CLAQR4(1) LAPACK auxiliary routine (version 3.2) CLAQR4(1)NAME
CLAQR4 - CLAQR4 compute the eigenvalues of a Hessenberg matrix H and,
optionally, the matrices T and Z from the Schur decomposition H = Z T
Z**H, where T is an upper triangular matrix (the Schur form), and Z is
the unitary matrix of Schur vectors
SYNOPSIS
SUBROUTINE CLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z,
LDZ, WORK, LWORK, INFO )
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
LOGICAL WANTT, WANTZ
COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
PURPOSE
CLAQR4 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**H, where T is an upper triangular matrix (the
Schur form), and Z is the unitary matrix of Schur vectors.
Optionally Z may be postmultiplied into an input unitary
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 unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
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 CGEBAL, and then passed to CGEHRD when the matrix output
by CGEBAL 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) COMPLEX 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 triangular matrix
T from the Schur decomposition (the Schur form). If INFO = 0 and
WANT is .FALSE., then the contents of H are unspecified on exit.
(The output value of H when INFO.GT.0 is given under the descrip‐
tion 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).
W (output) COMPLEX array, dimension (N)
The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
in W(ILO:IHI). 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 W(i) = H(i,i).
Z (input/output) COMPLEX 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) COMPLEX 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 CLAQR4 does a workspace query. In
this case, CLAQR4 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 a unitary matrix. The final value of H is upper Hes‐
senberg 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(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
where U is the unitary 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 CLAQR4(1)
