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Function File: condest (A)

Function File: condest (A, t)

Function File: [est, v] = condest (…)

Function File: [est, v] = condest (A, solve, solve_t, t)

Function File: [est, v] = condest (apply, apply_t, solve, solve_t, n, t)

Estimate the 1-norm condition number of a matrix A using t test vectors using a randomized 1-norm estimator.

If t exceeds 5, then only 5 test vectors are used.

If the matrix is not explicit, e.g., when estimating the condition number of A given an LU factorization, condest uses the following functions:

apply

A*x for a matrix x of size n by t.

apply_t

A'*x for a matrix x of size n by t.

solve

A \ b for a matrix b of size n by t.

solve_t

A' \ b for a matrix b of size n by t.

The implicit version requires an explicit dimension n.

condest uses a randomized algorithm to approximate the 1-norms.

condest returns the 1-norm condition estimate est and a vector v satisfying norm (A*v, 1) == norm (A, 1) * norm (v, 1) / est. When est is large, v is an approximate null vector.

References:

• N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. SIMAX vol 21, no 4, pp 1185-1201. http://dx.doi.org/10.1137/S0895479899356080
• N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. http://citeseer.ist.psu.edu/223007.html

See also: cond, norm, onenormest.

Demonstration 1

 N = 100;
 A = randn (N) + eye (N);
 condest (A)
 [L,U,P] = lu (A);
 condest (A, @(x) U \ (L \ (P*x)), @(x) P'*(L' \ (U'\x)))
 condest (@(x) A*x, @(x) A'*x, @(x) U \ (L \ (P*x)), @(x) P'*(L' \ (U'\x)), N)
 norm (inv (A), 1) * norm (A, 1)

ans =  979.54
ans =  979.54
ans =  936.81
ans =  979.54

Package: octave