gftool.basis.pole.gf_from_z

gftool.basis.pole.gf_from_z(z, gf_z, n_pole, moments=(), width=1.0, weight=None) → PoleFct

Find pole causal Green’s function fitting gf_z.

This function is only meaningful away from the real axis. Finds poles and weights for a pole Green’s function matching the given Green’s function gf_z.

Note that for an odd number of moments, the central pole is at z = 0, so the causal Green’s function g(0) diverges.

Parameters:

z(…, N_z) complex np.ndarray

Frequencies at which gf_z is given. Mind that the fit is only meaningful away from the real axis.

gf_z(…, N_z) complex np.ndarray

Causal Green’s function which is fitted.

n_poleint

Number of poles to fit.

moments(…, N) float array_like

Moments of the high-frequency expansion, where G(z) = moments / z**np.arange(N) for large z.

widthfloat or (…) float array_like or None, optional

Spread of the poles; they are in the interval [-width, width]. (default: 1.) width=1 are the normal Chebyshev nodes in the interval [-1, 1]. If width=None and the second moment moments[…, 1] is given, the largest pole will match the second moment, unless it is small (abs(moments[…, 1]) < 0.1), then we choose width=1.

weight(…, N_z) float np.ndarray, optional

Weighting of the fit. If an error σ of the input gf_z is known, this should be weight=1/σ. If high-frequency moments should be fitted correctly, weight=abs(z)**(N+1) is a good fit.

Returns:

gf.resids(…, N) float np.ndarray

Residues (or weight) of the poles.

gf.poles(N) or (…, N) float np.ndarray

Position of the poles, these are the Chebyshev nodes for degree N.

Raises:

ValueError

If more moments are given than poles are fitted (len(moments) > n_pole).

Notes

We employ the similarity of the relation betweens the moments and the poles and residues with polynomials and the Vandermond matrix. The poles are chooses as Chebyshev nodes, the residues are calculated accordingly.