?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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?lahqr

Computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Syntax

call slahqr( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info )

call dlahqr( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info )

call clahqr( wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info )

call zlahqr( wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info )

Include Files

mkl.fi

Description

The routine is an auxiliary routine called by ?hseqr to update the eigenvalues and Schur decomposition already computed by ?hseqr, by dealing with the Hessenberg submatrix in rows and columns ilo to ihi.

Input Parameters

wantt

LOGICAL.

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

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

wantz

LOGICAL.

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

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

n

INTEGER. The order of the matrix H (n≥ 0).

ilo, ihi

INTEGER.

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

Constraints:

1 ≤ ilo ≤ max(1,ihi); ihi ≤ n.

h, z

REAL for slahqr

DOUBLE PRECISION for dlahqr

COMPLEX for clahqr

DOUBLE COMPLEX for zlahqr.

Arrays:

h(ldh,*) contains the upper Hessenberg matrix h.

The second dimension of h must be at least max(1, n).

z(ldz,*)

If wantz = .TRUE., then, on entry, z must contain the current matrix z of transformations accumulated by ?hseqr. The second dimension of z must be at least max(1, n)

If wantz = .FALSE., then z is not referenced..

ldh

INTEGER. The leading dimension of h; at least max(1, n).

ldz

INTEGER. The leading dimension of z; at least max(1, n).

iloz, ihiz

INTEGER. Specify the rows of z to which transformations must be applied if wantz = .TRUE..

1 ≤ iloz ≤ ilo; ihi ≤ ihiz ≤ n.

Output Parameters

h

On exit, if info= 0 and wantt = .TRUE., then,

for slahqr/dlahqr, h is upper quasi-triangular in rows and columns ilo:ihi with any 2-by-2 diagonal blocks in standard form.
for clahqr/zlahqr, h is upper triangular in rows and columns ilo:ihi.

If info= 0 and wantt = .FALSE., the contents of h are unspecified on exit. If info is positive, see description of info for the output state of h.

wr, wi

REAL for slahqr

DOUBLE PRECISION for dlahqr

Arrays, DIMENSION at least max(1, n) each. Used with real flavors only. 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 = .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in h, with wr(i) = h(i,i), and, if h(i:i+1, i:i+1) is a 2-by-2 diagonal block, wi(i) = sqrt(h(i+1,i)*h(i,i+1)) and wi(i+1) = -wi(i).

w

COMPLEX for clahqr

DOUBLE COMPLEX for zlahqr.

Array, DIMENSION at least max(1, n). Used with complex flavors only. The computed eigenvalues ilo to ihi are stored in the corresponding elements of w.

If wantt = .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in h, with w(i) = h(i,i).

z

If wantz = .TRUE., then, on exit z has been updated; transformations are applied only to the submatrix z(iloz:ihiz, ilo:ihi).

info

INTEGER.

If info = 0, the execution is successful.

With info > 0,

if info = i, ?lahqr failed to compute all the eigenvalues ilo to ihi in a total of 30 iterations per eigenvalue; elements i+1:ihi of wr and wi (for slahqr/dlahqr) or w (for clahqr/zlahqr) contain those eigenvalues which have been successfully computed.
if wantt is .FALSE., then on exit the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ilo through info of the final output value of h.
if wantt is .TRUE., then on exit
(initial value of h)*u = u*(final value of h), (*)
where u is an orthognal matrix. The final value of h is upper Hessenberg and triangular in rows and columns info+1 through ihi.
if wantz is .TRUE., then on exit
(final value of z) = (initial value of z)* u,
where u is an orthognal matrix in (*) regardless of the value of wantt.

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?lahqr