PATTERSON_RULE
Gauss-Patterson Quadrature Rules


PATTERSON_RULE, a C++ program which generates a specific Gauss-Patterson quadrature rule, based on user input.

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

The Gauss-Patterson quadrature is a nested family which begins with the Gauss-Legendre rules of orders 1 and 3, and then succesively inserts one new abscissa in each subinterval. Thus, after the second rule, the Gauss-Patterson rules do not have the super-high precision of the Gauss-Legendre rules. They trade this precision in exchange for the advantages of nestedness. This means that Gauss-Patterson rules are only available for orders of 1, 3, 7, 15, 31, 63, 127, 255 or 511.

The standard Gauss-Patterson quadrature rule is used as follows:

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

The polynomial precision of a Gauss-Patterson rule can be checked numerically by the INT_EXACTNESS_LEGENDRE program. We should expect
IndexOrderFree+FixedExpected PrecisionActual Precision
011 + 02*1+0-1=11
133 + 02*3+0-1=55
274 + 32*4+3-1=1010 + 1 = 11
3158 + 72*8+7-1=2222 + 1 = 23
43116 + 152*16+15-1=4646 + 1 = 47
56332 + 312*32+31-1=9494 + 1 = 95
612764 + 632*64+63-1=190190 + 1 = 191
7255128 + 1272*128+127-1=382382 + 1 = 383
8511256 + 2552*256+255-1=766766 + 1 = 767
where the extra 1 degree of precision comes about because the rules are symmetric, and can integrate any odd monomial exactly. Thus, after the first rule, the precision is 3*2^index - 1.

Usage:

patterson_rule order a b filename
where

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

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

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.

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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.

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

KRONROD, a C++ library which can compute a Gauss and Gauss-Kronrod pair of quadrature rules of arbitrary order, by Robert Piessens, Maria Branders.

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

LATTICE_RULE, a C++ library which approximates M-dimensional integrals using lattice rules.

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_test

QUADRULE, a C++ library which defines 1-dimensional quadrature rules.

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. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.

Source Code:


Last revised on 30 March 2020.