DHSEQR(1) LAPACK driver routine (version 3.2) DHSEQR(1)NAME
DHSEQR - DHSEQR 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 DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ,
WORK, LWORK, INFO )
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
CHARACTER COMPZ, JOB
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
Z( LDZ, * )
PURPOSE
DHSEQR 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
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 orthog‐
onal 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 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 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) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H. On exit, if INFO = 0
and JOB = 'S', 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 eigenvalues)
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 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 DHSEQR, 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).
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N) The real and
imaginary parts, respectively, of the computed eigenvalues. 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 JOB = 'S', 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).
Z (input/output) DOUBLE PRECISION 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 orthogo‐
nal 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 orthogo‐
nal matrix generated by DORGHR after the call to DGEHRD 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) DOUBLE PRECISION 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 DHSEQR does a workspace query. In this case,
DHSEQR 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 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 COMPZ = 'V', then on exit (final value of
Z) = (initial value of Z)*U where U is the orthogonal 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 orthog‐
onal 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 DHSEQR(1)
