Dehint Dehint_r Deoint Deoint_r Qagi Qagi_r Qawf Qawf_r Qk15i Qk15i_r

Deoint() Sub Deoint ( F As  LongPtr, A As  Double, Omega As  Double, Integr As  Long, Result As  Double, Info As  Long, Optional Neval As  Long, Optional Err As  Double, Optional Eps As  Double = -1, Optional Phi As  Long = 0  ) 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 φ(t) can be selected from the following two functions by specifying the parameter. 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. Function F(X As Double) As Double F = f(X) End Function X should not be changed. [in] A Lower bound of integration interval a. [in] Omega ω value in cos or sin. [in] Integr Indicates which weight function is to be used. = 1: ∫ f(x) cosωx dx [0, +∞] = 2: ∫ f(x) sinωx dx [0, +∞] [out] Result Obtained approximation to the integral. [out] Info = 0: Successful exit = -4: The argument Integr had an illegal value (Integr <> 1 and Integr <> 2) = 1: Not converged [out] Neval (Optional) Number of evaluations of f(x). [out] Err (Optional) Estimated relative error of obtained Result. To be overestimated for many cases. This routine may return as successful exit even if Err > Eps. [in] Eps (Optional) Relative accuracy requested. (default = 1.0e-10) (If Eps <= 0 or Eps > 1, the default value will be used) [in] Phi (Optional) The transformation function can be specified. (default = 0) = 0: φ1(t) = 1: φ2(t) (For other values, the default value will be used.) 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. Example Program Compute the following integral. ∫ exp(-x^2)cos(x) dx [0, +∞] (= exp(-1/4)*1/2*√π) Function F1(X As Double) As Double F1 = Exp(-X ^ 2) End Function Sub Ex_Deoint() Dim A As Double, Omega As Double, Integr As Long, Result As Double Dim Info As Long, Neval As Long A = 0 Omega = 1 Integr = 1 Call Deoint(AddressOf F1, A, Omega, Integr, Result, Info) Debug.Print "S =", Result, "S(true) =", Exp(-1 / 4) * Dconst(17) / 2 Debug.Print "Info =", Info End Sub Example Results S = 0.690194223521232 S(true) = 0.690194223521572 Info = 0