ERA/Single-Point Loewner

Numerics.sploewner.build_exact_moments

(sigma, A, B, C, K, L=eye(size(C,1),size(C,1)), R=eye(size(B,2),size(B,2)))

Helper function to build generalized moments up to order K from left/right/two-sided quadrature samples at nodes \( z \) using quadrature weights \( w \).

Numerics.sploewner.sploewner_quadrature

(sigma, z, w, Ql, Qr, Qlr, K, m, options=struct("AbsTol",NaN))

Hankel/Single Point Loewner realization with two-sided quadrature samples. Given two-sided/left/right quadrature data Qlr/Ql/Qr, construction moments via contour integration approximated by a quadrature rule with nodes and weights \( ( z_k, w_k ) \).

Input arguments:

• sigma –

  shift value \( = \infty \Leftrightarrow \) Hankel, \( ! \infty \Leftrightarrow \) SPLoewner

• z –

  vector of quadrature nodes

• w –

  vector of quadrature weights

• Ql –

  vector of left-sided samples \( L^* T^{-1} \) at \( z_k \) in \(z\)

• Qr –

  vector of right-sided samples of \( T^{-1} R \) at \( z_k \) in \(z\)

• Qlr –

  vector of two-sided samples \( L^* T^{-1} R \) at \( z_k \) in \(z\)

• K –

  number of moments to use in data matrix construction

• m –

  number of poles to search for in \( \Omega \)

• options –

  options for realization

Numerics.sploewner.build_quadrature_moments

(sigma, z, w, Ql, Qr, Qlr, K)

Helper function to build generalized moments up to order K from left/right/two-sided quadrature samples at nodes \( z \) using quadrature weights \( w \).

Numerics.sploewner.sploewner_exact

(sigma, A, B, C, K, m, L=eye(size(C,1),size(C,2)), R=eye(size(B,2),size(B,1)), options=struct("AbsTol",NaN))

Hankel/Single Point Loewner realization given state, reachability, and observability matrices of an LTI system. Using these system matrices, (probed) generalzied moments are constructed up to order \( 2K \). We use these "exact" moments to construct the Hankel data matrices and the realize the system from the resulting matrix pencil.

Input arguments:

• sigma –

  shift value \( = \infty \Leftrightarrow \) Hankel, \( ! \infty \Leftrightarrow \) SPLoewner

• A –

  state matrix

• B –

  reachability matrix

• C –

  observability matrix

• K –

  half of the number of moments to use in data matrix construction

• m –

  number of poles to search for in \( \Omega \)

• L –

  left matrix of probing directions

• R –

  right matrix of probing directions

• options –

  options for realization

Numerics.sploewner.build_sploewner

(sigma, Ml, Mr, Mlr, K)

Helper function to build the data matrices for SPLoewner realization.