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Multi-Point Loewner

Numerics.mploewner.build_loewner 

(BB, CC, theta, sigma)

construct base and shifted Loewner matrices from left and right interpolation data

Numerics.mploewner.mploewner_quadrature 

(z, w, Ql, Qr, L, R, theta, sigma, m, PadStrategy=NaN, Verbose=true, AbsTol=NaN)

Multi-Point Loewner realization with one-sided quadrature samples. Given left/right quadrature data Ql/Qr, compute probed left/right transfer function samples at left/right interpolation points (theta/sigma) via contour integration approximated by a quadrature rule with nodes and weights \( ( z_k, w_k ) \).

Input arguments:

  • 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 \(z\)

  • L –

    \( n \times \ell \) matrix of left probing directions

  • R –

    \( n \times r \) matrix of right probing directions

  • theta –

    left interpolation points

  • sigma –

    right interpolation points

  • m –

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

Name-value arguments:

  • PadStrategy –

    padding strategy for construction of BB/CC (if the number of left/right tangential directions is less than the number of corresponding interpolation points)

  • Verbose –

    verbose output (or not)

  • AbsTol –

    absolute tolerance for base data matrix rank determination

Numerics.mploewner.build_exact_data 

(H, theta, sigma, L=randn(size(H(randn(1)),1),length(theta)), R=randn(size(H(randn(1)),1),length(sigma)), PadStrategy="cyclical", Verbose=true)

construct left/right interpolation data BB/CC from exact transfer function evaluations at theta/sigma

Numerics.mploewner.mploewner_exact_siso 

(H, theta, sigma, m, PadStrategy=NaN, Verbose=true, AbsTol=NaN)

Multi-Point Loewner realization with exact transfer function samples.

Input arguments:

  • H –

    transfer function

  • theta –

    left interpolation points

  • sigma –

    right interpolation points

  • m –

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

Name-value arguments:

  • PadStrategy –

    padding strategy for construction of BB/CC (if the number of left/right tangential directions is less than the number of corresponding interpolation points)

  • Verbose –

    verbose output (or not)

  • AbsTol –

    absolute tolerance for base data matrix rank determination

Numerics.mploewner.build_quadrature_data 

(z, w, Ql, Qr, L, R, theta, sigma, PadStrategy="cyclical", Verbose=true)

construct left/right interpolation data BB/CC from left/right quadrature evaluations given quadrature nodes/weights z and w

Numerics.mploewner.mploewner_exact 

(H, theta, sigma, L, R, m, PadStrategy=NaN, Verbose=true, AbsTol=NaN)

Multi-Point Loewner realization with exact transfer function samples.

Input arguments:

  • H –

    transfer function

  • theta –

    left interpolation points

  • sigma –

    right interpolation points

  • L –

    \( n \times \ell \) matrix of left probing directions

  • R –

    \( n \times r \) matrix of right probing directions

  • m –

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

Name-value arguments:

  • PadStrategy –

    padding strategy for construction of BB/CC (if the number of left/right tangential directions is less than the number of corresponding interpolation points)

  • Verbose –

    verbose output (or not)

  • AbsTol –

    absolute tolerance for base data matrix rank determination

Numerics.mploewner.build_exact_data_siso 

(H, theta, sigma)

construct left/right interpolation data BB/CC from exact transfer function evaluations at theta/sigma