MATLAB^®

condest - 1-norm condition number estimate

Syntax

c = condest(A) c = condest(A,t) [c,v] = condest(A)

Description

c = condest(A) computes a lower bound C for the 1-norm condition number of a square matrix A.

c = condest(A,t) changes t, a positive integer parameter equal to the number of columns in an underlying iteration matrix. Increasing the number of columns usually gives a better condition estimate but increases the cost. The default is t = 2, which almost always gives an estimate correct to within a factor 2.

[c,v] = condest(A) also computes a vector v which is an approximate null vector if c is large. v satisfies norm(A*v,1) = norm(A,1)*norm(v,1)/c.

Note condest invokes rand. If repeatable results are required then invoke rand('state',j), for some j, before calling this function.

This function is particularly useful for sparse matrices.

Algorithm

condest is based on the 1-norm condition estimator of Hager [1] and a block oriented generalization of Hager's estimator given by Higham and Tisseur [2]. The heart of the algorithm involves an iterative search to estimate without computing . This is posed as the convex, but nondifferentiable, optimization problem

subject to

Reference

[1] William W. Hager, "Condition Estimates," SIAM J. Sci. Stat. Comput. 5, 1984, 311-316, 1984.

[2] Nicholas J. Higham and Françoise Tisseur, "A Block Algorithm for Matrix 1-Norm Estimation with an Application to 1-Norm Pseudospectra, "SIAM J. Matrix Anal. Appl., Vol. 21, 1185-1201, 2000.

condeig

coneplot

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