qag qagi qk15

qagi() def qagi ( f  , bound  , inf  , epsabs  = 1.0e-10, epsrel  = 1.0e-10, limit  = 100  ) Semi-infinite/infinite interval adaptive quadrature (15-point Gauss-Kronrod rule) Purpose This routine computes I = integral of f(x) over [bound, +inf], [-inf, bound] or [-inf, +inf], satisfying the requested accuracy, where f(x) is a given function defined by a user supplied subroutine. 15-point Gauss-Kronrod rule is used, and the integration interval will be adaptively subdivided to satisfy the requested accuracy. The semi-infinite integration range is mapped onto the interval [0, 1], and then the integration rule is applied to compute the required integral. ∫ f(x)dx [bound, +∞] = ∫ f(bound + (1 - t)/t) / t^2 dt [0, 1] The infinite integral will be computed as the sum of two semi-infinite integrals. ∫ f(x)dx [-∞, +∞] = ∫ (f(x) + f(-x)) dx [0, +∞] Returns (result, abserr) result (float): Approximation to the integral. abserr (float): Estimate of the modulus of the absolute error, which should equal or exceed the true error. info (int): = 0: Successful exit = -1: The argument f is invalid = -6: The argument limit had an illegal value (limit < 1) = 1: Maximum number of subdivisions allowed has been achieved = 2: The occurrence of roundoff error is detected, which prevents the requested tolerance from being achieved = 3: Extremely bad integrand behaviour occurs at some points of the integration interval = 4: The algorithm does not converge. It is presumed that the requested tolerance cannot be achieved, and that the returned result is the best which can be obtained = 5: The integral is probably divergent, or slowly convergent Parameters [in] f The user supplied subroutine which calculates the integrand function f(x) defined as follows. _CODE def f(x): return computed function value f(x) _ENDCODE [in] bound The finite bound of original integration range. (Not referenced if interval is doubly infinite (inf = 2)) [in] inf The kind of integration range. = 1: Semi-infinite integral [bound, +∞] = -1: Semi-infinite integral [-∞, bound] = 2: Infinite integral [-∞, +∞] (If other value is specified, inf = 2 is assumed) [in] epsabs (Optional) Absolute accuracy requested. (default = 1.0e-10) The requested accuracy is assumed to be satisfied if abserr <= max(epsabs, epsrel*|result|)). [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 < 50*eps, epsrel is assumed to be 50*eps, where eps is the machine precision. [in] limit (Optional) Maximum number of subintervals in the partition of the given integration interval. (limit >= 1) (default = 100) Reference SLATEC (QUADPACK) Example Program Compute the following integral. ∫ 1/(1 + x^2) dx [0, +∞] (= π/2) def f(x): return 1/(1 + x**2) def TestQagi(): bound = 0.0 inf = 2 s, abserr, info = qagi(f, bound, inf) print('qagi [-inf, +inf]') print(s, abserr, info) inf = 1 s, abserr, info = qagi(f, bound, inf) print('qagi [0, +inf]') print(s, abserr, info) inf = -1 s, abserr, info = qagi(f, bound, inf) print('qagi [-inf, 0]') print(s, abserr, info) Example Results >>> TestQagi() qagi [-inf, +inf] 3.141592653589793 2.5779659138999894e-10 0 qagi [0, +inf] 1.5707963267948966 1.2889829569499947e-10 0 qagi [-inf, 0] 1.5707963267948966 1.2889829569499947e-10 0