Deiint Deiint_r

Deiint() Sub Deiint ( F As  LongPtr, Result As  Double, Info As  Long, Optional Neval As  Long, Optional Eps As  Double = -1  ) Infinite interval automatic quadrature (double exponential (DE) formula) Purpose This routine computes the integral of f(x) over [-∞, +∞], satisfying the requested accuracy, where f(x) is a given function defined by a user supplied subroutine f. The result is obtained by the automatic integration applying the double exponential (DE) formula. The double exponential (DE) formulas (see Defin) for infinite interval [-∞, +∞] is obtained by using the following transformation function. φ(t) = sinh((π/2)sinh(t)) This formula is suitable for the slowly decaying functions. 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. [out] Result Approximation to the integral. [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. (Eps too small, singularity within integral interval, etc.) [out] Neval (Optional) Number of integrand evaluations. [in] Eps (Optional) Absolute accuracy requested. (default = 1.0e-14) (If Eps <= 0, the default value will be used) Reference (Japanese book) Masatake Mori "FORTRAN77 Numerical Calculation Programming (augmented edition)" Iwanami Shoten (1987) Example Program Compute the following integral. ∫ 1/(1 + x^2) dx [-∞, +∞] (= π) Function F1(X As Double) As Double F1 = 1 / (1 + X ^ 2) End Function Sub Ex_Deiint() Dim Result As Double, Info As Long Call Deiint(AddressOf F1, Result, Info) Debug.Print "S =", Result, "S(true) =", Dconst(13) Debug.Print "Info =", Info End Sub Example Results S = 3.14159265358979 S(true) = 3.14159265358979 Info = 0