculation of the Weights of Interpolatory Quadratures,
   Numerische Mathematik,
   Volume 40, Number 3, October 1982, pages 407-422.
3. Roger Martin, James Wilkinson,
   The Implicit QL Algorithm,
   Numerische Mathematik,
   Volume 12, Number 5, December 1968, pages 377-383.

Source Code:

• cawiq.m computes quadrature weights for a given set of knots.
• cdgqf.m computes a Gauss quadrature formula with default A, B and simple knots.
• cegqf.m computes a quadrature formula and applies it to a function.
• cegqfs.m estimates an integral using a standard quadrature formula.
• ceiqf.m constructs and applies a quadrature formula based on user knots.
• ceiqfs.m computes and applies a quadrature formula based on user knots.
• cgqf.m computes knots and weights of a Gauss quadrature formula.
• cgqfs.m computes knots and weights of a Gauss quadrature formula.
• chkqf.m computes and prints the moments of a quadrature formula.
• chkqfs.m checks the polynomial accuracy of a quadrature formula.
• ciqf.m computes weights for a classical weight function and any interval.
• ciqfs.m computes some weights of a quadrature formula in the default interval.
• class_matrix.m computes the Jacobi matrix for a quadrature rule.
• cliqf.m computes a classical quadrature formula, with optional printing.
• cliqfs.m computes the weights of a quadrature formula in the default interval.
• cwiqd.m computes all the weights for a given knot.
• eiqf.m evaluates an interpolatory quadrature formula.
• eiqfs.m evaluates a quadrature formula defined by CLIQF or CLIQFS.
• i4_sign.m returns the sign of an I4.
• imtqlx.m diagonalizes a symmetric tridiagonal matrix.
• parchk.m checks parameters ALPHA and BETA for classical weight functions.
• r8_sign.m returns the sign of an R8.
• scmm.m computes moments of a classical weight function scaled to [A,B].
• scqf.m scales a quadrature formula to a nonstandard interval.
• sct.m rescales distinct knots to an interval [A,B].
• sgqf.m computes knots and weights of a Gauss Quadrature formula.
• timestamp.m prints the current YMDHMS date as a time stamp.
• wmn.m evaluates the first M moments of classical weight functions.
• wtfn.m evaluates the classical weight functions at given points.

Examples and Tests:

TOMS655_TEST is the test distributed with the code.

• toms655_test.m, a sample calling program;
• toms655_test_output.txt, the output file.
• toms655_test_check.txt, the expected output file.

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