# zlahqr.f man page

zlahqr.f

## Synopsis

### Functions/Subroutines

subroutine **zlahqr** (WANTT, WANTZ, **N**, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)**ZLAHQR** computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

## Function/Subroutine Documentation

### subroutine zlahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, integer ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer INFO)

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

**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.

**Parameters:**-
*WANTT*WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required.

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

*N*N is INTEGER The order of the matrix H. N >= 0.

*ILO*ILO is INTEGER

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

*W*W is 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*ILOZ is INTEGER

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

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

*INFO*INFO is INTEGER = 0: successful exit .GT. 0: if INFO = i, ZLAHQR failed to compute all the 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.)

**Author:**-
Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**December 2016

**Contributors:**

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).

Definition at line 197 of file zlahqr.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zlahqr(3) is an alias of zlahqr.f(3).

Tue Nov 14 2017 Version 3.8.0 LAPACK