Decomposition

decompose(p :: Polynomial{true,T},  rkf :: Function)

Decompose the homogeneous polynomial $p$ as $∑ ω_i (ξ_{i1} x_1 + ... + ξ_{in} x_n)ᵈ$ where $d$ is the degree of $p$.

The optional argument rkf is the rank function used to determine the numerical rank from the vector S of singular values. Its default value eps_rkf(1.e-6) determines the rank as the first i s.t. S[i+1]/S[i]< 1.e-6 where S is the vector of singular values.

If the rank function cst_rkf(r) is used, the SVD is truncated at rank r.

decompose(σ :: Series{T}, rkf :: Function)

Decompose the series $σ$ as a weighted sum of exponentials. Return $ω$, $Ξ$ where

• $ω$ is the vector of weights,
• $Ξ$ is the matrix of frequency points, stored per row.

The list of monomials of degree $\leq {d-1 \over 2}$ are used to construct the Hankel matrix, where $d$ is the maximal degree of the moments in $σ$.

The optional argument rkf is the rank function used to determine the numerical rank from the vector S of singular values. Its default value eps_rkf(1.e-6) determines the rank as the first i s.t. S[i+1]/S[i]< 1.e-6 where S is the vector of singular values.

If the rank function cst_rkf(r) is used, the SVD is truncated at rank r.