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# shseqr

```
NAME
SHSEQR - compute the eigenvalues of a real upper 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 SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI,
Z, LDZ, WORK, LWORK, INFO )

CHARACTER      COMPZ, JOB

INTEGER        IHI, ILO, INFO, LDH, LDZ, LWORK, N

REAL           H( LDH, * ), WI( * ), WORK( * ), WR( * ),
Z( LDZ, * )

PURPOSE
SHSEQR computes the eigenvalues of a real upper 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 factoriza-
tion 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 orthogonal matrix Q on entry, and
the product Q*Z is returned.

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 triangular in rows and columns 1:ILO-1
and IHI+1:N. ILO and IHI are normally set by a pre-
vious call to SGEBAL, and then passed to SGEHRD when

the matrix output by SGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set to 1 and N
respectively.  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 JOB = 'S', H contains the upper quasi-triangular
matrix T from the Schur decomposition (the Schur
form); 2-by-2 diagonal blocks (corresponding to com-
plex 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) < 0. If JOB = 'E', the contents of
H are unspecified on exit.

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 imaginary parts, respectively, of the computed
eigenvalues. If two eigenvalues are computed as a
complex conjugate pair, they are stored in consecu-
tive elements of WR and WI, say the i-th and
(i+1)th, with WI(i) > 0 and WI(i+1) < 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) REAL array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set, and on
exit, Z contains the orthogonal 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 Z contains Q*Z.  Nor-
mally Q is the orthogonal matrix generated by SORGHR
after the call to SGEHRD which formed the Hessenberg
matrix H.

LDZ     (input) INTEGER
The leading dimension of the array Z.  LDZ >=
max(1,N) if COMPZ = 'I' or 'V'; LDZ >= 1 otherwise.

WORK    (workspace) REAL array, dimension (N)

LWORK   (input) INTEGER
This argument is currently redundant.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
> 0:  if INFO = i, SHSEQR failed to compute all of
the eigenvalues in a total of 30*(IHI-ILO+1) itera-
tions; elements 1:ilo-1 and i+1:n of WR and WI con-
tain those eigenvalues which have been successfully
computed.
```