get_dftcorr_linear()

void fourier::get_dftcorr_linear	(	double	th,
		double *	corfac,
		cplx *	endcor
	)

Returns correction factor \(W(\theta)\) and boundary correction term \(\alpha_0(\theta)\) needed for linearly corrected DFT.

Purpose

For a function \(f(t)\), represented on an equidistant grid \(t_n= n h\), \(n=0,\dots,N\), we would like to compute the Fourier integral \(I(\omega) = \int^b_a dt\, e^{i \omega t}f(t)\). Using linear interpolation of \(f(t)\), the \(I(\omega)\) can be computed by by linearlt corrected discrete Fourier transformation (DFT). The algorithm is explained in

W. H. Press, S. A. Teukolosky, W. T. Vetterling, B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, 2007, chapter 13.9

get_dftcorr_linear computes the required correction factor \(W(\theta)\) and the boundary term \(\alpha_0(\theta)\), where \(\theta = \omega h\).

Parameters

th	[double] \(\theta = \omega h\) as explained above.
corfac	[double] on return, correction factor \(W(\theta)\)
endcor	[complex] on return, boundary term \(\alpha_0(\theta)\)

Definition at line 117 of file fourier.cpp.

Referenced by cntr::matsubara_ft().

{
    double ai,ar;
    double th2,th4,th6,cth,sth;
    
    if (fabs(th) < 5.0e-2) {
        th2=th*th;
        th4=th2*th2;
        th6=th4*th2;
        *corfac=1.0-(1.0/12.0)*th2+(1.0/360.0)*th4-(1.0/20160.0)*th6;
        ar=-0.5+th2/24.0-(1.0/720.0)*th4+(1.0/40320.0)*th6;
        ai=th*(1.0/6.0-(1.0/120.0)*th2+(1.0/5040.0)*th4-(1.0/362880.0)*th6);
    } else {
        cth=cos(th);
        sth=sin(th);
        th2=th*th;
        *corfac=2.0*(1.0-cth)/th2;
        ar=(cth-1.0)/th2;
        ai=(th-sth)/th2;
    }
    endcor[0]=cplx(ar,ai);
    endcor[1]=0;
}

Here is the caller graph for this function: