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### Singular Value Decomposition (SVD)

The singular value decomposition of an matrix is given by where and are orthogonal (unitary) and is an diagonal matrix with real diagonal elements, , such that The are the singular values of and the first columns of and are the left and right singular vectors of .
The singular values and singular vectors satisfy: where and are the columns of and 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 LA_GESVD computes all the singular values and (optionally) left and/or right singular vectors.
• a divide and conquer driver LA_GESDD 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 in the LAPACK Users' Guide).

 Type of Function and storage scheme Real/complex Complex problem Hermitian SEP simple driver LA_SYEV LA_HEEV divide and conquer driver LA_SYEVD LA_HEEVD expert driver LA_SYEVX LA_HEEVX RRR driver LA_SYEVR LA_HEEVR simple driver (packed storage) LA_SPEV LA_HPEV divide and conquer driver LA_SPEVD LA_HPEVD (packed storage) expert driver (packed storage) LA_SPEVX LA_HPEVX simple driver (band matrix) LA_SBEV LA_HBEV divide and conquer driver LA_SBEVD LA_HBEVD (band matrix) expert driver (band matrix) LA_SBEVX LA_HBEVX simple driver (tridiagonal matrix) LA_STEV divide and conquer driver LA_STEVD (tridiagonal matrix) (real only) expert driver (tridiagonal matrix) LA_STEVX RRR driver (tridiagonal matrix) LA_STEVR NEP simple driver for Schur factorization LA_GEES expert driver for Schur factorization LA_GEESX simple driver for eigenvalues/vectors LA_GEEV expert driver for eigenvalues/vectors LA_GEEVX SVD simple driver LA_GESVD divide and conquer driver LA_GESDD     Next: Generalized Eigenvalue and Singular Up: Standard Eigenvalue and Singular Previous: Nonsymmetric Eigenproblems (NEP)   Contents   Index
Susan Blackford 2001-08-19