NAME¶

ZLAHQR - ZLAHQR i an auxiliary routine called by CHSEQR to update the eigenvalues and Schur decomposition already computed by CHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI

SYNOPSIS¶

SUBROUTINE ZLAHQR(

WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO )

INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N LOGICAL WANTT, WANTZ COMPLEX*16 H( LDH, * ), W( * ), Z( LDZ, * )

PURPOSE¶

ZLAHQR is an auxiliary routine called by CHSEQR to update the
eigenvalues and Schur decomposition already computed by CHSEQR, by
dealing with the Hessenberg submatrix in rows and columns ILO to
IHI.

ARGUMENTS¶

WANTT (input) LOGICAL

= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.

WANTZ (input) LOGICAL

= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.

N (input) INTEGER

The order of the matrix H. N >= 0.

ILO (input) INTEGER

IHI (input) INTEGER It is assumed that H is already upper triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). ZLAHQR 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.

H (input/output) COMPLEX*16 array, dimension (LDH,N)

On entry, the upper Hessenberg matrix H. On exit, if INFO is zero and if WANTT is .TRUE., then H is upper triangular in rows and columns ILO:IHI. If INFO is zero and if WANTT is .FALSE., then the contents of H are unspecified on exit. The output state of H in case INF is positive is below under the description of INFO.

LDH (input) INTEGER

The leading dimension of the array H. LDH >= max(1,N).

W (output) COMPLEX*16 array, dimension (N)

The computed eigenvalues ILO to IHI are stored in the corresponding elements of W. If WANTT is .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).

ILOZ (input) INTEGER

IHIZ (input) INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

Z (input/output) COMPLEX*16 array, dimension (LDZ,N)

If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by CHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced.

LDZ (input) INTEGER

The leading dimension of the array Z. LDZ >= max(1,N).

INFO (output) INTEGER

= 0: successful exit
eigenvalues ILO to IHI in a total of 30 iterations per eigenvalue; elements i+1:ihi of W contain those eigenvalues which have been successfully computed. If INFO .GT. 0 and WANTT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ILO thorugh INFO of the final, output value of H. If INFO .GT. 0 and 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 INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regardless of the value of WANTT.)

FURTHER DETAILS¶

02-96 Based on modifications by
David Day, Sandia National Laboratory, USA
12-04 Further modifications by
Ralph Byers, University of Kansas, USA
This is a modified version of ZLAHQR from LAPACK version 3.0.
It is (1) more robust against overflow and underflow and
(2) adopts the more conservative Ahues & Tisseur stopping
criterion (LAWN 122, 1997).