Contents

# ?hseqr

Computes all eigenvalues and (optionally) the Schur factorization of a matrix reduced to Hessenberg form.

## Syntax

Include Files
• mkl.h
Description
The routine computes all the eigenvalues, and optionally the Schur factorization, of an upper Hessenberg matrix
H
:
H
=
Z
*
T
*
Z
H
, where
T
is an upper triangular (or, for real flavors, quasi-triangular) matrix (the Schur form of
H
), and
Z
is the unitary or orthogonal matrix whose columns are the Schur vectors
z
i
.
You can also use this routine to compute the Schur factorization of a general matrix
A
which has been reduced to upper Hessenberg form
H
:
A
=
Q
*
H
*
Q
H
, where
Q
is unitary (orthogonal for real flavors);
A
= (
QZ
)*
T
*(
QZ
)
H
.
In this case, after reducing
A
to Hessenberg form by gehrd, call orghr to form
Q
explicitly and then pass
Q
to
?hseqr
with
compz
=
'V'
.
You can also call gebal to balance the original matrix before reducing it to Hessenberg form by
?hseqr
, so that the Hessenberg matrix
H
will have the structure:
where
H
11
and
H
33
are upper triangular.
If so, only the central diagonal block
H
22
(in rows and columns
ilo
to
ihi
) needs to be further reduced to Schur form (the blocks
H
12
and
H
23
are also affected). Therefore the values of
ilo
and
ihi
can be supplied to
?hseqr
directly. Also, after calling this routine you must call gebak to permute the Schur vectors of the balanced matrix to those of the original matrix.
If
?gebal
has not been called, however, then
ilo
must be set to 1 and
ihi
to
n
. Note that if the Schur factorization of
A
is required,
?gebal
must not be called with
job
=
'S'
or
'B'
, because the balancing transformation is not unitary (for real flavors, it is not orthogonal).
?hseqr
uses a multishift form of the upper Hessenberg
QR
algorithm. The Schur vectors are normalized so that
||
z
i
||
2
= 1
, but are determined only to within a complex factor of absolute value 1 (for the real flavors, to within a factor
±
1).
Input Parameters
job
Must be
'E'
or
'S'
.
If
job
=
'E'
, then eigenvalues only are required.
If
job
=
'S'
, then the Schur form
T
is required.
compz
Must be
'N'
or
'I'
or
'V'
.
If
compz
=
'N'
, then no Schur vectors are computed (and the array
z
is not referenced).
If
compz
=
'I'
, then the Schur vectors of
H
are computed (and the array
z
is initialized by the routine).
If
compz
=
'V'
, then the Schur vectors of
A
are computed (and the array
z
must contain the matrix
Q
on entry).
n
The order of the matrix
H
(
n
0
).
ilo
,
ihi
If
A
has been balanced by
?gebal
, then
ilo
and
ihi
must contain the values returned by
?gebal
. Otherwise,
ilo
must be set to 1 and
ihi
to
n
.
h
,
z
Arrays:
h
(size max(1,
ldh
*
n
))
) The
n
-by-
n
upper Hessenberg matrix
H
.
z
(size max(1,
ldz
*
n
))
If
compz
=
'V'
, then
z
must contain the matrix
Q
from the reduction to Hessenberg form.
If
compz
=
'I'
, then
z
need not be set.
If
compz
=
'N'
, then
z
is not referenced.
ldh
h
; at least max(1,
n
).
ldz
z
;
If
compz
=
'N'
, then
ldz
1
.
If
compz
=
'V'
or
'I'
, then
ldz
max(1,
n
)
.
Output Parameters
w
Array, size at least max (1,
n
). Contains the computed eigenvalues, unless
info
>0. The eigenvalues are stored in the same order as on the diagonal of the Schur form
T
(if computed).
wr
,
wi
Arrays, size at least max (1,
n
) each.
Contain the real and imaginary parts, respectively, of the computed eigenvalues, unless
info
> 0
. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having positive imaginary part first. The eigenvalues are stored in the same order as on the diagonal of the Schur form
T
(if computed).
h
If
info
= 0
and
job
= 'S'
,
h
contains the upper quasi-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
> 0
is given under the description of
info
below.)
z
If
compz
=
'V'
and
info
= 0
, then
z
contains
Q
*
Z
.
If
compz
=
'I'
and
info
= 0
, then
z
contains the unitary or orthogonal matrix
Z
of the Schur vectors of
H
.
If
compz
=
'N'
, then
z
is not referenced.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, the
i
-th parameter had an illegal value.
If
info
=
i
,
?hseqr
failed to compute all of the eigenvalues. Elements 1,2, ...,
ilo
-1 and
i
+1,
i
+2, ...,
n
of the eigenvalue arrays (
wr
and
wi
for real flavors and
w
for complex flavors) contain the real and imaginary parts of those eigenvalues that have been successfully found.
If
info
> 0
, and
job
=
'E'
, then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns
ilo
through
info
of the final output value of
H
.
If
info
> 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 Hessenberg and triangular in rows and columns
info
+1
through
ihi
.
If
info
> 0
, and
compz
=
'V'
, then on exit (final value of
Z
) = (initial value of
Z
)*
U
, where
U
is the unitary matrix (regardless of the value of
job
).
If
info
> 0
, and
compz
=
'I'
, then on exit (final value of
Z
) =
U
, where
U
is the unitary matrix (regardless of the value of
job
).
If
info
> 0
, and
compz
=
'N'
, then
Z
is not accessed.
Application Notes
The computed Schur factorization is the exact factorization of a nearby matrix
H
+
E
, where
||
E
||
2
<
O
(
ε
) ||
H
||
2
/
s
i
, and
ε
is the machine precision.
If
λ
i
is an exact eigenvalue, and
μ
i
is the corresponding computed value, then
|
λ
i
-
μ
i
|
c
(
n
)*
ε
*||
H
||
2
/
s
i
, where
c
(
n
)
is a modestly increasing function of
n
, and
s
i
is the reciprocal condition number of
λ
i
. The condition numbers
s
i
may be computed by calling trsna.
The total number of floating-point operations depends on how rapidly the algorithm converges; typical numbers are as follows.
If only eigenvalues are computed:
7
n
3
for real flavors
25
n
3
for complex flavors.
If the Schur form is computed:
10
n
3
for real flavors
35
n
3
for complex flavors.
If the full Schur factorization is computed:
20
n
3
for real flavors
70
n
3
for complex flavors.

#### Product and Performance Information

1

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Notice revision #20110804