CHEBYSHEV1_RULE Gauss-Chebyshev Type 1 Quadrature Rules

CHEBYSHEV1_RULE is a MATLAB program which generates a specific Gauss-Chebyshev type 1 quadrature rule, based on user input.

The rule is written to three files for easy use as input to other programs.

The Gauss Chevbyshev type 1 quadrature rule is used as follows:

```        Integral ( A <= x <= B ) f(x) / sqrt ( ( x - A ) * ( B - x ) ) dx
```
is to be approximated by
```        Sum ( 1 <= i <= order ) w(i) * f(x(i))
```

Usage:

chebyshev1_rule ( order, a, b, 'filename' )
where
• order is the number of points in the quadrature rule.
• a is the left endpoint;
• b is the right endpoint.
• filename specifies the output files: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

Languages:

CHEBYSHEV1_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

ALPERT_RULE, a MATLAB library which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

CCN_RULE, a MATLAB program which defines a nested Clenshaw Curtis quadrature rule.

CHEBYSHEV_POLYNOMIAL, a MATLAB library which evaluates the Chebyshev polynomial and associated functions.

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

CLENSHAW_CURTIS_RULE, a MATLAB program which defines a Clenshaw Curtis 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 can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS_CHEBYSHEV1, a MATLAB program which checks the polynomial exactness of a Gauss-Chebyshev type 1 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 can compute and print a Gauss-Legendre quadrature rule.

LINE_FELIPPA_RULE, a MATLAB library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

PATTERSON_RULE, a MATLAB program which computes a Gauss-Patterson quadrature rule.

POWER_RULE, a MATLAB program which constructs a power rule, that is, a product quadrature rule from identical 1D factor rules.

QUADRATURE_RULES_CHEBVYSHEV1, a dataset directory which contains triples of files defining standard Gauss-Chebyshev tupe 1 quadrature rules.

TRUNCATED_NORMAL_RULE, a MATLAB program which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].

Reference:

1. Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
3. 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.
4. Jaroslav Kautsky, Sylvan Elhay,
Calculation of the Weights of Interpolatory Quadratures,
Numerische Mathematik,
Volume 40, 1982, pages 407-422.
5. Roger Martin, James Wilkinson,
The Implicit QL Algorithm,
Numerische Mathematik,
Volume 12, Number 5, December 1968, pages 377-383.
6. Arthur Stroud, Don Secrest,