Reference

build_Abase!(A1, s, poles)

Builds the base matrix with the 1 / (s - p), 1.0 and s coefficients. It is assumed that the poles were sorted by cplxpair.

cplxpair(x)

To be used to sort an array by real values, then complex conjugate pairs. The more positive reals will be first, then the pairs with smaller imaginary part.

Example

real_values = [-1, -3, -2]
complex_values = [-1 + 1im, -2 - 2im, -1 + 2im]
sorted_values = sort!([complex_values; conj(complex_values); real_values], by = cplxpair)

# output

9-element Vector{Complex{Int64}}:
 -1 + 0im
 -2 + 0im
 -3 + 0im
 -1 - 1im
 -1 + 1im
 -1 - 2im
 -1 + 2im
 -2 - 2im
 -2 + 2im

pole_identification(s, f, poles, weight, relaxed) -> new_poles

Stage 1 of the Vector Fitting.

See also vector_fitting, residue_identification.

rational(s, poles, residues, d, h) -> `f(s)`

Rational transfer function with complex frequencies s, a set of poles a_n, residues r_n and real constants d and h.

\[\sum_{n=1}^N \frac{r_n}{s - a_n} + d + s h\]

recommended_init_poles(s, Npoles) -> init_poles

Builds a vector of recommended initial poles sorted by cplxpair.

residue_identification(s, f, poles, weight) -> residues, d, h

Stage 2 of the Vector Fitting. This should be called separately for each column of f and weight when ndims(f) == 2.

See also vector_fitting, pole_identification.

vector_fitting(
    s,
    f,
    init_poles,
    weight = 1;
    relaxed = true,
    force_stable = true,
    maxiter = 5,
    tol = 1e-12,
) -> poles, residues, d, h, fitted, error_norm

Vector Fitting of the array f with complex frequency s using a set of initial poles init_poles.

f can be a matrix of dimensions (Ns, Nc) and the fitting will be over its columns using a set of common poles.

relaxed controls the nontriviality constraint. relaxed=true usually converges faster, but can be less stable for non-smooth functions.

force_stable controls if unstable poles should be reflected to the semi-left complex plane, that is, forced to have negative real part.

maxiter is the maximum of iterations that will be done to try to achieve a convergence with desired error_norm tolerance tol.

See also recommended_init_poles, rational, pole_identification, residue_identification.