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Function File: [est, v, w, iter] = onenormest (A, t)

Function File: [est, v, w, iter] = onenormest (apply, apply_t, n, t)

Apply Higham and Tisseur’s randomized block 1-norm estimator to matrix A using t test vectors.

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

If the matrix is not explicit, e.g., when estimating the norm of inv (A) given an LU factorization, onenormest applies A and its conjugate transpose through a pair of functions apply and apply_t, respectively, to a dense matrix of size n by t. The implicit version requires an explicit dimension n.

Returns the norm estimate est, two vectors v and w related by norm (w, 1) = est * norm (v, 1), and the number of iterations iter. The number of iterations is limited to 10 and is at least 2.

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: condest, norm, cond.

Demonstration 1

 N = 100;
 A = randn (N) + eye (N);
 [L,U,P] = lu (A);
 nm1inv = onenormest (@(x) U\(L\(P*x)), @(x) P'*(L'\(U'\x)), N, 30)
 norm (inv (A), 1)

nm1inv =  20.956
ans =  20.956

Package: octave