Condition number of singular value: zero-structured and patterned case

In this paper, the condition number of singular value is discussed. The expressions of the zero-structured and patterned conditioning are also studied. The numerical tests are reported to provide interesting information about the singular value sensitivity when the perturbations of the matrix have arbitrarily assigned zero-structured and patterned case.

[1]  Yimin Wei,et al.  Condition Numbers for Structured Least Squares Problems , 2006 .

[2]  J. H. Wilkinson The algebraic eigenvalue problem , 1966 .

[3]  Yi-min Wei,et al.  Condition Number for Under-Determined Toeplitz Systems , 2006 .

[4]  Richard J. Vaccaro,et al.  A Second-Order Perturbation Expansion for the SVD , 1994 .

[5]  G. Stewart A note on the perturbation of singular values , 1979 .

[6]  Ilse C. F. Ipsen Relative perturbation results for matrix eigenvalues and singular values , 1998, Acta Numerica.

[7]  Ren-Cang Li,et al.  Relative Perturbation Theory: II. Eigenspace and Singular Subspace Variations , 1996, SIAM J. Matrix Anal. Appl..

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

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

[10]  Silvia Noschese,et al.  Eigenvalue patterned condition numbers: Toeplitz and Hankel cases , 2007 .

[11]  G. Stewart A second order perturbation expansion for small singular values , 1984 .

[12]  Nicholas J. Higham,et al.  Algorithm 694: a collection of test matrices in MATLAB , 1991, TOMS.

[13]  Nicholas J. Higham,et al.  Structured Backward Error and Condition of Generalized Eigenvalue Problems , 1999, SIAM J. Matrix Anal. Appl..

[14]  Françoise Tisseur,et al.  A Chart of Backward Errors for Singly and Doubly Structured Eigenvalue Problems , 2002, SIAM J. Matrix Anal. Appl..

[15]  Volker Mehrmann,et al.  Numerical methods in control , 2000 .

[16]  Silvia Noschese,et al.  Eigenvalue condition numbers: zero-structured versus traditional , 2006 .

[17]  Daniel B. Szyld,et al.  An introduction to iterative Toeplitz solvers , 2009, Math. Comput..

[18]  Albrecht Böttcher,et al.  Structured condition numbers of large Toeplitz matrices are rarely better than usual condition numbers , 2005, Numer. Linear Algebra Appl..

[19]  Gene H. Golub,et al.  Ill-conditioned eigensystems and the computation of the Jordan canonical form , 1975, Milestones in Matrix Computation.

[20]  Nicholas J. Higham,et al.  Backward Error and Condition of Structured Linear Systems , 1992, SIAM J. Matrix Anal. Appl..

[21]  Michael Stewart,et al.  Perturbation of the SVD in the presence of small singular values , 2006 .