Acta Numerica 2004: Volume 13 (Acta Numerica) by Arieh Iserles

By Arieh Iserles

Acta Numerica surveys every year crucial advancements in numerical arithmetic and clinical computing. the themes and authors of the great survey articles are selected by means of a exclusive overseas editorial board to be able to document an important and well timed advancements in a way available to the broader neighborhood of pros with an curiosity in medical computing. Acta Numerica volumes have proved to be a beneficial device not just for researchers and pros wishing to increase their figuring out of numerical ideas and algorithms and stick to new advancements, but in addition as a sophisticated educating reduction at schools and universities. some of the unique articles were used because the best source for graduate classes. this actual quantity was once initially released in 2004.

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1 compute % , . . , x\ by back-substitution in the triangular systems ihiXi — Ci O{Z. Note that in steps (1) and (3) the computations can be performed in parallel on the M independent subsystems. There are many alternative ways to organize this algorithm. Cox (1990) considers the following modifications to reduce the storage requirement. By merging steps (1) and (2) it is not necessary to hold all blocks Tj simultaneously in memory. Even more storage can be saved by discarding R4 and Si after Tj has been computed in step (1).

1) is uniquely determined in the nondegenerate case. Therefore the two algorithms will, using exact arithmetic, produce the same bidiagonal decomposition. However, the Lanczos method is less stable numerically and is mainly of interest when A is a large and sparse matrix. 1. Bidiagonalization using Householder transformations With no loss of generality we assume that m > n. , we alternately multiply (b, A) from the left and right with Householder transformations Qk and Pk, respectively. Here Qk is chosen to zero the last m — k elements in the kth column of (b, A) and Pk is chosen to zero the last n — k elements in the A;th row of A.

Ostrouchov and D. Sorensen, eds (1995), LAPACK Users' Guide, 2nd edn, SIAM, Philadelphia. E. Anderssen, Z. Bai and J. Dongarra (1992), 'Generalized QR factorization and its applications', Linear Algebra Appl. 162-164, 243-271. M. Arioli (2000), 'The use of QR factorization in sparse quadratic programming and backward error issues', SIAM J. Matrix Anal. Appl. 21, 825-839. Z. Bai and J. W. Demmel (1993), 'Computing the generalized singular value decomposition', SIAM J. Sci. Comput. 14, 1464-1486.

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