Singular Value Decomposition (SVD)

The singular value decomposition of an $m \times n$ 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 $m \times n$ 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 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[1]).

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

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