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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 use RandStream to initialize the random number generator before calling this function.

s = RandStream('mt19937ar','Seed',0);
RandStream.setDefaultStream(s)

See the RandStream documentation for more information.

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.