p?laqr0

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/22/2024

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p?laqr0

Computes the eigenvalues of a Hessenberg matrix and optionally returns the matrices from the Schur decomposition.

Syntax

call pslaqr0( wantt, wantz, n, ilo, ihi, h, desch, wr, wi, iloz, ihiz, z, descz, work, lwork, iwork, liwork, info, reclevel )

call pdlaqr0( wantt, wantz, n, ilo, ihi, h, desch, wr, wi, iloz, ihiz, z, descz, work, lwork, iwork, liwork, info, reclevel )

Description

p?laqr0 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z*T*Z^T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors.

Optionally Z may be postmultiplied into an input orthogonal matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q * H * Q^T = (QZ) * T * (QZ)^T.

Input Parameters

wantt

(global ) LOGICAL

= .TRUE. : the full Schur form T is required;

= .FALSE. : only eigenvalues are required.

wantz

(global ) LOGICAL

= .TRUE. : the matrix of Schur vectors Z is required;

= .FALSE.: Schur vectors are not required.

n

(global ) INTEGER

The order of the Hessenberg matrix H (and Z if wantz). n≥ 0.

ilo, ihi

(global ) INTEGER

It is assumed that the matrix H is already upper triangular in rows and columns 1:ilo-1 and ihi+1:n. ilo and ihi are normally set by a previous call to p?gebal, and then passed to p?gehrd when the matrix output by ihi is reduced to Hessenberg form. Otherwise ilo and ihi should be set to 1 and n, respectively. If n > 0, then 1 ≤ilo≤ihi≤n.

If n = 0, then ilo = 1 and ihi = 0.

h

REAL for pslaqr0

DOUBLE PRECISION for pdlaqr0

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

The upper Hessenberg matrix H.

desch

(global and local ) INTEGER

Array of size dlen_.

The array descriptor for the distributed matrix H.

iloz, ihiz

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

REAL for pslaqr0

DOUBLE PRECISION for pdlaqr0

Array of size (lld_z, LOC_c(n)).

If wantz is .TRUE., contains the matrix Z.

If wantzis .FALSE., z is not referenced.

descz

(global and local ) INTEGER array of size dlen_.

The array descriptor for the distributed matrix Z.

work

REAL for pslaqr0

DOUBLE PRECISION for pdlaqr0

(local workspace) array of size lwork

lwork

(local ) INTEGER

The length of the workspace array work.

iwork

(local workspace) INTEGER array of size liwork

liwork

(local ) INTEGER

The length of the workspace array iwork.

reclevel

(local ) INTEGER

Level of recursion. reclevel = 0 must hold on entry.

OUTPUT Parameters

h

On exit, if wantt is .TRUE., the matrix H is upper quasi-triangular in rows and columns ilo:ihi, with 1-by-1 and 2-by-2 blocks on the main diagonal. The 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. If info = 0 and wanttis .FALSE., the contents of h are unspecified on exit.

wr, wi

REAL for pslaqr0

DOUBLE PRECISION for pdlaqr0

The real and imaginary parts, respectively, of the computed eigenvalues ilo to ihi are 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 h.

z

Updated matrix with transformations applied only to the submatrix Z(ilo:ihi,ilo:ihi).

If COMPZ = 'I', on exit, if info = 0, z contains the orthogonal matrix Z of the Schur vectors of H.

If wantz is .TRUE., then Z(ilo:ihi,iloz:ihiz) is replaced by Z(ilo:ihi,iloz:ihiz)*U, when U is the orthogonal/unitary Schur factor of H(ilo:ihi,ilo:ihi).

If wantzis .FALSE., then z is not defined.

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.

info

INTEGER

> 0: if info = i, then the routine failed to compute all the eigenvalues. Elements 1:ilo-1 and i+1:n of wr and wi contain those eigenvalues which have been successfully computed.

> 0: if wanttis .FALSE., then the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ilo through ihi of the final output value of h.

> 0: if wantt is .TRUE., then (initial value of H)*U = U*(final value of H), where U is an orthogonal/unitary matrix. The final value of H is upper Hessenberg and quasi-triangular/triangular in rows and columns info+1 through ihi.

> 0: if wantz is .TRUE., then (final value of Z(ilo:ihi,iloz:ihiz))=(initial value of Z(ilo:ihi,iloz:ihiz))*U, where U is the orthogonal/unitary matrix in the previous expression (regardless of the value of wantt).

> 0: if wantzis .FALSE., then z is not accessed.

Parent topic: ScaLAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

p?laqr0

Syntax

Description

Input Parameters

OUTPUT Parameters

See Also