Abstract
New lower bounds on the minimal condition numbers of a matrix with respect to both one-sided and two-sided scaling by diagonal matrices are obtained. These bounds improve certain results obtained by F. L. Bauer.
- 1 BAUER, F.L. Optimally scMed matrices. Numer.Math. 5 (1963),73-87.Google Scholar
- 2 BAUER, F.L. Some aspects of scaling invariance. Colloq. Internat. C.N.R.S., No. 165t 1966, pp. 37-47.Google Scholar
- 3 BAUER, F.L. Remarks on optimally scMed matrices. Numer. Math. 13 (1969), 1-3.Google Scholar
- 4 BAUER, F. L., STOER, J., AND WIZZGALL, C. Absolute and monotonic norms. Numer. Math. $ (1961), 257-264.Google Scholar
- 5 BUSINGER, P.A. Matrices which can be optimally scaled. Numer. Math. 12 (1968), 346-348.Google Scholar
- 6 BUSINGEI~, P. A. Extremal properties of balanced tri-diagonal matrices. Math. Comput. ~$ (1969), 193-195.Google Scholar
- 7 FADDEEVA, V.N. Some extremum problems for matrix norms. Zh. Vychisl. Mat. i Mat. Fiz. 7 (1967), 401-404.Google Scholar
- 8 FADDEEV, D. K., KUBLANOVSKAYA, V. N., AND FADDEEVA, V.N. Sur les syst~mes lin~aires alg6briques de matrices rectangulaires et mal-conditiondes. Colloq. Internat. C.N.R.S., No. 165, 1966, pp. 161-170.Google Scholar
- 9 FENNER, T. I., AND LOIZOU, G. Matrix bounds on the spectral condition number. Linear Alge. bra & Its Appls. 8 (1974), 157-178.Google Scholar
- 10 FOCKE, J. ~ber die multiplikativit~t yon matrizennormen. Numer. Math. 7 (1965), 251-254.Google Scholar
- 11 OSBORNE, E.E. On pre-conditioning of matrices. J. ACM 7 (1960), 338-345. Google Scholar
- 12 STOER, J., AND WXTZGALL, C. Transformations by diagonal matrices in a normed space. Numer. Math. ~ (1962), 158-171.Google Scholar
- 13 VAN DER SLUIS, A.Condition numbers and equilibration of matrices. Numer. Math. 14 (1969), 14-23.Google Scholar
- 14 VAN DER SLUIS, A. Stability of solutions of linear algebraic systems. Numer. Math. 14 (1970), 246-251.Google Scholar
- 15 VAN hER SLU~S, A. Condition, equilibration and pivoting in linear algebraic systems. Numer. Math. 15 (1970), 74-86.Google Scholar
- 16 WXLKINSON, J.H. Rounding Errors in A{gebraic Processes. H.M.S.O., London, 1963. Google Scholar
Index Terms
Some New Bounds on the Condition Numbers of Optimally Scaled Matrices
Recommendations
Eigenvalue condition numbers and pseudospectra of Fiedler matrices
The aim of the present paper is to analyze the behavior of Fiedler companion matrices in the polynomial root-finding problem from the point of view of conditioning of eigenvalues. More precisely, we compare: (a) the condition number of a given root $${\...
Optimally scaled and optimally conditioned Vandermonde and Vandermonde-like matrices
AbstractVandermonde matrices with real nodes are known to be severely ill-conditioned. We investigate numerically the extent to which the condition number of such matrices can be reduced, either by row-scaling or by optimal configurations of nodes. In the ...
Condition Numbers of Random Triangular Matrices
Let Ln be a lower triangular matrix of dimension n each of whose nonzero entries is an independent N(0,1) variable, i.e., a random normal variable of mean 0 and variance 1. It is shown that kn, the 2-norm condition number of Ln, satisfies \begin{...
Comments