TOMS655
Weights for Interpolatory Quadrature

TOMS655 is a C++ library which computes weights for interpolatory quadrature schemes, by Sylvan Elhay and Jaroslav Kautsky.

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 C++ program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

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

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

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

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

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

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

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

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

QUADMOM, a C++ library which computes a Gaussian quadrature rule for a weight function rho(x) based on the Golub-Welsch procedure that only requires knowledge of the moments of rho(x).

QUADRULE a C++ 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:

• toms655.cpp, the source code;
• toms655.hpp, the include file;
• toms655.sh, commands to compile the source code;

Examples and Tests:

TOMS655_PRB tests various routines in the package.

• toms655_prb.cpp, a sample calling program;
• toms655_prb.sh, commands to compile, link and run the test program;
• toms655_prb_output.txt, the output file.

List of Routines:

• CAWIQ computes quadrature weights for a given set of knots.
• CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
• CEGQF computes a quadrature formula and applies it to a function.
• CEGQFS estimates an integral using a standard quadrature formula.
• CEIQF constructs and applies a quadrature formula based on user knots.
• CEIQFS computes and applies a quadrature formula based on user knots.
• CGQF computes knots and weights of a Gauss quadrature formula.
• CGQFS computes knots and weights of a Gauss quadrature formula.
• CHKQF computes and prints the moments of a quadrature formula.
• CHKQFS checks the polynomial accuracy of a quadrature formula.
• CLIQF computes a classical quadrature formula, with optional printing.
• CIQF computes weights for a classical weight function and any interval.
• CIQFS computes some weights of a quadrature formula in the default interval.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• CLIQFS computes the weights of a quadrature formula in the default interval.
• CWIQD computes all the weights for a given knot.
• EIQF evaluates an interpolatory quadrature formula.
• EIQFS evaluates a quadrature formula defined by CLIQF or CLIQFS.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• I4_SIGN returns the sign of an I4.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• PARCHK checks parameters ALPHA and BETA for classical weight functions.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_MAX returns the maximum of two R8's.
• R8_MIN returns the minimum of two R8's.
• R8_SIGN returns the sign of an R8.
• SCMM computes moments of a classical weight function scaled to [A,B].
• SCQF scales a quadrature formula to a nonstandard interval.
• SCT rescales distinct knots to an interval [A,B].
• SGQF computes knots and weights of a Gauss Quadrature formula.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• WM evaluates the first M moments of classical weight functions.
• WTFN evaluates the classical weight functions at given points.

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