TOMS655_ORIGINAL
Weights for Interpolatory Quadrature

TOMS655_ORIGINAL is a FORTRAN77 library which computes weights for interpolatory quadrature schemes, by Sylvan Elhay and Jaroslav Kautsky. This is the original version of ACM TOMS Algorithm 655.

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.

Languages:

TOMS655_ORIGINAL is available in a FORTRAN77 version.

Related Data and Programs:

CHEBYSHEV1_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

GEGENBAUER_RULE, a FORTRAN90 program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a FORTRAN90 program which computes a Gauss-Hermite quadrature rule.

JACOBI_RULE, a FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a FORTRAN90 program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a FORTRAN90 program which computes a Gauss-Legendre quadrature rule.

QUADRULE, a FORTRAN90 library which contains information about quadrature rules, both as tabulated values, and as computational procedures.

TOMS655, a FORTRAN77 library which computes the weights for interpolatory quadrature rules; this library is commonly called IQPACK, by Sylvan Elhay and Jaroslav Kautsky. This version of ACM TOMS algorithm 655 has been extensively modified.

TOMS793, a FORTRAN77 library which carries out Gauss quadrature for rational functions, by Walter Gautschi; this is ACM TOMS algorithm 793.

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_original.f, the source code;
• toms655_original.sh, commands to compile the source code;

Examples and Tests:

TOMS655_ORIGINAL_PRB is the test distributed with the code.

• toms655_original_prb.f, a sample calling program;
• toms655_original_prb.sh, commands to compile, link and run the test program;
• toms655_original_prb_output.txt, the output file.
• check.txt, the expected output file.

You can go up one level to the FORTRAN77 source codes.