◆ qag()

Finite interval adaptive quadrature (15/21/31/41/51/61 point Gauss-Kronrod rule)

Purpose: This routine computes I = integral of f over [a, b], satisfying the requested accuracy, where f is a given function defined by a user supplied subroutine.
15, 21, 31, 41, 51 or 61 point Gauss-Kronrod rule is used, and the integration interval will be adaptively subdivided to satisfy the requested accuracy.

Returns: (result, abserr, info)

result (Float64):
Approximation to I = integral of f over [a, b].

abserr (Float64):
Estimate of the modulus of the absolute error, which should equal or exceed the true error.

info (Int32):
= 0: Successful exit
= -1: The argument f is invalid
= -7: The argument limit had an illegal value (limit < 1)
= 1: Maximum number of subdivisions allowed has been reached
= 2: Requested tolerance cannot be achieved due to roundoff error
= 3: Bad integrand behavior found in the integration interval

Parameters

[in]	f	The user supplied subroutine which calculates the integrand function f(x) defined as follows. _CODE function f(x) returns computed function value f(x). end _ENDCODE
[in]	a	Lower limit of integration.
[in]	b	Upper limit of integration.
[in]	epsabs	(Optional) Absolute accuracy requested. The requested accuracy is assumed to be satisfied if abserr <= max(epsabs, epsrel*\|result\|)). (default = 1.0e-10)
[in]	epsrel	(Optional) Relative accuracy requested. (default = 1.0e-10) The requested accuracy is assumed to be satisfied if abserr <= max(epsabs, epsrel\|result\|)). If epsabs <= 0 and epsrel < 50eps, epsrel is assumed to be 50*eps, where eps is the machine precision.
[in]	key	(Optional) Key for choice of local integration rule. (default = 1) = 1: qk15 = 2: qk21 = 3: qk31 = 4: qk41 = 5: qk51 = 6: qk61 If key < 1, key = 1 is assumed. If key > 6, key = 6 is assumed.
[in]	limit	(Optional) Maximum number of subintervals in the partition of the given integration interval [a, b]. (limit >= 1) (default = 100)

Example Program: Compute the following integral.
∫ 1/(1 + x^2) dx [0, 4] (= atan(4))

function TestQag()

f(x) = 1/(1 + x^2)

a = 0

b = 4

s, abserr, info = qag(f, a, b)

println(s, " ", abserr, " ", info)

end

qag
function qag(f::Function, a::Real, b::Real, epsabs::Real=1.0e-10, epsrel::Real=1.0e-10, key::Integer=1, limit::Integer=100)
Finite interval adaptive quadrature (15/21/31/41/51/61 point Gauss-Kronrod rule)