RGSD an algorithm for computing the Kronecker structure and reducing subspaces of singular A-lB pencils

An algorithm (RGSVD) for computing the structure elements associated with the Kronecker canonical form (KCF) of a matrix pencil $A - \lambda B$, where A and B are complex m by n matrices, is presented. RGSVD is based on repeated generalized singular value decompositions (or more precisely cosine-sine decompositions of partitioned orthonormal matrices). It extracts the structures of the zero and/or the infinite eigenvalues together with the left (row) or right (column) minimal indices of $A - \lambda B$. By accumulating equivalence transformations, RGSVD also produces pairs of reducing subspaces associated with e.g. the zero structure and the right Kronecker indices.

[1]  J. Demmel.  A Numerical Analyst's Jordan Canonical Form. , 1983 .

[2]  V. Kublanovskaya,et al.  An approach to solving the spectral problem of A-λB , 1983 .

[3]  V. Kublanovskaya.  AB-Algorithm and its modifications for the spectral problems of linear pencils of matrices , 1984 .

[4]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[5]  Ji-guang Sun.  Perturbation Analysis for the Generalized Singular Value Problem , 1983 .

[6]  Paul Van Dooren,et al.  Algorithm 590: DSUBSP and EXCHQZ: FORTRAN Subroutines for Computing Deflating Subspaces with Specified Spectrum , 1982, TOMS.

[7]  P. Dooren.  Reducing subspaces: Definitions, properties and algorithms , 1983 .

[8]  C. Loan.  Generalizing the Singular Value Decomposition , 1976 .

[9]  Bo Kågström,et al.  An Algorithm for Numerical Computation of the Jordan Normal Form of a Complex Matrix , 1980, TOMS.

[10]  M. Saunders,et al.  Towards a Generalized Singular Value Decomposition , 1981 .

[11]  P. Dooren.  The generalized eigenstructure problem in linear system theory , 1981 .

[12]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[13]  C. Loan.  Computing the CS and the generalized singular value decompositions , 1985 .

[14]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[15]  James Hardy Wilkinson,et al.  Kronecker''s canonical form and the QZ algorithm , 1979 .

[16]  James Hardy Wilkinson,et al.  Linear Differential Equations and Kronecker's Canonical Form , 1978 .

[17]  W. Kahan.  Conserving Confluence Curbs Ill-Condition , 1972 .

[18]  Bo Kågström,et al.  The generalized singular value decomposition and the general (A-λB)-problem , 1984 .

[19]  Ludwig Elsner,et al.  Perturbation thèorems for the generalized eigenvalue problem , 1982 .

[20]  P. Dooren.  The Computation of Kronecker's Canonical Form of a Singular Pencil , 1979 .

[21]  Bo Kågström,et al.  Algorithm 560: JNF, An Algorithm for Numerical Computation of the Jordan Normal Form of a Complex Matrix [F2] , 1980, TOMS.

[22]  J. Demmel.  The condition number of equivalence transformations that block diagonalize matrix pencils , 1983 .

[23]  Charles Van Loan,et al.  A General Matrix Eigenvalue Algorithm , 1975 .

[24]  V. Kublanovskaya,et al.  On a method of solving the complete eigenvalue problem for a degenerate matrix , 1966 .

[25]  G. Stewart.  A method for computing the generalized singular value decomposition , 1983 .

[26]  Bo Kågström,et al.  On computing the Kronecker canonical form of regular (A-λB)-pencils , 1983 .

[27]  G. Stewart.  Computing theCS decomposition of a partitioned orthonormal matrix , 1982 .

[28]  C. Paige,et al.  A Note on a Result of Sun Ji-Guang: Sensitivity of the CS and GSV Decompositions , 1984 .

[29]  Ji-guang Sun,et al.  Perturbation analysis for the generalized eigenvalue and the generalized singular value problem , 1983 .