defint defint_r qag qag_r qags qags_r qng qng_r

defint() void defint ( double(*)(double)  f, double  a, double  b, double  eps, int  l, double *  result, int *  neval, int *  info  ) Finite interval automatic quadrature (double exponential (DE) formula) Purpose This routine computes the integral of f(x) over [a, b], satisfying the requested accuracy, where f(x) is a given function defined by the user supplied function f. The result is obtained by the automatic integration applying the double exponential (DE) formula. The integral interval [-1, 1] can be transformed into the infinite interval [-∞, ∞] by the variable transformation. ∫ f(x)dx [-1, 1] = ∫ g(t)dt [-∞, ∞] where x=φ(t), g(t) = f(φ(t))φ'(t) φ(t) is the monotone function, and it satisfies φ(-∞) = -1, φ(∞) = 1. By choosing suitable φ(t), g(t) can have the double exponential asymptotic behavior and can be integrated efficiently by trapezoidal rule. The following transformation function is used in this routine. φ(t) = tanh((π/2)sinh(t)) The DE formula can be applied even if the singularities exist at the end points. Parameters [in] f The user supplied function which computes the integrand function f(x) normally (when l = 0) defined as follows. double f(double x) { return computed f(x) value; } When f(x) has singularities at endpoints, f may define the function f2(y) instead of f(x) and defint may be called with l = 1 so that the integral of f(x) on [a, b] can be obtained with better precision, where f2(y) is defined as follows.   f2(y) = f(a - y) (-(b - a)/2 < y < 0)   f2(y) = f(b - y) (0 < y <= (b - a)/2) [in] a Lower limit of integration. [in] b Upper limit of integration. [in] eps Absolute accuracy requested. max(|eps|, 1.0e-32) is used as the tolerance. [in] l = 0: The user supplied subroutine f computes f(x). = 1: The user supplied subroutine f computes f2(y). (For other values, l = 0 is assumed) [out] result Approximation to integral of f(x) over [a, b]. [out] neval Number of integrand evaluations. [out] info = 0: Successful exit = 1: Slow decay on negative side = 2: Slow decay on positive side = 3: Both of above = 4: Insufficient mesh refinement Reference Masatake Mori "FORTRAN77 Numerical Calculation Programming (augmented edition)" Iwanami Shoten, 1987. (Japanese book)