gftool.basis.pole.gf_from_moments

gftool.basis.pole.gf_from_moments(moments, width=1.0) → PoleFct

Find pole Green’s function matching given moments.

Finds poles and weights for a pole Green’s function matching the given high frequency moments for large z: g(z) = np.sum(weights / (z - poles)) = moments / z**np.arange(N)

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

Parameters:

moments(…, N) float array_like

Moments of the high-frequency expansion, where G(z) = moments / z**np.arange(1, N+1) for large z.

widthfloat or (…) float array_like or None, optional

Spread of the poles; they are in the interval [-width, width]. 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.

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.

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.