# ?lasd3

Finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by ?bdsdc.

## Syntax

call slasd3( nl, nr, sqre, k, d, q, ldq, dsigma, u, ldu, u2, ldu2, vt, ldvt, vt2, ldvt2, idxc, ctot, z, info )

call dlasd3( nl, nr, sqre, k, d, q, ldq, dsigma, u, ldu, u2, ldu2, vt, ldvt, vt2, ldvt2, idxc, ctot, z, info )

## Description

The routine ?lasd3 finds all the square roots of the roots of the secular equation, as defined by the values in D and Z.

It makes the appropriate calls to ?lasd4 and then updates the singular vectors by matrix multiplication.

The routine ?lasd3 is called from ?lasd1.

## Input Parameters

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-(n r+1) rectangular matrix. The bidiagonal matrix has `n = nl + nr + 1` rows and `m = n + sqre ≥ n` columns.

k

  INTEGER.The size of the secular equation, `1 ≤ `k ≤ n``.

q

  REAL for slasd3

  DOUBLE PRECISION for dlasd3

  Workspace array, DIMENSION at least (ldq, k).

ldq

  INTEGER. The leading dimension of the array Q.

  `ldq ≥ k`.

dsigma

  REAL for slasd3

  DOUBLE PRECISION for dlasd3

  Array, DIMENSION (k). The first k elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.

ldu

  INTEGER. The leading dimension of the array u.

  `ldu ≥ n`.

u2

  REAL for slasd3

  DOUBLE PRECISION for dlasd3

  Array, DIMENSION (ldu2, n).

  The first k columns of this matrix contain the non-deflated left singular vectors for the split problem.

ldu2

  INTEGER. The leading dimension of the array u2.

  `ldu2 ≥ n`.

ldvt

  INTEGER. The leading dimension of the array vt.

  `ldvt ≥ n`.

vt2

  REAL for slasd3

  DOUBLE PRECISION for dlasd3

  Array, DIMENSION (ldvt2, n).

  The first k columns of vt2' contain the non-deflated right singular vectors for the split problem.

ldvt2

  INTEGER. The leading dimension of the array vt2.

  `ldvt2 ≥ n`.

idxc

  INTEGER. Array, DIMENSION (n).

  The permutation used to arrange the columns of u (and rows of vt) into three groups: the first group contains non-zero entries only at and above (or before) `nl +1`; the second contains non-zero entries only at and below (or after) `nl+2`; and the third is dense. The first column of u and the row of vt are treated separately, however. The rows of the singular vectors found by ?lasd4 must be likewise permuted before the matrix multiplies can take place.

ctot

  INTEGER. Array, DIMENSION (4). A count of the total number of the various types of columns in u (or rows in vt), as described in idxc.

  The fourth column type is any column which has been deflated.

z

  REAL for slasd3

  DOUBLE PRECISION for dlasd3

  Array, DIMENSION (k). The first k elements of this array contain the components of the deflation-adjusted updating row vector.

## Output Parameters

d

  REAL for slasd3

  DOUBLE PRECISION for dlasd3

  Array, DIMENSION (k). On exit the square roots of the roots of the secular equation, in ascending order.

u

  REAL for slasd3

  DOUBLE PRECISION for dlasd3

  Array, DIMENSION (ldu, n).

  The last n - k columns of this matrix contain the deflated left singular vectors.

vt

  REAL for slasd3

  DOUBLE PRECISION for dlasd3

  Array, DIMENSION (ldvt, m).

  The last `m - k` columns of vt' contain the deflated right singular vectors.

vt2

  Destroyed on exit.

z

  Destroyed on exit.

info

  INTEGER.

  If `info = 0`): successful exit.

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

  If `info = 1`, an singular value did not converge.

## Application Notes

This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

Parent topic: Auxiliary Routines