New Fast and Accurate Jacobi SVD Algorithm. I

This paper is the result of contrived 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 a general dense matrix. 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. Despite its sound numerical qualities, the Jacobi SVD is not included in the state of the art matrix computation libraries and it is even considered obsolete by some leading researches. 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, and it outperforms not only the best implementations of the one–sided Jacobi algorithm but also the QR algorithm. Moreover, it seems that the potential of the new approach is yet to be fully exploited.