Module rgsl::integration[−][src]

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Introduction

Each algorithm computes an approximation to a definite integral of the form,

I = \int_a^b f(x) w(x) dx where w(x) is a weight function (for general integrands w(x)=1). The user provides absolute and relative error bounds (epsabs, epsrel) which specify the following accuracy requirement,

|RESULT - I| <= max(epsabs, epsrel |I|)

where RESULT is the numerical approximation obtained by the algorithm. The algorithms attempt to estimate the absolute error ABSERR = |RESULT

I| in such a way that the following inequality holds,

|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)

In short, the routines return the first approximation which has an absolute error smaller than epsabs or a relative error smaller than epsrel.

Note that this is an either-or constraint, not simultaneous. To compute to a specified absolute error, set epsrel to zero. To compute to a specified relative error, set epsabs to zero. The routines will fail to converge if the error bounds are too stringent, but always return the best approximation obtained up to that stage.

The algorithms in QUADPACK use a naming convention based on the following letters,

Q - quadrature routine

N - non-adaptive integrator A - adaptive integrator

G - general integrand (user-defined) W - weight function with integrand

S - singularities can be more readily integrated P - points of special difficulty can be supplied I - infinite range of integration O - oscillatory weight function, cos or sin F - Fourier integral C - Cauchy principal value The algorithms are built on pairs of quadrature rules, a higher order rule and a lower order rule. The higher order rule is used to compute the best approximation to an integral over a small range. The difference between the results of the higher order rule and the lower order rule gives an estimate of the error in the approximation.

QNG non-adaptive Gauss-Kronrod integration

The QNG algorithm is a non-adaptive procedure which uses fixed Gauss-Kronrod-Patterson abscissae to sample the integrand at a maximum of 87 points. It is provided for fast integration of smooth functions.

QAG adaptive integration

The QAG algorithm is a simple adaptive integration procedure. The integration region is divided into subintervals, and on each iteration the subinterval with the largest estimated error is bisected. This reduces the overall error rapidly, as the subintervals become concentrated around local difficulties in the integrand. These subintervals are managed by a gsl_integration_workspace struct, which handles the memory for the subinterval ranges, results and error estimates.

QAGS adaptive integration with singularities

The presence of an integrable singularity in the integration region causes an adaptive routine to concentrate new subintervals around the singularity. As the subintervals decrease in size the successive approximations to the integral converge in a limiting fashion. This approach to the limit can be accelerated using an extrapolation procedure. The QAGS algorithm combines adaptive bisection with the Wynn epsilon-algorithm to speed up the integration of many types of integrable singularities.

References and Further Reading

The following book is the definitive reference for QUADPACK, and was written by the original authors. It provides descriptions of the algorithms, program listings, test programs and examples. It also includes useful advice on numerical integration and many references to the numerical integration literature used in developing QUADPACK.

R. Piessens, E. de Doncker-Kapenga, C.W. Ueberhuber, D.K. Kahaner. QUADPACK A subroutine package for automatic integration Springer Verlag, 1983. The CQUAD integration algorithm is described in the following paper:

P. Gonnet, “Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants”, ACM Transactions on Mathematical Software, Volume 37 (2010), Issue 3, Article 26.

Functions

qawf

This function attempts to compute a Fourier integral of the function f over the semi-infinite interval [a,+\infty).

qk