CHSEQR(1) LAPACK driver routine (version 3.2) CHSEQR(1)NAME
CHSEQR - CHSEQR 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 CHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK,
LWORK, INFO )
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
CHARACTER COMPZ, JOB
COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
PURPOSE
CHSEQR 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
JOB (input) CHARACTER*1
= 'E': compute eigenvalues only;
= 'S': compute eigenvalues and the Schur form T. COMPZ (input)
CHARACTER*1
= 'N': no Schur vectors are computed;
= 'I': Z is initialized to the unit matrix and the matrix Z of
Schur vectors of H is returned; = 'V': Z must contain an unitary
matrix Q on entry, and the product Q*Z is returned.
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. 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 respec‐
tively. 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 JOB = 'S', H contains the upper triangular matrix T from the
Schur decomposition (the Schur form). If INFO = 0 and JOB = 'E',
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.)
Unlike earlier versions of CHSEQR, this subroutine may explicitly
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. If JOB = 'S', 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,N)
If COMPZ = 'N', Z is not referenced. If COMPZ = 'I', on entry Z
need not be set and on exit, if INFO = 0, Z contains the unitary
matrix Z of the Schur vectors of H. If COMPZ = 'V', on entry Z
must contain an N-by-N matrix Q, which is assumed to be equal to
the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On
exit, if INFO = 0, Z contains Q*Z. Normally Q is the unitary
matrix generated by CUNGHR after the call to CGEHRD which formed
the Hessenberg matrix H. (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 COMPZ = 'I' or COMPZ =
'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.
WORK (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns an estimate of the optimal
value for LWORK. LWORK (input) INTEGER The dimension of the
array WORK. LWORK .GE. max(1,N) is sufficient and delivers very
good and sometimes optimal performance. However, LWORK as large
as 11*N may be required for optimal performance. A workspace
query is recommended to determine the optimal workspace size. If
LWORK = -1, then CHSEQR does a workspace query. In this case,
CHSEQR checks the input parameters and estimates the optimal
workspace size for the given values of N, ILO and IHI. The esti‐
mate 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
value
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 JOB = 'E', 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 JOB = 'S', 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 COMPZ = 'V', then on exit (final value of Z) =
(initial value of Z)*U where U is the unitary matrix in (*)
(regard- less of the value of JOB.) If INFO .GT. 0 and COMPZ =
'I', then on exit (final value of Z) = U where U is the unitary
matrix in (*) (regard- less of the value of JOB.) If INFO .GT. 0
and COMPZ = 'N', then Z is not accessed.
LAPACK driver routine (version 3November 2008 CHSEQR(1)