?lasda

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?lasda

Computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by ?bdsdc.

Syntax

call slasda( icompq, smlsiz, n, sqre, d, e, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info )

call dlasda( icompq, smlsiz, n, sqre, d, e, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info )

Include Files

mkl.fi

Description

Using a divide and conquer approach, ?lasda computes the singular value decomposition (SVD) of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e, where m = n + sqre.

The algorithm computes the singular values in the SVDB = U*S*VT. The orthogonal matrices U and VT are optionally computed in compact form. A related subroutine ?lasd0 computes the singular values and the singular vectors in explicit form.

Input Parameters

icompq

INTEGER.

Specifies whether singular vectors are to be computed in compact form, as follows:

= 0: Compute singular values only.

= 1: Compute singular vectors of upper bidiagonal matrix in compact form.

smlsiz

INTEGER.

The maximum size of the subproblems at the bottom of the computation tree.

n

INTEGER. The row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array d.

sqre

INTEGER. Specifies the column dimension of the bidiagonal matrix.

If sqre = 0: the bidiagonal matrix has column dimension m = n

If sqre = 1: the bidiagonal matrix has column dimension m = n + 1.

d

REAL for slasda

DOUBLE PRECISION for dlasda.

Array, DIMENSION (n). On entry, d contains the main diagonal of the bidiagonal matrix.

e

REAL for slasda

DOUBLE PRECISION for dlasda.

Array, DIMENSION ( m - 1 ). Contains the subdiagonal entries of the bidiagonal matrix. On exit, e is destroyed.

ldu

INTEGER. The leading dimension of arrays u, vt, difl, difr, poles, givnum, and z. ldu≥n.

ldgcol

INTEGER. The leading dimension of arrays givcol and perm. ldgcol≥n.

work

REAL for slasda

DOUBLE PRECISION for dlasda.

Workspace array, DIMENSION (6n+(smlsiz+1)²).

iwork

INTEGER.

Workspace array, Dimension must be at least (7n).

Output Parameters

d

On exit d, if info = 0, contains the singular values of the bidiagonal matrix.

u

REAL for slasda

DOUBLE PRECISION for dlasda.

Array, DIMENSION (ldu, smlsiz) if icompq =1.

Not referenced if icompq = 0.

If icompq = 1, on exit, u contains the left singular vector matrices of all subproblems at the bottom level.

vt

REAL for slasda

DOUBLE PRECISION for dlasda.

Array, DIMENSION ( ldu, smlsiz+1 ) if icompq = 1, and not referenced if icompq = 0. If icompq = 1, on exit, vt' contains the right singular vector matrices of all subproblems at the bottom level.

k

INTEGER.

Array, DIMENSION (n) if icompq = 1 and

DIMENSION (1) if icompq = 0.

If icompq = 1, on exit, k(i) is the dimension of the i-th secular equation on the computation tree.

difl

REAL for slasda

DOUBLE PRECISION for dlasda.

Array, DIMENSION ( ldu, nlvl ),

where nlvl = floor(log₂(n/smlsiz)).

difr

REAL for slasda

DOUBLE PRECISION for dlasda.

Array,

DIMENSION ( ldu, 2 nlvl ) if icompq = 1 and

DIMENSION (n) if icompq = 0.

If icompq = 1, on exit, difl(1:n, i) and difr(1:n,2i -1) record distances between singular values on the i-th level and singular values on the (i -1)-th level, and difr(1:n, 2i ) contains the normalizing factors for the right singular vector matrix. See ?lasd8 for details.

z

REAL for slasda

DOUBLE PRECISION for dlasda.

Array,

DIMENSION ( ldu, nlvl ) if icompq = 1 and

DIMENSION (n) if icompq = 0. The first k elements of z(1, i) contain the components of the deflation-adjusted updating row vector for subproblems on the i-th level.

poles

REAL for slasda

DOUBLE PRECISION for dlasda.

Array, DIMENSION(ldu, 2*nlvl)

if icompq = 1, and not referenced if icompq = 0. If icompq = 1, on exit, poles(1, 2i - 1) and poles(1, 2i) contain the new and old singular values involved in the secular equations on the i-th level.

givptr

INTEGER. Array, DIMENSION (n) if icompq = 1, and not referenced if icompq = 0. If icompq = 1, on exit, givptr( i ) records the number of Givens rotations performed on the i-th problem on the computation tree.

givcol

INTEGER .

Array, DIMENSION(ldgcol, 2*nlvl) if icompq = 1, and not referenced if icompq = 0. If icompq = 1, on exit, for each i, givcol(1, 2 i - 1) and givcol(1, 2 i) record the locations of Givens rotations performed on the i-th level on the computation tree.

perm

INTEGER. Array, DIMENSION ( ldgcol, nlvl ) if icompq = 1, and not referenced if icompq = 0. If icompq = 1, on exit, perm (1, i) records permutations done on the i-th level of the computation tree.

givnum

REAL for slasda

DOUBLE PRECISION for dlasda.

Array DIMENSION ( ldu, 2*nlvl ) if icompq = 1, and not referenced if icompq = 0. If icompq = 1, on exit, for each i, givnum(1, 2 i - 1) and givnum(1, 2 i) record the C- and S-values of Givens rotations performed on the i-th level on the computation tree.

c

REAL for slasda

DOUBLE PRECISION for dlasda.

Array,

DIMENSION (n) if icompq = 1, and

DIMENSION (1) if icompq = 0.

If icompq = 1 and the i-th subproblem is not square, on exit, c(i) contains the C-value of a Givens rotation related to the right null space of the i-th subproblem.

s

REAL for slasda

DOUBLE PRECISION for dlasda.

Array,

DIMENSION (n) icompq = 1, and

DIMENSION (1) if icompq = 0.

If icompq = 1 and the i-th subproblem is not square, on exit, s(i) contains the S-value of a Givens rotation related to the right null space of the i-th subproblem.

info

INTEGER.

= 0: successful exit.

< 0: if info = -i, the i-th argument had an illegal value

> 0: If info = 1, an singular value did not converge

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?lasda