dehint dehint_r deoint deoint_r qagi qagi_r qawf qawf_r qk15i qk15i_r

deoint() void deoint ( double(*)(double)  f, double  a, double  omega, int  iflag, double  eps, double *  result, int *  neval, double *  err, int *  info  ) Semi-infinite interval automatic quadrature for Fourier integrals (double exponential (DE) formula) Purpose This routine computes the following Fourier integrals by the automatic integration applying the double exponential (DE) formula. Ic = ∫ f(x) cosωx dx [a, +∞] or Is = ∫ f(x) sinωx dx [a, +∞] where f(x) is a given function defined by a user supplied subroutine f. Ic or Is is transformed to the intergral on the infinite interval [-∞, +∞] by using the following transformation function, and the integral value is computed by the trapezoidal rule. Ic: x = Mφ(t - π/2M)/ω Is: x = Mφ(t)/ω The following φ(t) is used. φ1(t) = t/(1 - exp(-2t -α(1 - exp(-t)) - β(exp(t) - 1))), β = 1/4, α = β/sqrt(1 + Mlog(1 + M)/4π) or φ2(t) = t/(1 - exp(-ksinh(t))), k = 2π The step size h and the constant M are selected to satisfy Mh = π. Please refer to the reference below for more details. Parameters [in] f The user supplied subroutine which calculates the integrand function f(x) defined as follows. double f(double x) { return computed f(x) value } [in] a Lower bound of integration interval. [in] omega ω value in cos or sin. [in] iflag Choose the type of integral. = 0: ∫ f(x) cosωx dx [0, +∞] = 1: ∫ f(x) sinωx dx [0, +∞] Further, the transformation function can be selected as follows. iflag += 0: φ1(t) (recommended) iflag += 2: φ2(t) [in] eps Relative accuracy requested. (0 < eps <= 1) [out] result Obtained approximation to the integral. [out] neval Number of evaluations of f(x). [out] err Estimated relative error of obtained v. To be overestimated for many cases. This routine may return as successful exit even if err > eps. [out] info = 0: Successful exit = -4: The argument iflag had an illegal value (iflag != 0 to 3) = -5: The argument eps had an illegal value (eps <= 0 or eps > 1) = 1: Not converged Reference Ooura and Mori, "The double exponential formula for oscillatory functions over the half infinite interval", J. Comput. Appl. Math., 38 (1991), 353-360. Ooura and Mori, "A robust double exponential formula for Fourier-type integrals", J. Comput. Appl. Math., 112 (1999), 229-241.