p?laqr3

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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p?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 pslaqr3( wantt, wantz, n, ktop, kbot, nw, h, desch, iloz, ihiz, z, descz, ns, nd, sr, si, v, descv, nh, t, desct, nv, wv, descw, work, lwork, iwork, liwork, reclevel )

call pdlaqr3( wantt, wantz, n, ktop, kbot, nw, h, desch, iloz, ihiz, z, descz, ns, nd, sr, si, v, descv, nh, t, desct, nv, wv, descw, work, lwork, iwork, liwork, reclevel )

Description

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

Input Parameters

wantt

(global ) LOGICAL

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

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

wantz

(global ) LOGICAL

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

If .FALSE., then z is not referenced.

n

(global ) INTEGER

The order of the matrix H and (if wantz is .TRUE.), the order of the orthogonal matrix Z.

ktop

(global ) INTEGER

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

kbot

(global ) INTEGER

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

nw

(global ) INTEGER

Deflation window size. 1 ≤nw≤ (kbot-ktop+1).

h

REAL for pslaqr3

DOUBLE PRECISION for pdlaqr3

(local ) array of size (lld_h,LOC_c(n))

The initial n-by-n section of H stores the Hessenberg matrix undergoing aggressive early deflation.

desch

(global and local) INTEGER array of size dlen_.

The array descriptor for the distributed matrix H.

iloz, ihiz

(global ) INTEGER

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

z

REAL for pslaqr3

DOUBLE PRECISION for pdlaqr3

Array of size (lld_z,LOC_c(n))

If wantz is .TRUE., then on output, the orthogonal similarity transformation mentioned above has been accumulated into the matrix Z(iloz:ihiz,kbot:ktop) from the right.

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

descz

(global and local) INTEGER array of size dlen_.

The array descriptor for the distributed matrix Z.

v

REAL for pslaqr3

DOUBLE PRECISION for pdlaqr3

(global workspace) array of size (lld_v,LOC_c(nw))

An nw-by-nw distributed work array.

descv

(global and local) INTEGER array of size dlen_.

The array descriptor for the distributed matrix V.

nh

INTEGER scalar

The number of columns of t. nh≥nw.

t

REAL for pslaqr3

DOUBLE PRECISION for pdlaqr3

(global workspace) array of size (lld_t,LOC_c(nh))

desct

(global and local) INTEGER array of size dlen_.

The array descriptor for the distributed matrix T.

nv

(global ) INTEGER

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

wv

(global workspace) REAL array of size (lld_w,LOC_c(nw))

descw

(global and local) INTEGER array of size dlen_.

The array descriptor for the distributed matrix wv.

work

(local workspace) REAL array of size lwork.

lwork

(local ) INTEGER

The size of the work array work (lwork≥1). lwork = 2*nw suffices, but greater efficiency may result from larger values of lwork.

If lwork = -1, then a workspace query is assumed; p?laqr3 only estimates the optimal workspace size for the given values of n, nw, ktop and kbot. The estimate is returned in work(1). No error message related to lwork is issued by xerbla. Neither h nor z are accessed.

iwork

(local workspace) INTEGER array of size liwork

liwork

(local ) INTEGER

The length of the workspace array iwork (liwork≥1).

If liwork=-1, then a workspace query is assumed.

OUTPUT Parameters

h

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

z

IF wantz is .TRUE., then on output, the orthogonal similarity transformation mentioned above has been accumulated into the matrix Z(iloz:ihiz,kbot:ktop) from the right.

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

ns

(global ) INTEGER

The number of unconverged (that is, approximate) eigenvalues returned in sr and si that may be used as shifts by the calling subroutine.

nd

(global ) INTEGER

The number of converged eigenvalues uncovered by this subroutine.

sr, si

REAL for pslaqr3

DOUBLE PRECISION for pdlaqr3

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

work(1)

On exit, if info = 0, work(1) returns the optimal lwork

iwork(1)

On exit, if info = 0, iwork(1) returns the optimal liwork

Parent topic: ScaLAPACK Auxiliary Routines

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