# ?lals0

Applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by ?gelsd.

## Syntax

call slals0 ( icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, info )

call dlals0 ( icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, info )

call clals0( icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, rwork, info )

call zlals0( icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, rwork, info )

• mkl.fi

## Description

The routine applies back the multiplying factors of either the left or right singular vector matrix of a diagonal matrix appended by a row to the right hand side matrix B in solving the least squares problem using the divide-and-conquer SVD approach.

For the left singular vector matrix, three types of orthogonal matrices are involved:

(1L) Givens rotations: the number of such rotations is givptr;the pairs of columns/rows they were applied to are stored in givcol;and the c- and s-values of these rotations are stored in givnum.

(2L) Permutation. The (nl+1)-st row of B is to be moved to the first row, and for j=2:n, perm(j)-th row of B is to be moved to the j-th row.

(3L) The left singular vector matrix of the remaining matrix.

For the right singular vector matrix, four types of orthogonal matrices are involved:

(1R) The right singular vector matrix of the remaining matrix.

(2R) If `sqre = 1`, one extra Givens rotation to generate the right null space.

(3R) The inverse transformation of (2L).

(4R) The inverse transformation of (1L).

## Input Parameters

icompq

INTEGER. Specifies whether singular vectors are to be computed in factored form:

If `icompq = 0`: Left singular vector matrix.

If `icompq = 1`: Right singular vector matrix.

nl

INTEGER. The row dimension of the upper block.

`nl ≥ 1`.

nr

INTEGER. The row dimension of the lower block.

`nr ≥ 1`.

sqre

INTEGER.

If `sqre = 0`: the lower block is an nr-by-nr square matrix.

If `sqre = 1`: the lower block is an nr-by-(nr+1) rectangular matrix. The bidiagonal matrix has row dimension `n = nl + nr + 1`, and column dimension `m = n + sqre`.

nrhs

INTEGER. The number of columns of B and bx.

Must be at least 1.

b

REAL for slals0

DOUBLE PRECISION for dlals0

COMPLEX for clals0

DOUBLE COMPLEX for zlals0.

Array, DIMENSION ( ldb, nrhs ).

Contains the right hand sides of the least squares problem in rows 1 through m.

ldb

INTEGER. The leading dimension of b.

Must be at least `max(1,max( m, n ))`.

bx

REAL for slals0

DOUBLE PRECISION for dlals0

COMPLEX for clals0

DOUBLE COMPLEX for zlals0.

Workspace array, DIMENSION ( ldbx, nrhs ).

ldbx

INTEGER. The leading dimension of bx.

perm

INTEGER. Array, DIMENSION (n).

The permutations (from deflation and sorting) applied to the two blocks.

givptr

INTEGER. The number of Givens rotations which took place in this subproblem.

givcol

INTEGER. Array, DIMENSION ( ldgcol, 2 ). Each pair of numbers indicates a pair of rows/columns involved in a Givens rotation.

ldgcol

INTEGER. The leading dimension of givcol, must be at least n.

givnum

REAL for slals0/clals0

DOUBLE PRECISION for dlals0/zlals0

Array, DIMENSION ( ldgnum, 2 ). Each number indicates the c or s value used in the corresponding Givens rotation.

ldgnum

INTEGER. The leading dimension of arrays difr, poles and givnum, must be at least k.

poles

REAL for slals0/clals0

DOUBLE PRECISION for dlals0/zlals0

Array, DIMENSION ( ldgnum, 2 ). On entry, poles(1:k, 1) contains the new singular values obtained from solving the secular equation, and poles(1:k, 2) is an array containing the poles in the secular equation.

difl

REAL for slals0/clals0

DOUBLE PRECISION for dlals0/zlals0

Array, DIMENSION ( k ). On entry, difl(i) is the distance between i-th updated (undeflated) singular value and the i-th (undeflated) old singular value.

difr

REAL for slals0/clals0

DOUBLE PRECISION for dlals0/zlals0

Array, DIMENSION ( ldgnum, 2 ). On entry, difr(i, 1) contains the distances between i-th updated (undeflated) singular value and the i+1-th (undeflated) old singular value. And difr(i, 2) is the normalizing factor for the i-th right singular vector.

z

REAL for slals0/clals0

DOUBLE PRECISION for dlals0/zlals0

Array, DIMENSION ( k ). Contains the components of the deflation-adjusted updating row vector.

K

INTEGER. Contains the dimension of the non-deflated matrix. This is the order of the related secular equation. `1 ≤ k ≤ n`.

c

REAL for slals0/clals0

DOUBLE PRECISION for dlals0/zlals0

Contains garbage if `sqre =0` and the c value of a Givens rotation related to the right null space if `sqre = 1`.

s

REAL for slals0/clals0

DOUBLE PRECISION for dlals0/zlals0

Contains garbage if `sqre =0` and the s value of a Givens rotation related to the right null space if `sqre = 1`.

work

REAL for slals0

DOUBLE PRECISION for dlals0

Workspace array, DIMENSION ( k ). Used with real flavors only.

rwork

REAL for clals0

DOUBLE PRECISION for zlals0

Workspace array, DIMENSION (k*(1+nrhs) + 2*nrhs). Used with complex flavors only.

## Output Parameters

b

On exit, contains the solution X in rows 1 through n.

info

INTEGER.

If `info = 0`: successful exit.

If `info = -i < 0`, the i-th argument had an illegal value.

For more complete information about compiler optimizations, see our Optimization Notice.