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.

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.)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

     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).