p?lahqr

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?lahqr

Computes the Schur decomposition and/or eigenvalues of a matrix already in Hessenberg form.

Syntax

call pslahqr(wantt, wantz, n, ilo, ihi, a, desca, wr, wi, iloz, ihiz, z, descz, work, lwork, iwork, ilwork, info)

call pdlahqr(wantt, wantz, n, ilo, ihi, a, desca, wr, wi, iloz, ihiz, z, descz, work, lwork, iwork, ilwork, info)

call pclahqr(wantt, wantz, n, ilo, ihi, a, desca, w, iloz, ihiz, z, descz, work, lwork, iwork, ilwork, info)

call pzlahqr(wantt, wantz, n, ilo, ihi, a, desca, w, iloz, ihiz, z, descz, work, lwork, iwork, ilwork, info)

Include Files

Description

This is an auxiliary routine used to find the Schur decomposition and/or eigenvalues of a matrix already in Hessenberg form from columns ilo and ihi.

NOTE:

These restrictions apply to the use of p?lahqr:

The code requires the distributed block size to be square and at least 6.
The code requires A and Z to be distributed identically and have identical contexts.
The matrix A must be in upper Hessenberg form. If elements below the subdiagonal are non-zero, the resulting transformations can be nonsimilar.
All eigenvalues are distributed to all the nodes.

Input Parameters

wantt

(global) LOGICAL

If wantt= .TRUE., the full Schur form T is required;

If wantt = .FALSE., only eigenvalues are required.

wantz

(global) LOGICAL.

If wantz= .TRUE., the matrix of Schur vectors Z is required;

If wantz = .FALSE., Schur vectors are not required.

n

(global) INTEGER. The order of the Hessenberg matrix A (and z if wantz). n≥0.

ilo, ihi

(global) INTEGER.

It is assumed that A is already upper quasi-triangular in rows and columns ihi+1:n, and that A(ilo, ilo-1) = 0 (unless ilo = 1). p?lahqr works primarily with the Hessenberg submatrix in rows and columns ilo to ihi, but applies transformations to all of H if wantt is .TRUE.. 1≤ilo≤max(1,ihi); ihi ≤ n.

a

(global)

REAL for pslahqr

DOUBLE PRECISION for pdlahqr

COMPLEX for pclahqr

COMPLEX*16 for pzlahqr

Array, of size (lld_a,*). On entry, the upper Hessenberg matrix A.

desca

(global and local) INTEGER array of size dlen_. The array descriptor for the distributed matrix A.

iloz, ihiz

(global) INTEGER. Specify the rows of the matrix Z to which transformations must be applied if wantz is .TRUE.. 1≤iloz≤ilo; ihi≤ihiz≤n.

z

(global )REAL for pslahqr

DOUBLE PRECISION for pdlahqr

COMPLEX for pclahqr

COMPLEX*16 for pzlahqr

Array. If wantz is .TRUE., on entry z must contain the current matrix Z of transformations accumulated by pdhseqr. If wantz is .FALSE., z is not referenced.

descz

(global and local) INTEGER array of size dlen_. The array descriptor for the distributed matrix Z.

work

(local)

REAL for pslahqr

DOUBLE PRECISION for pdlahqr

COMPLEX for pclahqr

COMPLEX*16 for pzlahqr

Workspace array with size lwork.

lwork

(local) INTEGER. The size of work. lwork is assumed big enough so that lwork≥3*n + max(2*max(lld_z,lld_a) + 2*LOCq(n), 7*ceil(n/hbl)/lcm(NPROW,NPCOL))).

If lwork = -1, then work(1) gets set to the above number and the code returns immediately.

iwork

(global and local) INTEGER array of size ilwork. Not referenced and can be NULL pointer.

ilwork

(local) INTEGER This holds some of the iblk integer arrays. Not referenced and can be NULL pointer.

Output Parameters

a

On exit, if wantt is .TRUE., A is upper quasi-triangular in rows and columns ilo:ihi, with any 2-by-2 or larger diagonal blocks not yet in standard form. If wantt is .FALSE., the contents of A are unspecified on exit.

work(1)

On exit work(1) contains the minimum value of lwork required for optimum performance.

wr, wi

(global replicated output)

REAL for pslahqr

DOUBLE PRECISION for pdlahqr

Arrays of size n each. The real and imaginary parts, respectively, of the computed eigenvalues ilo to ihiare stored in the corresponding elements of wr and wi. If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of wr and wi, say the i-th and (i+1)-th, with wi(i)> 0 and wi(i+1) < 0. If wantt is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in A. A may be returned with larger diagonal blocks until the next release.

w

(global replicated output)

COMPLEX for pclahqr

COMPLEX*16 for pzlahqr

Array of size n. The computed eigenvalues ilo to ihi are stored in the corresponding elements of w. If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of w, say the i-th and (i+1)-th, with w(i)> 0 and w(i+1) < 0. If wantt is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in A. A may be returned with larger diagonal blocks until the next release.

z

On exit z has been updated; transformations are applied only to the submatrix Z(iloz:ihiz, ilo:ihi).

info

(global) INTEGER.

= 0: the execution is successful.

< 0: the parameter number - info is incorrect or inconsistent

> 0: p?lahqr failed to compute all the eigenvalues ilo to ihi in a total of 30*(ihi-ilo+1) iterations; if info = i, elements i+1: ihi of wr and wi contain the eigenvalues that have been successfully computed.

Parent topic: Nonsymmetric Eigenvalue Problems: ScaLAPACK Computational Routines

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