# ?laqr3

Performs the orthogonal/unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggresive early deflation).

## Syntax

call slaqr3( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork )

call dlaqr3( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork )

call claqr3( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork )

call zlaqr3( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork )

## Description

This routine accepts as input an upper Hessenberg matrix H and performs an orthogonal/unitary similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been overwritten by a new Hessenberg matrix that is a perturbation of an orthogonal/unitary similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries.

## Input Parameters

wantt

  LOGICAL.

  If `wantt = .TRUE.`, then the Hessenberg matrix H is fully updated so that the quasi-triangular/triangular Schur factor may be computed (in cooperation with the calling subroutine).

  If `wantt = .FALSE.`, then only enough of H is updated to preserve the eigenvalues.

wantz

  LOGICAL.

  If `wantz = .TRUE.`, then the orthogonal/unitary matrix Z is updated so that the orthogonal/unitary Schur factor may be computed (in cooperation with the calling subroutine).

  If `wantz = .FALSE.`, then Z is not referenced.

n

  INTEGER. The order of the Hessenberg matrix H and (if `wantz = .TRUE.`) the order of the orthogonal/unitary matrix Z.

ktop

  INTEGER.

  It is assumed that either `ktop=1` or `h(ktop,ktop-1)=0`. ktop and kbot together determine an isolated block along the diagonal of the Hessenberg matrix.

kbot

  INTEGER.

  It is assumed without a check that either `kbot=n` or `h(kbot+1,kbot)=0`. ktop and kbot together determine an isolated block along the diagonal of the Hessenberg matrix.

nw

  INTEGER.

  Size of the deflation window. 1≤nw≤`(kbot-ktop+1)`.

h

  REAL for slaqr3

  DOUBLE PRECISION for dlaqr3

  COMPLEX for claqr3

  COMPLEX*16 for zlaqr3.

  Array, DIMENSION (ldh, n), on input the initial n-by-n section of h stores the Hessenberg matrix H undergoing aggressive early deflation.

ldh

  INTEGER. The leading dimension of the array h just as declared in the calling subroutine. `ldh≥n`.

iloz, ihiz

  INTEGER. Specify the rows of Z to which transformations must be applied if wantz is `.TRUE.. 1≤iloz≤ihiz≤n`.

z

  REAL for slaqr3

  DOUBLE PRECISION for dlaqr3

  COMPLEX for claqr3

  COMPLEX*16 for zlaqr3.

  Array, DIMENSION (ldz, ihi), contains the matrix Z if wantz is .TRUE.. If wantz is .FALSE., then z is not referenced.

ldz

  INTEGER. The leading dimension of the array z just as declared in the calling subroutine. `ldz≥1`.

v

  REAL for slaqr3

  DOUBLE PRECISION for dlaqr3

  COMPLEX for claqr3

  COMPLEX*16 for zlaqr3.

  Workspace array with dimension `(ldv, nw)`. An nw-by-nw work array.

ldv

  INTEGER. The leading dimension of the array v just as declared in the calling subroutine. `ldv≥nw`.

nh

  INTEGER. The number of column of t. `nh≥nw`.

t

  REAL for slaqr3

  DOUBLE PRECISION for dlaqr3

  COMPLEX for claqr3

  COMPLEX*16 for zlaqr3.

  Workspace array with dimension `(ldt, nw)`.

ldt

  INTEGER. The leading dimension of the array t just as declared in the calling subroutine. `ldt≥nw`.

nv

  INTEGER. The number of rows of work array wv available for workspace. `nv≥nw`.

wv

  REAL for slaqr3

  DOUBLE PRECISION for dlaqr3

  COMPLEX for claqr3

  COMPLEX*16 for zlaqr3.

  Workspace array with dimension `(ldwv, nw)`.

ldwv

  INTEGER. The leading dimension of the array wv just as declared in the calling subroutine. `ldwv≥nw`.

work

  REAL for slaqr3

  DOUBLE PRECISION for dlaqr3

  COMPLEX for claqr3

  COMPLEX*16 for zlaqr3.

  Workspace array with dimension lwork.

lwork

  INTEGER. The dimension of the array work.

  `lwork=2*nw)` is sufficient, but for the optimal performance a greater workspace may be required.

  If `lwork=-1,`then the routine performs a workspace query: it estimates the optimal workspace size for the given values of the input parameters n, nw, ktop, and kbot. The estimate is returned in `work(1)`. No error messages related to the lwork is issued by xerbla. Neither H nor Z are accessed.

## Output Parameters

h

  On output h has been transformed by an orthogonal/unitary similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries.

`work(1)`

  On exit `work(1)` is set to an estimate of the optimal value of lwork for the given values of the input parameters n, nw, ktop, and kbot.

z

  If wantz is .TRUE., then the orthogonal/unitary similarity transformation is accumulated into z(iloz:ihiz, ilo:ihi) from the right.

  If wantz is .FALSE., then z is unreferenced.

nd

  INTEGER. The number of converged eigenvalues uncovered by the routine.

ns

  INTEGER. The number of unconverged, that is approximate eigenvalues returned in sr, si or in sh that may be used as shifts by the calling subroutine.

sh

  COMPLEX for claqr3

  COMPLEX*16 for zlaqr3.

  Arrays, DIMENSION (kbot).

  The approximate eigenvalues that may be used for shifts are stored in the `sh(kbot-nd-ns+1)`through the `sh(kbot-nd)`.

  The converged eigenvalues are stored in the `sh(kbot-nd+1)`through the `sh(kbot)`.

sr, si

  REAL for slaqr3

  DOUBLE PRECISION for dlaqr3

  Arrays, DIMENSION (kbot) each.

  The real and imaginary parts of the approximate eigenvalues that may be used for shifts are stored in the `sr(kbot-nd-ns+1)`through the `sr(kbot-nd)`, and `si(kbot-nd-ns+1)` through the `si(kbot-nd)`, respectively.

  The real and imaginary parts of converged eigenvalues are stored in the `sr(kbot-nd+1)`through the `sr(kbot)`, and `si(kbot-nd+1)` through the `si(kbot)`, respectively.

Parent topic: Auxiliary Routines