# zbMATH — the first resource for mathematics

A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra. (English) Zbl 0959.65061
The authors propose a new matrix $$1$$-norm estimation algorithm which is applicable to real and complex $$(n \times n)$$ matrices. The algorithm is a block generalization of the $$1$$-norm power method [see, e.g., W. W. Hager, SIAM J. Sci. Stat. Comput. 5, 311-316 (1984; Zbl 0542.65023); J. Higham, ACM Trans. Math. Softw. 14, No. 4, 381-396 (1988; Zbl 0665.65043)]. Instead of one starting vector a $$(n \times t)$$ matrix is used, where the last $$t-1$$ columns are randomly chosen. For $$t=1$$ the new algorithm is similar to the original algorithm. There are some improvements leading to a higher efficiency.
The behaviour of the algorithm is demonstrated by numerical examples. The accuracy and reliability of the estimates increase generally with $$t$$. The number of iterations of the algorithm required for convergence is essentially independent of $$t$$.
Finally, the application of the new algorithm to the computation of $$1$$-norm pseudospectra is discussed.

##### MSC:
 65F35 Numerical computation of matrix norms, conditioning, scaling
##### Software:
condest; LAPACK; LINPACK; mctoolbox
Full Text: