General-purpose routines

The two general-purpose routines most suitable for use without further analysis of the integrand are QAGS for integration over a finite interval and QAGI for integration over an infinite interval.^[7] These two routines are used in GNU Octave (the quad command)^[5] and R (the integrate function).^[9]

QAGS

uses global adaptive quadrature based on 21-point Gauss–Kronrod quadrature within each subinterval, with acceleration by Peter Wynn's epsilon algorithm.^[7][10]

QAGI

is the only general-purpose routine for infinite intervals, and maps the infinite interval onto the semi-open interval (0,1] using a transformation then uses the same approach as QAGS, except with 15-point rather than 21-point Gauss–Kronrod quadrature.^[2] For an integral over the whole real line, the transformation used is ${\displaystyle x=(1-t)/t}$:^[2]

$${\displaystyle \int _{-\infty }^{+\infty }f(x)dx=\int _{0}^{1}{dt \over t^{2}}\left(f\left({\frac {1-t}{t}}\right)+f\left(-{\frac {1-t}{t}}\right)\right)\;.}$$

This is not the best approach for all integrands: another transformation may be appropriate, or one might prefer to break up the original interval and use QAGI only on the infinite part.^[7]

Brief overview of the other automatic routines

QNG

simple non-adaptive integrator

QAG

simple adaptive integrator

QAGP

similar to QAGS but allows user to specify locations of internal singularities, discontinuities etc.

QAWO

integral of cos(ωx) f(x) or sin(ωx) f(x) over a finite interval

QAWF

Fourier transform

QAWS

integral of w(x) f(x) from a to b, where f is smooth and w(x) = (x–a)^α (b–x)^β log^k(x–a) log^l(b–x), with k, l = 0 or 1 and α, β > –1

QAWC

Cauchy principal value of the integral of f(x)/(x–c) for user-specified c and f ^[2]