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### Singular Value Decomposition (SVD)

The singular value decomposition of an m-by-n matrix A is given by

where U and V are orthogonal (unitary) and is an m-by-n diagonal matrix with real diagonal elements, , such that

The are the singular values of A and the first min(m,n) columns of U and V are the left and right singular vectors of A.

The singular values and singular vectors satisfy:

where ui and vi are the ith columns of U and V respectively.

There are two types of driver routines for the SVD. Originally LAPACK had just the simple driver described below, and the other one was added after an improved algorithm was discovered.

• a simple driver xGESVD computes all the singular values and (optionally) left and/or right singular vectors.
• a divide and conquer driver xGESDD solves the same problem as the simple driver. It is much faster than the simple driver for large matrices, but uses more workspace. The name divide-and-conquer refers to the underlying algorithm (see sections 2.4.4 and 3.4.3).

 Type of Function and storage scheme Single precision Double precision problem real complex real complex SEP simple driver SSYEV CHEEV DSYEV ZHEEV divide and conquer driver SSYEVD CHEEVD DSYEVD ZHEEVD expert driver SSYEVX CHEEVX DSYEVX ZHEEVX RRR driver SSYEVR CHEEVR DSYEVR ZHEEVR simple driver (packed storage) SSPEV CHPEV DSPEV ZHPEV divide and conquer driver SSPEVD CHPEVD DSPEVD ZHPEVD (packed storage) expert driver (packed storage) SSPEVX CHPEVX DSPEVX ZHPEVX simple driver (band matrix) SSBEV CHBEV DSBEV ZHBEV divide and conquer driver SSBEVD CHBEVD DSBEVD ZHBEVD (band matrix) expert driver (band matrix) SSBEVX CHBEVX DSBEVX ZHBEVX simple driver (tridiagonal matrix) SSTEV DSTEV divide and conquer driver SSTEVD DSTEVD (tridiagonal matrix) expert driver (tridiagonal matrix) SSTEVX DSTEVX RRR driver (tridiagonal matrix) SSTEVR DSTEVR NEP simple driver for Schur factorization SGEES CGEES DGEES ZGEES expert driver for Schur factorization SGEESX CGEESX DGEESX ZGEESX simple driver for eigenvalues/vectors SGEEV CGEEV DGEEV ZGEEV expert driver for eigenvalues/vectors SGEEVX CGEEVX DGEEVX ZGEEVX SVD simple driver SGESVD CGESVD DGESVD ZGESVD divide and conquer driver SGESDD CGESDD DGESDD ZGESDD

Next: Generalized Eigenvalue and Singular Up: Standard Eigenvalue and Singular Previous: Nonsymmetric Eigenproblems (NEP)   Contents   Index
Susan Blackford
1999-10-01