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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.

65F35 Numerical computation of matrix norms, conditioning, scaling
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