toms655, a FORTRAN77 code 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
| Index | Name | Interval | Weight function |
|---|---|---|---|
| 1 | Legendre | (a,b) | 1.0 |
| 2 | Chebyshev Type 1 | (a,b) | ((b-x)*(x-a))^(-0.5) |
| 3 | Gegenbauer | (a,b) | ((b-x)*(x-a))^alpha |
| 4 | Jacobi | (a,b) | (b-x)^alpha*(x-a)^beta |
| 5 | Laguerre and Generalized Laguerre | (a,+oo) | (x-a)^alpha*exp(-b*(x-a)) |
| 6 | Hermite and Generalized Hermite | (-oo,+oo) | |x-a|^alpha*exp(-b*(x-a)^2) |
| 7 | Exponential | (a,b) | |x-(a+b)/2.0|^alpha |
| 8 | Rational | (a,+oo) | (x-a)^alpha*(x+b)^beta |
| 9 | Chebyshev Type 2 | (a,b) | ((b-x)*(x-a))^(+0.5) |
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.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
toms655 is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.
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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.
QUADMOM, a FORTRAN77 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).
QUADRATURE_WEIGHTS, a FORTRAN90 library which illustrates techniques for computing the weights of a quadrature rule, assuming that the points have been specified.
QUADRULE, a FORTRAN90 library which contains information about quadrature rules, both as tabulated values, and as computational procedures.