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

Source: R/bigalgebra.R
dgesdd.Rd

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
)

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.

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.

Examples

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)
#> [1] 0
S
#>           [,1]
#> [1,] 2.5683690
#> [2,] 1.9865047
#> [3,] 0.1726739
U
#>            [,1]        [,2]       [,3]       [,4]
#> [1,]  0.1711751  0.09221058 0.78855772  0.5834150
#> [2,]  0.8605882 -0.33631329 0.12096045 -0.3628360
#> [3,]  0.2092122  0.91796420 0.08793762 -0.3253290
#> [4,] -0.4316449 -0.18902993 0.59650002 -0.6497215
VT
#>            [,1]        [,2]        [,3]
#> [1,]  0.2496213 -0.04136007  0.96745984
#> [2,]  0.1202120 -0.99003537 -0.07334196
#> [3,] -0.9608529 -0.13460796  0.24216198

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)
#> [1] 0
S[,]
#> [1] 2.6819327 1.2859400 0.8128895
U[,]
#>            [,1]       [,2]       [,3]        [,4]
#> [1,] -0.1658242 -0.5205485 -0.3243924 -0.77220537
#> [2,]  0.4356896 -0.6461869 -0.3770659  0.50043820
#> [3,] -0.8648680 -0.1078115 -0.3028043  0.38560286
#> [4,] -0.1862263 -0.5475842  0.8129578  0.06760851
VT[,]
#>           [,1]       [,2]       [,3]
#> [1,] 0.5814564  0.0745909  0.8101510
#> [2,] 0.4372976  0.8110586 -0.3885289
#> [3,] 0.6860607 -0.5801897 -0.4389768

rm(A,S,U,VT)
gc()
#>           used (Mb) gc trigger  (Mb) limit (Mb) max used  (Mb)
#> Ncells 1145916 61.2    2286118 122.1         NA  2286118 122.1
#> Vcells 2052436 15.7    8388608  64.0      65536  5285191  40.4