Singular Value Decomposition (SVD)

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

$$A = U \Sigma V ^T, \quad (A=U\Sigma V ^H \quad \mbox{in the complex case})$$

where U and V are orthogonal (unitary) and $\Sigma$ is an m-by-n diagonal matrix with real diagonal elements, $\sigma _ i$, such that

$$\sigma_1 \ge \sigma_2 \ge \ldots \ge \sigma_{\min (m,n)} \ge 0 .$$

The $\sigma _ i$ 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:

$$A v_i = \sigma_i u_i \quad \mbox{and} \quad A^T u_i = \sigma_i v_i \quad ({\rm or} \quad A^H u_i = \sigma_i v_i \quad )$$

where u_i and v_i are the i^th 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).

Table 2.5: Driver routines for standard eigenvalue and singular value problems

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