?laqr3

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?laqr3

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

Include Files

mkl.fi

Description

The 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

DOUBLE COMPLEX 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

DOUBLE COMPLEX for zlaqr3.

Array, DIMENSION (ldz, n), 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

DOUBLE COMPLEX 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

DOUBLE COMPLEX 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

DOUBLE COMPLEX 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

DOUBLE COMPLEX 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

DOUBLE COMPLEX 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: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?laqr3