DLAQR0(1) LAPACK auxiliary routine (version 3.2) DLAQR0(1)NAME
DLAQR0 - DLAQR0 compute the eigenvalues of a Hessenberg matrix H and,
optionally, the matrices T and Z from the Schur decomposition H = Z T
Z**T, where T is an upper quasi-triangular matrix (the Schur form),
and Z is the orthogonal matrix of Schur vectors
SYNOPSIS
SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ,
IHIZ, Z, LDZ, WORK, LWORK, INFO )
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
LOGICAL WANTT, WANTZ
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
Z( LDZ, * )
PURPOSE
DLAQR0 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**T, where T is an upper quasi-triangular matrix (the
Schur form), and Z is the orthogonal matrix of Schur vectors.
Optionally Z may be postmultiplied into an input orthogonal
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
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 .GE. 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that H is already upper tri‐
angular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous
call to DGEBAL, and then passed to DGEHRD when the matrix output
by DGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI
should be set to 1 and N, respectively. If N.GT.0, then
1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.
H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H. On exit, if INFO = 0
and WANTT is .TRUE., then H contains the upper quasi-triangular
matrix T from the Schur decomposition (the Schur form); 2-by-2
diagonal blocks (corresponding to complex conjugate pairs of ei‐
genvalues) are returned in standard form, with H(i,i) =
H(i+1,i+1) and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
.FALSE., then the contents of H are unspecified on exit. (The
output value of H when INFO.GT.0 is given under the description
of INFO below.) This subroutine may explicitly set H(i,j) = 0
for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
LDH (input) INTEGER
The leading dimension of the array H. LDH .GE. max(1,N).
WR (output) DOUBLE PRECISION array, dimension (IHI)
WI (output) DOUBLE PRECISION array, dimension (IHI) The real
and imaginary parts, respectively, of the computed eigenvalues of
H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
and WI(ILO:IHI). 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) .GT. 0 and WI(i+1) .LT.
0. If WANTT is .TRUE., then 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 .LE. ILOZ
.LE. ILO; IHI .LE. IHIZ .LE. N.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI)
If WANTZ is .FALSE., then Z is not referenced. If WANTZ is
.TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value
of Z when INFO.GT.0 is given under the description of INFO
below.)
LDZ (input) INTEGER
The leading dimension of the array Z. if WANTZ is .TRUE. then
LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
WORK (workspace/output) DOUBLE PRECISION array, dimension LWORK
On exit, if LWORK = -1, WORK(1) returns an estimate of the opti‐
mal value for LWORK. LWORK (input) INTEGER The dimension of the
array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typi‐
cally as large as 6*N may be required for optimal performance. A
workspace query to determine the optimal workspace size is recom‐
mended. If LWORK = -1, then DLAQR0 does a workspace query. In
this case, DLAQR0 checks the input parameters and estimates the
optimal workspace size for the given values of N, ILO and IHI.
The estimate is returned in WORK(1). No error message related to
LWORK is issued by XERBLA. Neither H nor Z are accessed.
INFO (output) INTEGER
= 0: successful exit
the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain
those eigenvalues which have been successfully computed. (Fail‐
ures are rare.) If INFO .GT. 0 and WANT is .FALSE., then on
exit, the remaining unconverged eigenvalues are the eigen- values
of the upper Hessenberg matrix rows and columns ILO through 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 orthogonal matrix. The final value of H is upper
Hessenberg and quasi-triangular in rows and columns INFO+1 through
IHI. If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final
value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of
Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal matrix in (*)
(regard- less of the value of WANTT.) If INFO .GT. 0 and WANTZ is
.FALSE., then Z is not accessed.
LAPACK auxiliary routine (versioNovember 2008 DLAQR0(1)
