DLAHQR - i 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

SUBROUTINE DLAHQR(

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

LOGICAL WANTT, WANTZ

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

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

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.

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

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

> 0: DLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30*(IHI-ILO+1) iterations; if INFO = i, elements i+1:ihi of WR and WI contain those eigenvalues which have been successfully computed.

2-96 Based on modifications by

David Day, Sandia National Laboratory, USA