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Singular Value Decomposition


Let A be a general real m-by-n matrix. The singular value decomposition (SVD) of A is the factorization tex2html_wrap_inline1373 , where U and V are orthogonal, and tex2html_wrap_inline1379 , tex2html_wrap_inline1381 , with tex2html_wrap_inline1383 . If A is complex, then its SVD is tex2html_wrap_inline1387 where U and V are unitary, and tex2html_wrap_inline1393 is as before with real diagonal elements. The tex2html_wrap_inline1395 are called the singular values, the first r columns of V the right singular vectors and the first r columns of U the left singular vectors.

The SVD and symmetric eigendecompositions are entirely analogous, so that any algorithm for one has a counterpart for the other. As soon as the final version of the symmetric eigenvalue algorithm has been developed, we will produce an SVD version. In the meantime, we plan to release an SVD code based on serial QR iteration, where each processor redundantly runs QR iteration on a bidiagonal matrix, but updates a subset of the rows of the U and V in an embarrassingly parallel fashion.

Susan Blackford
Thu Jul 25 15:38:00 EDT 1996