Documentation

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 boundcfor the 1-norm condition number of a square matrixA.

c = condest(A,t)changest, 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 ist = 2, which almost always gives an estimate correct to within a factor 2.

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

    Note:condestinvokesrand. If repeatable results are required then userngto set the random number generator to its startup settings before usingcondest.

    rng('default')

Tips

This function is particularly useful for sparse matrices.

Algorithms

condestis 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 ‖ A − 1 ‖ 1 without computingA−1. This is posed as the convex but nondifferentiable optimization problem max ‖ A − 1 x ‖ 1 subject to ‖ x ‖ 1 = 1

References

[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.

See Also

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Introduced before R2006a

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