PolyMatAnalyticEigValues

PolyMatAnalyticEigValues(R);
  [L,Perm,prec,xi2] = PolyMatAnalyticEigValues(R) returns the extracted analytic
  eigenvalues L of the polynomial matrix R [1,2] as described in [3]. The algorithm
  operated in discrete frequency bins and iterates until a defined bound for xi2 
  (see [3]) is reached, or the FFT length exceeds Nmax. Compared to a spectrally 
  majorised ordering, the permutations of the eigenvalues in each frequency bin 
  are returned in Perm. An approximate approximation error is returned in EVPrecision,
  which is a mix of truncation and time-domain aliasing. 

  Input parameters
     R         MxMxL parahermitian matrix
     Nmax      maximum FFT length (optional; default 2^10)
     PrecLimit precision limit for iteration

  Output parameters
     L         MxK matrix containing the extracted analytic eigenvalues in
               its rows
     Perm      Permutations compared to the spectrally majorised solution
     prec      time domain precision
     xi2       convergence/divergence metric

  References
  [1] S Weiss, J Pestana and IK Proulder: "On the existence and uniqueness of
      the eigenvalue decomposition of a parahermitian matrix," IEEE Trans. on
      Signal Processing, 66(10):2659-2672, May 2018.
  [2] S Weiss, J Pestana, IK Proulder, and FK Coutts: "Corrections to `On the 
      existence and uniqueness of the eigenvalue decomposition of a para-
      hermitian matrix'," IEEE Trans. on Signal Processing, 66(23):6325-6327, 
      Dec. 2018.
  [3] S Weiss, IK Proulder, and FK Coutts: "Parahermitian matrix eigenvalue 
      decomposition: extraction of analytic eigenvalues," IEEE Trans. on 
      Signal Processing, 69:722-737, Jan. 2021

0001 function [L_analytic,L_permutation,EVPrecision,xi2] = PolyMatAnalyticEigValues(R,Nmax,PrecLimit);
0002 %PolyMatAnalyticEigValues(R);
0003 %  [L,Perm,prec,xi2] = PolyMatAnalyticEigValues(R) returns the extracted analytic
0004 %  eigenvalues L of the polynomial matrix R [1,2] as described in [3]. The algorithm
0005 %  operated in discrete frequency bins and iterates until a defined bound for xi2
0006 %  (see [3]) is reached, or the FFT length exceeds Nmax. Compared to a spectrally
0007 %  majorised ordering, the permutations of the eigenvalues in each frequency bin
0008 %  are returned in Perm. An approximate approximation error is returned in EVPrecision,
0009 %  which is a mix of truncation and time-domain aliasing.
0010 %
0011 %  Input parameters
0012 %     R         MxMxL parahermitian matrix
0013 %     Nmax      maximum FFT length (optional; default 2^10)
0014 %     PrecLimit precision limit for iteration
0015 %
0016 %  Output parameters
0017 %     L         MxK matrix containing the extracted analytic eigenvalues in
0018 %               its rows
0019 %     Perm      Permutations compared to the spectrally majorised solution
0020 %     prec      time domain precision
0021 %     xi2       convergence/divergence metric
0022 %
0023 %  References
0024 %  [1] S Weiss, J Pestana and IK Proulder: "On the existence and uniqueness of
0025 %      the eigenvalue decomposition of a parahermitian matrix," IEEE Trans. on
0026 %      Signal Processing, 66(10):2659-2672, May 2018.
0027 %  [2] S Weiss, J Pestana, IK Proulder, and FK Coutts: "Corrections to `On the
0028 %      existence and uniqueness of the eigenvalue decomposition of a para-
0029 %      hermitian matrix'," IEEE Trans. on Signal Processing, 66(23):6325-6327,
0030 %      Dec. 2018.
0031 %  [3] S Weiss, IK Proulder, and FK Coutts: "Parahermitian matrix eigenvalue
0032 %      decomposition: extraction of analytic eigenvalues," IEEE Trans. on
0033 %      Signal Processing, 69:722-737, Jan. 2021