TOMS726
Orthogonal Polynomials and Quadrature Rules

TOMS726 is a FORTRAN77 library which computes recursion relationships for various families of orthogonal polynomials, as well as the abscissas and weights of related quadrature rules; the library is commonly called ORTHPOL, and is by Walter Gautschi.

Languages:

TOMS726 is available in a FORTRAN77 version and a FORTRAN90 version.

Related Data and Programs:

TOMS655, a FORTRAN77 library which computes the weights for interpolatory quadrature rules.

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

Reference:

1. William Cody, Kenneth Hillstrom,
   Chebyshev Approximations for the Natural Logarithm of the Gamma Function,
   Mathematics of Computation,
   Volume 21, Number 98, April 1967, pages 198-203.
2. Walter Gautschi,
   On Generating Orthogonal Polynomials,
   SIAM Journal on Scientific and Statistical Computing,
   Volume 3, Number 3, 1982, pages 289-317.
3. Walter Gautschi,
   Algorithm 726: ORTHPOL - A Package of Routines for Generating Orthogonal Polynomials and Gauss-Type Quadrature Rules,
   ACM Transactions on Mathematical Software,
   Volume 20, Number 1, March 1994, pages 21-62.

Source Code:

• toms726.f, the source code.
• toms726.sh, commands to compile the source code.

Examples and Tests:

• toms726_prb.f, a sample calling program.
• toms726_prb.sh, commands to compile and run the sample program.
• toms726_prb_output.txt, the output file.

List of Routines:

• ALGA_R4 evaluates the logarithm of the gamma function.
• ALGA_R8 evaluates the logarithm of the gamma function.
• CHEB_R4 generates recursion coefficients ALPHA and BETA.
• CHEB_R8 generates recursion coefficients ALPHA and BETA.
• CHRI_R4 implements the Christoffel or generalized Christoffel theorem.
• CHRI_R8 implements the Christoffel or generalized Christoffel theorem.
• FEJER_R4 generates a Fejer quadrature rule.
• FEJER_R8 generates a Fejer quadrature rule.
• GAMMA_R4 evaluates the gamma function for real positive X.
• GAMMA_R8 evaluates the gamma function for real positive argument.
• GAUSS_R4 generates an N-point Gaussian quadrature formula.
• GAUSS_R8 generates an N-point Gaussian quadrature formula.
• GCHRI_R4 implements the generalized Christoffel theorem.
• GCHRI_R8 implements the generalized Christoffel theorem.
• KERN_R4 generates the kernels in the Gauss quadrature remainder term.
• KERN_R8 generates the kernels in the Gauss quadrature remainder term.
• KNUM_R4 integrates certain rational polynomials.
• KNUM_R8 is a double-precision version of the routine KNUM_R4.
• LANCZ_R4 applies Stieltjes's procedure, using the Lanczos method.
• LANCZ_R8 is a double-precision version of the routine LANCZ_R4.
• LOB_R4 generates a Gauss-Lobatto quadrature rule.
• LOB_R8 generates a Gauss-Lobatto quadrature rule.
• MCCHEB_R4 is a multiple-component discretized modified Chebyshev algorithm.
• MCCHEB_R8 is a double-precision version of the routine MCCHEB_R4.
• MCDIS_R4 is a multiple-component discretization procedure.
• MCDIS_R8 is a double-precision version of the routine MCDIS_R4.
• NU0HER estimates a starting index for recursion with the Hermite measure.
• NU0JAC estimates a starting index for recursion with the Jacobi measure.
• NU0LAG estimates a starting index for recursion with the Laguerre measure.
• QGP_R4 is a general-purpose discretization routine.
• QGP_R8 is a double-precision version of the routine QGP_R4.
• RADAU_R4 generates a Gauss-Radau quadrature formula.
• RADAU_R8 generates a Gauss-Radau quadrature formula.
• RECUR_R4 generates recursion coefficients for orthogonal polynomials.
• RECUR_R8 is a double-precision version of the routine RECUR_R4.
• STI_R4 applies Stieltjes's procedure.
• STI_R8 is a double-precision version of the routine STI_R4.
• SYMTR_R4 maps T in [-1,1] to X in (-oo,oo).
• SYMTR_R8 maps T in [-1,1] to X in (-oo,oo).
• T_FUNCTION solves Y = T * log ( T ) for T, given nonnegative Y.
• TR_R4 maps T in [-1,1] to X in [0,oo).
• TR_R8 maps T in [-1,1] to X in [0,oo).

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