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New fast and accurate Jacobi SVD algorithm. I

This paper is the result of concerted efforts to break the barrier between numerical accuracy and run time efficiency in computing the fundamental decomposition of numerical linear algebra -- the singular value decomposition (SVD) of general dense matrices. It is an unfortunate fact that the numerically most accurate one--sided Jacobi SVD algorithm is several times slower than generally less accurate bidiagonalization based methods such as the QR or the divide and conquer algorithm. Our quest for a highly accurate and efficient SVD algorithm has led us to a new, superior variant of the Jacobi algorithm. The new algorithm has inherited all good high accuracy properties of the Jacobi algorithm, and it can outperform the QR algorithm.

Jacobi method; singular value decomposition; eigenvalues

Ovaj rad je rezultat istraživanja započetih i velikim dijelom završenih na prethodnom projektu 0037120. Drugi dio rada je objavljen u istom časopisu. Kako se radilo o dugoročnom opsežnom istraživanju, oba rada su preliminarno objavljena kako LAPACK Working notes pod brojevima 169 i 170 (http://www.netlib.org/lapack/lawns/downloads/). Proces recenziranja je trajao dugo i radovi su nakon finalne revizije konačno tiskani u siječnju 2008. Ovaj rad je nagradjen prestižnom strukovnom nagradom SIAG/Linear Algebra Prize, za najbolji rad u području primijenjene linearne algebre u trogodišnjem periodu, http://www.siam.org/prizes/sponsored/siagla.php O nagradjenim radovima se može pročitati u SIAM News na URL adresi http://www.siam.org/pdf/news/1696.pdf