### 19.3 Introduction to QUADPACK

QUADPACK is a collection of functions for the numerical computation of one-dimensional definite integrals. It originated from a joint project of R. Piessens 1, E. de Doncker 2, C. Ueberhuber 3, and D. Kahaner 4.

The QUADPACK library included in Maxima is an automatic translation (via the program `f2cl`) of the Fortran source code of QUADPACK as it appears in the SLATEC Common Mathematical Library, Version 4.1 5. The SLATEC library is dated July 1993, but the QUADPACK functions were written some years before. There is another version of QUADPACK at Netlib 6; it is not clear how that version differs from the SLATEC version.

The QUADPACK functions included in Maxima are all automatic, in the sense that these functions attempt to compute a result to a specified accuracy, requiring an unspecified number of function evaluations. Maxima’s Lisp translation of QUADPACK also includes some non-automatic functions, but they are not exposed at the Maxima level.

Further information about QUADPACK can be found in the QUADPACK book 7.

#### 19.3.1 Overview

``quad_qag``

  Integration of a general function over a finite interval. `quad_qag` implements a simple globally adaptive integrator using the strategy of Aind (Piessens, 1973). The caller may choose among 6 pairs of Gauss-Kronrod quadrature formulae for the rule evaluation component. The high-degree rules are suitable for strongly oscillating integrands.

``quad_qags``

  Integration of a general function over a finite interval. `quad_qags` implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

``quad_qagi``

  Integration of a general function over an infinite or semi-infinite interval. The interval is mapped onto a finite interval and then the same strategy as in `quad_qags` is applied.

``quad_qawo``

  Integration of \(\cos(\omega x) f(x)\) or \(\sin(\omega x) f(x)\) over a finite interval, where \(\omega\) is a constant. The rule evaluation component is based on the modified Clenshaw-Curtis technique. `quad_qawo` applies adaptive subdivision with extrapolation, similar to `quad_qags`.

``quad_qawf``

  Calculates a Fourier cosine or Fourier sine transform on a semi-infinite interval. The same approach as in `quad_qawo` is applied on successive finite intervals, and convergence acceleration by means of the Epsilon algorithm (Wynn, 1956) is applied to the series of the integral contributions.

``quad_qaws``

  Integration of \(w(x)f(x)\) over a finite interval [a, b], where w is a function of the form \((x-a)^\alpha (b-x)^\beta v(x)\) and v(x) is 1 or \(\log(x-a)\) or \(\log(b-x)\) or

  \(\log(x-a)\log(b-x)\)

  , and \(\alpha > -1\) and \(\beta > -1\).

  A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on the subintervals which contain a or b.

``quad_qawc``

  Computes the Cauchy principal value of f(x)/(x - c) over a finite interval (a, b) and specified c. The strategy is globally adaptive, and modified Clenshaw-Curtis integration is used on the subranges which contain the point x = c.

``quad_qagp``

  Basically the same as `quad_qags` but points of singularity or discontinuity of the integrand must be supplied. This makes it easier for the integrator to produce a good solution.

Categories:Integral calculus ·Numerical methods ·Share packages ·Package quadpack ·

#### Footnotes

### (1)

Applied Mathematics and Programming Division, K.U. Leuven

### (2)

Applied Mathematics and Programming Division, K.U. Leuven

### (3)

Institut für Mathematik, T.U. Wien

### (4)

National Bureau of Standards, Washington, D.C., U.S.A

### (5)

https://www.netlib.org/slatec

### (6)

https://www.netlib.org/quadpack

### (7)

R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, and D.K. Kahaner. QUADPACK: A Subroutine Package for Automatic Integration. Berlin: Springer-Verlag, 1983, ISBN 0387125531.