CHEBYSHEV1_RULE Gauss-Chebyshev Type 1 Quadrature Rules

CHEBYSHEV1_RULE is a C++ program which can generate 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:

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

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

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

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

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

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

LINE_NCC_RULE, a C++ library which computes a Newton Cotes Closed (NCC) quadrature rule for the line, that is, for an interval of the form [A,B], using equally spaced points which include the endpoints.

LINE_NCO_RULE, a C++ library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

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

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_CHEBYSHEV1, a dataset directory of triples of files defining standard Gauss-Chebyshev type 1 quadrature rules.

TANH_SINH_RULE, a C++ program which computes and writes out a tanh-sinh quadrature rule of given order.

TRUNCATED_NORMAL_RULE, a C++ 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,
Prentice Hall, 1966,
LC: QA299.4G3S7.

Examples and Tests:

• cheby1_o5_r.txt, the region file created by the command
```
chebyshev1_rule 5 -1 +1 cheby1_o5
```
• cheby1_o5_w.txt, the weight file created by the command
```
chebyshev1_rule 5 -1 +1 cheby1_o5
```
• cheby1_o5_x.txt, the abscissa file created by the command
```
chebyshev1_rule 5 -1 +1 cheby1_o5
```

List of Routines:

• MAIN is the main program for CHEBYSHEV1_RULE.
• CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
• CGQF computes knots and weights of a Gauss quadrature formula.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• 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_SIGN returns the sign of an R8.
• R8MAT_WRITE writes an R8MAT file with no header.
• RULE_WRITE writes a quadrature rule to three files.
• SCQF scales a quadrature formula to a nonstandard interval.
• SGQF computes knots and weights of a Gauss Quadrature formula.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last revised on 23 February 2010.