# Documentation

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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 `rng` to set the random number generator to its startup settings before using `condest`.`rng('default')`

## Tips

This function is particularly useful for sparse matrices.

## Algorithms

`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 ${‖{A}^{-1}‖}_{1}$ without computing A−1. This is posed as the convex but nondifferentiable optimization problem $\mathrm{max}{‖{A}^{-1}x‖}_{1}$ 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.