p?laqr2

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?laqr2

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 pslaqr2( wantt, wantz, n, ktop, kbot, nw, a, desca, iloz, ihiz, z, descz, ns, nd, sr, si, t, ldt, v, ldv, wr, wi, work, lwork )

call pdlaqr2( wantt, wantz, n, ktop, kbot, nw, a, desca, iloz, ihiz, z, descz, ns, nd, sr, si, t, ldt, v, ldv, wr, wi, work, lwork )

Description

p?laqr2 accepts as input an upper Hessenberg matrix A and performs an orthogonal similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output Ais overwritten by a new Hessenberg matrix that is a perturbation of an orthogonal similarity transformation of A. It is to be hoped that the final version of A has many zero subdiagonal entries.

This routine handles small deflation windows which is affordable by one processor. Normally, it is called by p?laqr1. All the inputs are assumed to be valid without checking.

Product and Performance Information
Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex. Notice revision #20201201

Product and Performance Information

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.

Notice revision #20201201

Input Parameters

wantt

(global ) LOGICAL

If .TRUE., then the Hessenberg matrix A 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 A 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 A and (if wantz is .TRUE.) the order of the orthogonal matrix Z.

ktop, kbot

(global ) INTEGER

It is assumed without a check that either kbot = n or A(kbot+1,kbot)=0. kbot and ktop together determine an isolated block along the diagonal of the Hessenberg matrix. However, A(ktop,ktop-1)=0 is not essentially necessary if wantt is .TRUE. .

nw

(global ) INTEGER

Deflation window size. 1 ≤nw≤ (kbot-ktop+1). Normally nw≥ 3 if p?laqr2 is called by p?laqr1.

a

REAL for pslaqr2

DOUBLE PRECISION for pdlaqr2

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

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

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 z to which transformations must be applied if wantz is .TRUE.. 1 ≤iloz≤ihiz≤n.

z

REAL for pslaqr2

DOUBLE PRECISION for pdlaqr2

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 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.

t

REAL for pslaqr2

DOUBLE PRECISION for pdlaqr2

(local workspace) array of size ldt*nw.

ldt

(local ) INTEGER

The leading dimension of the array t. ldt≥nw.

v

REAL for pslaqr2

DOUBLE PRECISION for pdlaqr2

(local workspace) array of size ldv*nw.

ldv

(local ) INTEGER

The leading dimension of the array v. ldv≥nw.

wr, wi

REAL for pslaqr2

DOUBLE PRECISION for pdlaqr2

(local workspace) array of size kbot.

work

REAL for pslaqr2

DOUBLE PRECISION for pdlaqr2

(local workspace) array of size lwork.

lwork

(local ) INTEGER

work(lwork) is a local array and lwork is assumed big enough so that lwork≥nw*nw.

OUTPUT Parameters

a

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

z

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 pslaqr2

DOUBLE PRECISION for pdlaqr2

(global ) array of size kbot

On output, 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.

On processor #0, 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. On other processors, these entries are set to zero.

Parent topic: ScaLAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

p?laqr2

Syntax

Description

Input Parameters

OUTPUT Parameters

See Also