SLAHQR(1) LAPACK auxiliary routine (version 3.2) SLAHQR(1)NAME
SLAHQR - SLAHQR i 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
SYNOPSIS
SUBROUTINE SLAHQR( 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
REAL H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
PURPOSE
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.
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). 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 (input/output) 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 con‐
tents 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) REAL array, dimension (N)
WI (output) REAL array, dimension (N) The real and imagi‐
nary 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 trans‐
formations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <=
ILO; IHI <= IHIZ <= N.
Z (input/output) 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 refer‐
enced.
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 eigen‐
value; 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 ei‐
genvalues 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 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).
LAPACK auxiliary routine (versioNovember 2008 SLAHQR(1)
