bigalgebra (version 1.1.2)

dgesdd: DGESDD computes the singular value decomposition (SVD) of a real matrix.

Description

DGESDD computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and right singular vectors. If singular vectors are desired, it uses a divide-and-conquer algorithm.

The SVD is written

A = U * SIGMA * transpose(V)

where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A.

Note that the routine returns VT = V**T, not V.

Usage

dgesdd(
  JOBZ = "A",
  M = NULL,
  N = NULL,
  A,
  LDA = NULL,
  S,
  U,
  LDU = NULL,
  VT,
  LDVT = NULL,
  WORK = NULL,
  LWORK = NULL
)

Value

IWORK an integer matrix dimension of (8*min(M,N)) A is updated.

if

JOBZ = 'O', A is overwritten with the first N columns of U (the left singular vectors, stored columnwise) if M >= N; A is overwritten with the first M rows of V**T (the right singular vectors, stored rowwise) otherwise.

if

JOBZ .ne. 'O', the contents of A are destroyed.

INFO an integer

= 0:

successful exit.

< 0:

if INFO = -i, the i-th argument had an illegal value.

> 0:

DBDSDC did not converge, updating process failed.

Arguments

JOBZ

a character. Specifies options for computing all or part of the matrix U:

= 'A':

all M columns of U and all N rows of V**T are returned in the arrays U and VT;

= 'S':

the first min(M,N) columns of U and the first min(M,N) rows of V**T are returned in the arrays U and VT;

= 'O':

If M >= N, the first N columns of U are overwritten on the array A and all rows of V**T are returned in the array VT; otherwise, all columns of U are returned in the array U and the first M rows of V**T are overwritten in the array A;

= 'N':

no columns of U or rows of V**T are computed.

M

an integer. The number of rows of the input matrix A. M >= 0.

N

an integer. The number of columns of the input matrix A. N >= 0.

A

the M-by-N matrix A.

LDA

an integer. The leading dimension of the matrix A. LDA >= max(1,M).

S

a matrix of dimension (min(M,N)). The singular values of A, sorted so that S(i) >= S(i+1).

U

U is a matrx of dimension (LDU,UCOL)

UCOL = M if

JOBZ = 'A' or JOBZ = 'O' and M < N; UCOL = min(M,N) if JOBZ = 'S'.

If

JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M orthogonal matrix U;

if

JOBZ = 'S', U contains the first min(M,N) columns of U (the left singular vectors, stored columnwise);

if

JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

LDU

an integer. The leading dimension of the matrix U. LDU >= 1; if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

VT

VT is matrix of dimension (LDVT,N)

If

JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the N-by-N orthogonal matrix V**T;

if

JOBZ = 'S', VT contains the first min(M,N) rows of V**T (the right singular vectors, stored rowwise);

if

JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

LDVT

an integer. The leading dimension of the matrix VT. LDVT >= 1; if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N; if JOBZ = 'S', LDVT >= min(M,N).

WORK

a matrix of dimension (MAX(1,LWORK))

LWORK

an integer. The dimension of the array WORK. LWORK >= 1. If LWORK = -1, a workspace query is assumed. The optimal size for the WORK array is calculated and stored in WORK(1), and no other work except argument checking is performed.

Let mx = max(M,N) and mn = min(M,N).

If

JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).

If

JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).

If

JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.

If

JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.

These are not tight minimums in all cases; see comments inside code. For good performance, LWORK should generally be larger; a query is recommended.

Examples

if (FALSE) {
set.seed(4669)
A = matrix(rnorm(12),4,3)
S = matrix(0,nrow=3,ncol=1)
U = matrix(0,nrow=4,ncol=4)
VT = matrix(0,ncol=3,nrow=3)
dgesdd(A=A,S=S,U=U,VT=VT)
S
U
VT

rm(A,S,U,VT)

A = as.big.matrix(matrix(rnorm(12),4,3))
S = as.big.matrix(matrix(0,nrow=3,ncol=1))
U = as.big.matrix(matrix(0,nrow=4,ncol=4))
VT = as.big.matrix(matrix(0,ncol=3,nrow=3))
dgesdd(A=A,S=S,U=U,VT=VT)
S[,]
U[,]
VT[,]

rm(A,S,U,VT)
gc()
}