Параллельные вычисления в ИММ УрО РАН
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Further Details: Error Bounds for the Singular Value DecompositionThe usual error analysis of the SVD algorithm PxGESVD in ScaLAPACK (see subsection 3.2.3)is as follows [71]:
The SVD algorithm is backward stable. This means that the computed SVD, , is nearly the exact SVD of A+E where , and p(m,n) is a modestly growing function of m and n. This means is the true SVD, so that and are both orthogonal, where , and . Each computed singular value differs from true by at most There is a small possibility that PxGESVD will fail to achieve the above error bounds on a heterogeneous network of processors for reasons discussed in section 6.2. On a homogeneous network, PxGESVD is as robust as the corresponding LAPACK routine xGESVD. A future release will attempt to detect heterogeneity and warn the user to use an alternative algorithm. In the special case of bidiagonal matrices, the singular values and singular vectors may be computed much more accurately. A bidiagonal matrix B has nonzero entries only on the main diagonal and the diagonal immediately above it (or immediately below it). PxGESVD computes the SVD of a general matrix by first reducing it to bidiagonal form B, and then calling xBDSQR (subsection 3.3.6) to compute the SVD of B. Reduction of a dense matrix to bidiagonal form B can introduce additional errors, so the following bounds for the bidiagonal case do not apply to the dense case. For the error analysis of xBDSQR, see the LAPACK manual.
Next: Error Bounds for the Up: Error Bounds for the Previous: Error Bounds for the Susan Blackford Tue May 13 09:21:01 EDT 1997 |