SLAHQR is an auxiliary routine called by SHSEQR to update the


    eigenvalues and Schur decomposition already computed by SHSEQR, 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 quasi-triangular in


          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless


          ILO = 1). SLAHQR 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 REAL array, dimension (LDH,N)


          On entry, the upper Hessenberg matrix H.


          On exit, if INFO is zero and if WANTT is .TRUE., H is upper


          quasi-triangular in rows and columns ILO:IHI, with any


          2-by-2 diagonal blocks in standard form. If INFO is zero


          and WANTT is .FALSE., the contents of H are unspecified on


          exit.  The output state of H if INFO is nonzero is given


          below under the description of INFO.

LDH

          LDH is INTEGER


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

WR

          WR is REAL array, dimension (N)

WI

          WI is REAL array, dimension (N)


          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 is .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).

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 REAL array, dimension (LDZ,N)


          If WANTZ is .TRUE., on entry Z must contain the current


          matrix Z of transformations accumulated by SHSEQR, 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, SLAHQR 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


                  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 SLAHQR 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).