SYNOPSIS

SUBROUTINE DLAHQR(

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

    

INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N

    

LOGICAL WANTT, WANTZ

    

DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )

PURPOSE

   DLAHQR is an auxiliary routine called by DHSEQR to update the
   eigenvalues and Schur decomposition already computed by DHSEQR, 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 quasi-triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). DLAHQR 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) DOUBLE PRECISION 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 (input) INTEGER

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

WR (output) DOUBLE PRECISION array, dimension (N)

WI (output) DOUBLE PRECISION 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 (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) DOUBLE PRECISION array, dimension (LDZ,N)

If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by DHSEQR, 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 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.)

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 DLAHQR 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).