Eigenproblems and Matrix Studies

A discussion during an informal seminar led to a proposal for a quick estimation of the largest eigenvalue of a matrix [J05]. A minor improvement on condition number estimation was given in [J07]. The accurate computation of eigenvalues of arrowhead matrices was considered in [J31]. Pete Stewart and I also studied the use of a modified Rayleigh quotient iteration for finding eigenvalues [J49].

Matrix scaling was studied in [J65], and factorizations of symmetric tridiagonal and triadic matrices were studied in [J77], for later use in modified Newton methods for optimization.

[J05]

Dianne P. O'Leary, G. W. Stewart, and James S. Vandergraft, ``Estimating the largest eigenvalue of a positive definite matrix,'' Mathematics of Computation 33 (1979) 1289-1292.

[J07]

Dianne P. O'Leary, ``Estimating matrix condition numbers,'' SIAM J. on Scientific and Statistical Computing 1 (1980) 205-209.

[J31]

Dianne P. O'Leary and G.W. Stewart, ``Computing the eigenvalues and eigenvectors of symmetric arrowhead matrices,'' Journal of Computational Physics 90 (1990) 497-505.

[J49]

Dianne P. O'Leary and G. W. Stewart, ``On the Convergence of a New Rayleigh Quotient Method with Applications to Large Eigenproblems," Electronic Transactions on Numerical Analysis, 7 (1998), http://etna.mcs.kent.edu/

[J65]

Dianne P. O'Leary, ``Scaling Symmetric Positive Definite Matrices to Prescribed Row Sums", Linear Algebra and Its Applications, 370 (2003) 185-191.

[J77]

Haw-ren Fang and Dianne P. O'Leary, ``Stable Factorizations of Symmetric Tridiagonal and Triadic Matrices," SIAM J. on Matrix Analysis and Applications, 28 (2006), pp. 576-595.