ETNA
Electronic Transactions on Numerical Analysis.
Volume 26, pp. 1-33, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
A STRUCTURED STAIRCASE ALGORITHM FOR
SKEW-SYMMETRIC/SYMMETRIC PENCILS
RALPH BYERS , VOLKER MEHRMANN , AND HONGGUO XU
Abstract. We present structure preserving algorithms for the numerical computation of structured staircase
forms of skew-symmetric/symmetric matrix pencils along with the Kronecker indices of the associated skew-symmetric/symmetric Kronecker-like canonical form. These methods allow deflation of the singular structure and deflation
of infinite eigenvalues with index greater than one. Two algorithms are proposed: one for general skew-symmetric/symmetric pencils and one for pencils in which the skew-symmetric matrix is a direct sum of and
. We show how to use the structured staircase form to solve boundary value problems arising in control
applications and present numerical examples.
Key words. structured staircase form, linear-quadratic control,
control, structured Kronecker canonical
form, skew-symmetric/symmetric pencil, skew-Hamiltonian/Hamiltonian pencil
AMS subject classifications. 65F15, 15A21, 93B40
1. Introduction. In this paper we study structure preserving numerical methods for the
computation of the structural information associated with the singular and infinite eigenvalue
parts of the Kronecker canonical form of real skew-symmetric/symmetric matrix pencils
where !#"$%&'")(+*-,/. , and 0 1234(6587#5 . Here by *-,/. 9 we denote the set
of real : 7; matrices. In the following we adopt the notation of [31] and call pencils of this
form even pencils, since replacing 0 $%3 by 0 !$%3 and transposing yields the same pencil.
Even pencils occur in the context of linear quadratic optimal control problems (see, e.g.,
[34, 38, 39, 45]), =< control problems, (see, e.g., [4, 18, 37, 46]), and other applications
(see, e.g., [31, 35]).
For control problems of the form
>@?
ABDC!AFEHGJIKFMLNAO
(1.2)
it has been shown in [34] that the solution of the linear quadratic optimal control problem
? TU
leads to the boundary value problem
A ??
A TU
QPR S DQPR S
(1.3)
I
I
(S is an auxiliary vector, typically it is a vector of Lagrange multipliers) with boundary conditions
>
A
Z
3
[
A
]
_
\
_
^
`
(1.4)
e
0WVYX X aWb < " S 0WV 3Z[cd
(1.1)
Received April 11, 2005. Accepted for publication June 27, 2006. Recommended by P. Van Dooren.
Department of Mathematics, University of Kansas, Lawrence, KS 44045, USA. This author was partially supported by National Science Foundation grants 0098150, 0112375, 9977352 and by Deutsche Forschungsgemeinschaft through the DFG Research Center M ATHEON Mathematics for key technologies in Berlin.
Institut für Mathematik, TU Berlin, Str.
des 17.
Juni 136, D-10623 Berlin, FRG
(mehrmann@math.tu-berlin.de). This author was partially supported by Deutsche Forschungsgemeinschaft,
through the DFG Research Center M ATHEON Mathematics for Key Technologies in Berlin.
Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA (xu@math.ukans.edu).
This author was partially supported by National Science Foundation grant 0314427, and the University of Kansas
General Research Fund allocation # 2301717.
1
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2
R. BYERS, V. MEHRMANN AND H. XU
>
where the matrix pencil associated with the boundary value problem
jC " k TU
+Of[&RP " c c gPRihC l G (1.5)
c c c
k
" GJ" m
" , lnMlg" and mMDm4" .)
is even, see [34]. (In particular, h
h
The solution of the boundary value problem can be obtained via the computation of a
structured Schur form of (1.5). Similar matrix pencils arise in the solution of optimal <
>
>
control problems; see [4, 46]. If the control problem comes from an ordinary differential
o and if it comes from a differential-algebraic equation, then is a
equation, then
>c
c TU
singular matrix.
For both theoretical and computational purposes, the pencil (1.5) should be regular and
of index at most . In order to check this property numerically and to remove singular parts
and higher index infinite eigenvalue parts we need a staircase form. We discuss this topic in
detail in Section 5.
We derive numerical methods to compute the characteristic quantities of the Kronecker
canonical form of
under structure preserving congruence transformations
p
+O
rq +fq Ds " #sMs " #sut
The motivation for preserving the even structure comes from the special properties of such
pencils. For example, even pencils have the Hamiltonian eigensymmetry, i.e., the finite eigenpairs and
quadruples for non-real eigenvalues of real
values occur in
pencils; see, e.g., [33, 34, 35].
As suggested by having eigenvalues with Hamiltonian symmetry, even pencils are closely
related to skew-Hamiltonian/Hamiltonian pencils. Let
, where is the
identity matrix. (We leave off the subscript , if the dimension is clear from the context.) A
matrix
is called Hamiltonian if
. A matrix
is called
skew-Hamiltonian if
. A matrix pencil
is called skew-Hamiltonian/Hamiltonian if is skew-Hamiltonian and is Hamiltonian. If the dimension of the even
pencil
is even, then the pencil is equivalent to the skew-Hamiltonian/Hamiltonian
.
pencil
Furthermore, if
, then
and we have a standard eigenvalue problem for
the Hamiltonian matrix
. It is well-known (see [28, 34]) that similarity transformations with symplectic matrices preserve the Hamiltonian and skew-Hamiltonian structure.
(A matrix
is called symplectic if
.) It was shown in [29], that if the
Hamiltonian matrix possesses a Hamiltonian Jordan form under symplectic similarity, then
it also admits a Hamiltonian Schur form under orthogonal symplectic transformations. This
work has been extended in [33] to skew-Hamiltonian/Hamiltonian pencils. For even pencils
there exist well-known structured Kronecker forms; see, e.g., [43]. We briefly review these
forms in Section 2.
It is the topic of this paper to construct a structured staircase form for even pencils that
displays the invariants of the structured Kronecker form, while working only with unitary
(orthogonal) transformations.
We could in theory also use an unstructured numerical method like the
or the GUPTRI
algorithm to obtain this information, but this would destroy the symmetry structure in even
pencils and introduce unnecessary unstructured rounding errors. The following example illustrates how such unstructured rounding errors may give misleading or even mathematically
impossible computed “eigenvalues” and Kronecker structure.
v wyvx
v z v { v x w|vx
o, : 7 :
} , f~ X €ƒ‚ € !X ‚ „
: 3%"+
0ˆ…]} …] } ‹ ‰ (#*-†‡,/. †%,
‰
(#*-†%,/. †‡, 32"8
0Š‰[}
‰[}
‰
'O
‰ … D } "ŒO } "
f } } "'[o
 } "
Ž (+*-†%,. †%,
Ž}=Ž "r }
hJ
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A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
‘7‘
3
E XAMPLE 1.1. As mentioned above, eigenvalues of even pencils have Hamiltonian
even pencil has at least one infinite eigenvalue. The other two eigenvalues
pairing. A
may be either both infinite, form a
pair of finite, real eigenvalues or form a complex
conjugate pair of finite eigenvalues with zero real part. If
is an eigenvalue, then it has
multiplicity two.
Consider a
even pencil with matrices
0’v w v 3
‘7‘
c p c UT "
f RP p c“c c c h
h c “
h
v 8c
c“c p UT "
RP c p c h p c”c h
where is a random real orthogonal matrix generated as described in [40]. The pencil is
congruent to the pencil
c p c TU
c”c TU
p
8PR c“c p +gPR c p c
c“c”c
c“c p
so it has a triple eigenvalue at • with geometric multiplicity p and algebraic multiplicity ‘ .
We calculated the eigenvalues of –)O for several different randomly generated
orthogonal matrices using the
in M
[32] version 6.0.0.88 (R12) with
š™2› algorithmreturns
unit round roughly — h t —˜— 7 p c h|.  M
strikingly different approximations of
eigenvalues for different randomly generated orthogonal matrices . For each example, we
h
took care that was exactly symmetric and exactly skew symmetric.
Out of 1000 examples, the eig function built into Matlab version 6.0.0.88 (R12) reports
that 644 have no finite eigenvalues (which is the correct result), 75 have one finite eigenvalue
™2œ
of magnitude roughly p c , 120 have two finite eigenvalues of magnitude roughly p c/ , and
œ
61 have three finite eigenvalues of magnitude roughly p c . None of the computed sets of
ATLAB
ATLAB
approximate eigenvalues that included finite eigenvalues was the set of eigenvalues of an
even pencil; none had Hamiltonian eigenvalue pairing. Often, there was a singleton finite
eigenvalue.
The
algorithm is numerically stable in the sense that the computed eigenvalues
are exactly correct for some rounding-error-small perturbation of the original data matrices.
However, this rounding-error-small perturbation is not necessarily an even perturbation of an
even pencil. The unstructured rounding errors are sufficient to destroy the Hamiltonian pairing and return entirely unrealistic sets of eigenvalue approximations and Kronecker structures
that do not occur in even pencils.
Recently, in [5, 11], numerical methods were developed to compute the Hamiltonian
Schur form for Hamiltonian matrices and the methods were extended to the regular pencil
case with nonsingular matrix in [4].
An important remaining issue is a structure preserving method to compute the structural
invariants under congruence associated with the infinite eigenvalues and the singular part
of the pencil. This is of particular importance in the case of optimal control problems for
descriptor systems, where is a singular matrix, [34], since typical numerical methods for
computing optimal feedback controls require the pencils to be regular and of index at most
one. If this is not the case, then the singular part and the part associated with higher index
singular blocks must be deflated first; see Section 5.
In Section 3 we derive structure preserving algorithms for the computation of structured
staircase forms for arbitrary even pencils. In particular we show how to determine the Kronecker indices associated with singular Kronecker blocks and with Kronecker blocks corresponding to the eigenvalue infinity. The staircase form also provides a structure preserving
hJ
>
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4
>
R. BYERS, V. MEHRMANN AND H. XU
p
way to deflate these blocks. Section 4 treats the computation of eigenvalues and deflating
subspaces for regular even pencils of index .
If
, then
is the direct sum of
and . In this case it has been shown
in [10] how to preserve not only skew-symmetry but the whole
structure.
The results and algorithms of this paper also adapt to symmetric/symmetric and Hermitian/Hermitian pencils for which a similar structured Kronecker canonical form is known; see,
e.g., [36, 42]. In a similar way, the results and algorithms also adapt to skew-Hermitian/Hermitian pencils and to complex skew-symmetric/symmetric pencils. For brevity, however, we
will not discuss such variations here.
It should be noted that some of the ideas presented in this paper have been observed and
discussed for special cases in [12]. Similar forms for a special case of symmetric/symmetric
pencils have recently been proposed in [30].
[o
f } ,jž c
},
c
} ,4ž c
>
2. Kronecker and staircase forms. In this section we review the Kronecker canonical
form and staircase forms for unstructured, asymmetric pencils.
T HEOREM 2.1. Kronecker Canonical Form [19, 24]. Let
. Then there
exist nonsingular matrices
and
such that
(Œ5-,/. ,
s(Œ5
j. ‡C (Ÿ*- ¡. ,
>
h
s 0 +OCj3
D¢^¤£{¥ 0§¦4¨ %©h ª§«¬wtztwtw%©ªƒ­®2© "¯ « wtztwtz2© "¯§° ‰]± «wtztwtz ‰]±z² }
³ «´wtwtztw }
³¶µ 3¶
(2.1)
where . . .
1. ¦4¨ ·c ¨ +Oc ¨ is an ¸ 7 ¸ block of zeros;
2. each © ªˆ¹ is an º2» 7 0Wº2» E p 3 right singular block with right minimal index º‡» and
form
¼@½¾
 à ½¾
ÂÃ
¿BÀ Á
ČÅÇÆ ¿Á À
Ä=È
Á block with left minimal
Á À index É » and form
3. each ©"¯ ¹ is a 0’É » E p 37 É » left À singular
½¼ ¾¾¾
½
¾
 ÃÃà ¾¾¾
 ÃÃÃ
¾À
à Á
Ã
¿Á
Ä ÅÇÆ ¿ À
Ä=Ê
ÀÁ
Á
À
¹
7
4. each ‰ ± is a Ë»
˽¾ » infinite eigenvalue block½¾ with index ËÌ» and form
 ÃÃà ¾¾¾
 ÃÃÃ
¼ ¾¾¾
à Á
Ã
À Á
¿
Ä ÅÇÆ ¿
ÄÈ
Á
À
¹
7
5. each } ³ is a ͘»
ͽ¾ ˜» Jordan block with finite½¾ eigenvalue vd» Á(5 and form
 ÃÃà ¾¾¾ δÏ
 ÃÃÃ
¼ ¾¾¾
Ã
Ã
Á
Á
¿
Ä Å‹Æ ¿
δÁ Ï Ä=Ð
Á
..
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5
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
>
rC
The Kronecker canonical form is unique up to permutation of the blocks, i. e., the kind, size
. It is more common
and number of the blocks are characteristic for the pencil
to express
as a combination of
and
blocks. Here, we display
explicitly to
emphasize the similarities between Theorem 2.1 and the structured form in Theorem 2.3
below.
There also exists a real version of the Kronecker canonical form, where the blocks
are in real Jordan form and the transformation matrices are real. A similar result also holds
for complex pencils, [19, 20].
D EFINITION 2.2.
i) An
matrix pencil
is called regular, if
for some
. Otherwise the pencil is called singular. (Singular pencils are those
whose Kronecker canonical form has either an
block with
or an
block
with
or an
block with
.)
ii) If
is regular, then a pair of complex numbers
is an eigenvalue of
, if
. If
is a singular pencil, then, for
our purposes in this paper, its eigenvalues are the eigenvalues of the regular blocks
in its Kronecker canonical form, i.e., the union of the eigenvalues of the
and
blocks in Theorem 2.1. We identify eigenvalues
with
with the finite
eigenvalue
. Eigenvalues
with
are called infinite eigenvalues.
iii) The index of a regular matrix pencil
is the size of the largest block
in
Theorem 2.1. It is denoted by
.
iv) The inertia index of a symmetric matrix is the triple In
, where
is the number of positive eigenvalues of , is the number of negative eigenvalues,
and is the number of zero eigenvalues.
Arbitrarily small rounding errors can radically change the kind and number of the Kronecker blocks. Consequently, it is problematic to compute the Jordan or Kronecker canonical
form with a numerical algorithm in finite precision arithmetic [41]. Among the most successful compromises in the nearly-impossible problem of calculating Kronecker canonical forms
are the staircase algorithms. Using a sequence of rank decisions, orthogonal matrix multiplications, and small perturbations, staircase algorithms transform a pencil into staircase or
generalized upper triangular (GUPTRI) form [13, 14, 15, 44]. The rank decisions and perturbations have the effect of determining the essential invariants in the Kronecker canonical
form of a “least generic” pencil within a tolerated perturbation. (A formal definition of the
term “least generic” is surprisingly complicated. See [16, 17] for a detailed discussion and a
recently developed interactive tool.) Since the GUPTRI form is built on a sequence of rank
decisions and tolerated perturbations with a built-in bias toward a nearby least generic pencil,
the computed invariants may not always agree with the invariants of the original pencil.
Example 1.1 demonstrates that otherwise excellent numerical methods can give unsatisfactory results when applied to even pencils, because the eigenvalues of even pencils have
a special structure that is not necessarily preserved by unstructured rounding errors. In fact,
even pencils have a special even Kronecker-like canonical form described by the following
theorem.
T HEOREM 2.3. [43] If
with
, then there exists a
nonsingular matrix
such that
¦¨
Ñ1X
Ñ X"
¦¨
}³ ¹
>
>
7: :
+OC
¢ÒÔÓ 0 +OCj3NDÕ c
0 123(Œ5@7F5
¦Ö¨
¸]× c ©uª
> º!× c ©"ª
c
> º×
>
+O> C
123Ö Õ 0 ċcØ3
ÇOC ¢ÒwÓ 0 Ç OCN3$Dc BOC 0
‰ ±¹
¹} ³
$
%
3
)
·
Õ
c
0
v D
Ù
0> 1 2>3 ŒOC [c
‰ ±¹
^ŠÚd¢ 0 ‡Cj3
0 '3¡ 0ˆÛ ÝÜ2ޘ3
Û
Ü
Þ
(2.2)
where
dß()*-,/. , àá!'"
%àâ#"
/
,
.
,
Œ
(
5
ã
ã " 0 +O'3 ã D¢^_£˜¥ 0Wä
å äæ ä$ç ä
è 3Ô
ä å [¢Ì^_£˜¥ 0ƒ¦ ¨ êŽ é «´wtwtztÔ Žêé ­ 3¶
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6
R. BYERS, V. MEHRMANN AND H. XU
ä æ DD¢¢^_^_£˜£˜¥¥ 0ìë † ª§«î«îí í ™ wwtztztwtwtÔtÔ ë † ªƒï_í ™ í ë † ¯ «´wtz«´twwtÔtz twë tԆ ¯Yð 3 3š
ä$ç D¢^_£˜¥ 0Wñ † ±«wtztw™ tz ñ 2õ̆ ±¬ö² «w™ twtzñ tÔî† õ̳ ö/÷/3 ñ † ³¶µ
äè
0ˆòBó òBóô
and the blocks have the following properties.
1. ¦ ¨ ·c ¨ +Oc ¨ ;
2. each Žêé ¹ is a 0’— Þ » E p 3u7 0’— Þ » E p 3 block that combines a right singular block and
a left singular block, both of minimal index Þ » . It has the form
½¾¾¾
½¾¾¾
Â
Ã
 ÃÃÃ
Ã
Ã
¾¼ ¾¾
¾
Á À ÃÃÃÃ ¾¾¾¾
À Á ÃÃÃÃ
¾¾
à ÅBÆ
ÃÈ
Å Á À
À
Á
Á
ÀÁ
¿Å
Ä ¿
Ä
À
Á
ÀÁ
À
W
ª
2
¹
í
E
{
3
7
E
3
™ is a 0ƒ—{º2» p 0ƒ—{º2» p block that contains a single block corresponding
3. each ë †
to the eigenvalue • with index —{º%» E p . It has the form
½¾¾¾
½¾¾¾
Â
Ã
 ÃÃÃ
Ã
Ã
¾¼ ¾¾
¾
Á À ÃÃÃÃ ¾¾¾¾
À Á ÃÃÃÃ
¾¾
à ÅBÆ
ÃÊ
Å Á À
À
Á
Á À
ÀÁ ø
¿Å
Ä ¿
Ä
À
Á
ÀÁ
À
+
(
ú
w
where ù ¯
p p˜û is the sign-index or sign-characteristic of the block;
4. each ë † ¹ is a üØɶ» 7 ü®ÉÝ» block that combines two —˜É¶» 7 —˜ÉÝ» infinite eigenvalue blocks
of index É » . It has the form
½¾¾¾
 ÃÃà ½¾¾
 ÃÃÃ
¾¼ ¾¾
Ã
¾
Ã
Á
À
¾
ÃÃÃ
ÃÃÃ ¾¾
¾¾¾
Á
ÃÃ ÅBÆ
Á
¾
È
Å
À
Á
Ä
ÁýÀ
Á
¿Å
Ä ¿
Á
Á
À
5. each ñ † ± ¹ í ™ is a 0WüØË » E — 3d7 0WüØË » E — 3 block that combines two 0ƒ—{Ë » E p 3d7 0ƒ—{Ë » E p 3
Jordan blocks corresponding to the eigenvalue c . It has the form
½¾¾¾
 ÃÃÃ
½¾¾¾
Â
Ã
¾
ÃÃÃ ¾¾¾
ÃÃÃ
¼ ¾¾
Á
À
ÃÃÃ
Á ÃÃ ¾¾¾
¾
Á
ÅÇÆ
ÃÈ
Å Á
À
¿
Ä
Á
Á À
¿
Ä
Å
Á
ÀÁ
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7
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
ñ † ³ ¹ ´—c Í » 7 —´Í »
½¾¾¾
 ÃÃÃ
½¾¾¾
Â
Ã
¾
ÃÃÃ
ÃÃÃ ¾¾¾
¼ ¾¾
Á
À
ÃÃÃ
Á ÃÃ ¾¾¾
¾
Á
ÅBÆ
ÃÊ
Å Á
ø
À
¿
Ä
Á
Á À
Ä
¿
Å
Á
Á
À
where ù (+ú p w p{û is the sign characteristic of this block;
7. each òBó ¹ is a —ØþÌ» 7 —˜þÌ» block that combines two þd» 7 þ» Jordan blocks corresponding to nonzero real eigenvalues ÿ » and ÿ » . It has the form
½¾¾¾
Ï Â ÃÃÃ
½¾¾¾
Â
Ã
¾
ÃÃÃ ¾¾¾
ÃÃÃ
¼ ¾¾
Á
ÃÃÃ
Á ÃÃ ¾¾¾
Ï
¾
Á
Ï
B
Å
Æ
ÃÐ
Å Á
¿
Ä
Á
Á
¿ Ï
Ä
Å
Á
Á
¹
Ì
õ
ö
8. The entries
take two slightly different forms.
(a) One possibility is that õdö ¹ is a — » 7 — » block combining two » 7
» Jordan
blocks with purely imaginary eigenvalues » z » ( » × c ). In this case it has the
form
½¾¾¾
 ÃÃÃ
Ï
½¾¾¾
Â
Ã
¾
ÃÃÃ ¾¾¾
ÃÃÃ
¼ ¾¾
Á
ÃÃÃ
Á ÃÃ ¾¾¾ Á Ï
¾
Ï
Ç
Å
Æ
ÃÊ
Å Á
Ä
¿
ø
Á
Á Ä
¿
Å
Y
Ï
Á
Á
where ù (+ú p w p{û is the sign characteristic.
(b) The other possibility is that õ ö ¹ is a ü » 7 ü » block combining » 7 » Jordan
blocks for each of the complex eigenvalues ÿ®» E ‡» ÿ˜» %» w ÿ˜» E ‡» z ÿ˜» %»
Õ c and ‡» DÕ c ). In this case it has form
(with ÿ˜» D
½¾¾¾
šÏ  ÃÃÃ
½¾¾¾
Â
Ã
¾
ÃÃÃ ¾¾¾
ÃÃÃ
¼ ¾¾
ÃÃ ¾¾¾
ÃÃÃ
šÏ
¾
š
Ï
‹
Å
Æ
Ã
Å
¿
Ä ¿
Ä
Å
Ï
c p and » %» ÿ˜» .
with p c
ÿ˜» %»
is a
block that contains a single Jordan block corresponding to
6. each
the eigenvalue . It has the form
.
..
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.
.
.
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ETNA
Kent State University
etna@mcs.kent.edu
8
R. BYERS, V. MEHRMANN AND H. XU
This structured Kronecker canonical form is unique up to permutation of the blocks, i.e., the
kind, size and number of the blocks as well as the sign characteristics are characteristic of
.
the pencil
6O
for complex even pencils
AOcorresponding
with ‡ structured
(]5 ,/. , andKronecker
á&! Jform
%fDis also, known
see [43].
It was shown in [33] that the existence of the canonical form in Theorem 2.3 guarantees
that corresponding condensed forms under orthogonal transformations also exist, see also
[29].
The computation of the canonical form in Theorem 2.3 faces similar difficulties to those
discussed above for the general Kronecker canonical form. Example 1.1 and experience
with the unstructured Kronecker canonical form suggest that a successful numerical method
for computing the characteristic indices and sign characteristics should use a staircase-like
condensed form under unitary transformations that preserve the even structure of the pencil.
This is the topic of this paper.
3. Staircase algorithms for even pencils. In this section we discuss staircase algorithms for even pencils of the form (1.1). One may distinguish two cases. The first method
that we discuss here deals with pencils where is a general skew-symmetric matrix and the
second method which is discussed in [10] treats the important special case that
with
~ X XX˜„
~  X € X®€ „ .
The procedures for computing staircase forms are built on a sequence of numerical rank
decisions. This is also true for the procedures for even pencils that we present below. For
general matrices the rank can be determined by the rank revealing QR factorization [21,
Sec. 5.4] or the singular value decomposition (SVD) [21, Sec. 8.6]. For more details on
determining numerical ranks, see, for example, [6, 21].
For symmetric and skew-symmetric matrices the rank can be determined via the appropriate Schur forms [21, Chapter 8]. An inexpensive way is the following modified rank
revealing
–factorization method. Let be symmetric or skew-symmetric. Compute the
rank revealing QR factorization
h
m
C
" C mc™ h
m ™ is of full row rank, is real orthogonal, and is a permutation. Compute
h
" C m ™ c j" m c ™%™ cc t
h h
h
The zero 0îp — 3 block follows from the symmetry or skew-symmetry of C . Note also that m ™‡™
C
m ™%™
where
must be nonsingular. When
is skew-symmetric,
must have even order.
3.1. Even staircase form. For a general even pencil we construct a symmetric variation
of the staircase form of [44]. The staircase form displays the regular, index 1 part of the
pencil. Moreover, we show below that the staircase form also displays the characteristic
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9
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
(]* ,/ . , 8
quantities describing the singular part and the eigenvalue infinity of Theorem 2.3.
T HEOREM 3.1. Even staircase form. For a matrix pencil
with
, there exists a real orthogonal matrix , such that
!
½¾¾¾
¾¾¾
¾¾¾
¾¾¾
¾¾¾
¾
" ‡D " (Œ* ,/. ,
 ÃÃÃ
!
ÅÅ
..
.
..
.
.
..
.
..
" % )+&
6
¿
&
..
6
.
ÐÝÝÐÐÝÐÝÐÝÐÝÐÐРж¶ÐжÝÐÐÝÝÐ ÐÐÐ
..
.
..
.
..
6
..
.
..
.
&8% &
6
&0
% &('-)
..
.
..
.
"% &('-)
..
.
,&
,&0% &('1"
,&('"% &('"
"% &('"
"% &2'-)
6
..
.
..
.
&8% &('"
6
&0
% &('1"
..
.
..
.
#"% )+&
..
6
..
.
..
.
&0% &('*)
ÐÝЇРжÐÝÐ
..
 ÃÃÃ
"% )+&('1"
6
ÃÃÃ
3 "
ÃÃÃ
3 &
4
ÃÃÃ
.
ÃÃÃ
ÃÃÃ
Ä
.
ÃÃÃ
3 &
4
..
.
..
.
ÃÃÃ
Ã
5 &
..
.
..
.
Ê
5 "
† ¹ ¹: † « tztwt : ,
(+* , . p
p
c cc 8 " (Œ* † ´. † » c (]* , ¹ . ¹ » (Œ* , ¹ . , ¹ p
"™‡™ ™ †‡™ †† ™%™ "™‡™ (]* † ´. † †‡† "†‡† (Œ*  † {.  † †
and the blocks †%† and and » , p wtztwtÔ are nonsingular.
Proof. A formal, constructive proof is given by Algorithm 1 in Appendix A, but for
™ :™
» . †% í ™¶ »
í ™ . í ™
» . †% í †  »
í ™ . í ™
#:
;:
:
9
:
:
9
>=
<
$?7@?BA
DC
C
ML
L
L
L
<
H
L
H
C
E E
C
!I
GF;H
L
:
@
J?7@?KA
E E
L
L
ON
L
E N
E
A
ease of explication, we present a less formal construction here. Both the formal algorithm
and the less formal construction described here are explicit but recursive procedures. During
the construction, we note the inertias of certain symmetric submatrices that will be used by
Theorem 3.3. Note also that some blocks in the partitioned matrices may be void, i.e., they
may have zero rows or zero columns or both.
Let
be an even pencil. If
, then the pencil is singular and trivially
in even staircase form. If is nonsingular, then this is a regular pencil of index and thus
trivially in even staircase form. If is singular, then determine an rank revealing factorization
/P
or skew-symmetric Schur decomposition , with orthogonal and C
iO
fDDc
™ " ™ ~ X XX؄
™
c
..
.
5 "
(3.1)
where 9
5 &
5 )
..
.
..
.
.
..
ÃÃÃ
3 "
ÃÃÃ
À
&('"% &('"
6
À
.
Ä
6
6
#"% &('*)
À
"% &
ÐÝÐÝÐ
#"% &('"
.
..
" % )+&('"
,&0% &
0
& % &('"
.
.
..
..
.
..
.
Å
"% &('-)
..
..
.
..
.
" % &('*)
6
././.../.
" % &('"
6
.
./..
""
" % &
6
¾¾¾
À
$"% &
ÐÝжРŠÝРжÐ
..
.
¾¾¾
¾¾¾
./..
./..
6
¾¾¾
" % &
" % &2'-)
7
¾¾¾
..
ÐÝжÐ
.
" % &2'1"
Å
6
..
¿ Å
½¾¾¾
ÐÝжÐ
#""
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10
R. BYERS, V. MEHRMANN AND H. XU
nonsingular. Perform a pencil equivalence
™ " 0 O3 ™ D c cc + …… "™%™ †™ …… †‡™ †† t
If … †%† is nonsingular, then the pencil is regular, of index at most p with £˜Ú … †‡† infinite
eigenvalues and the even staircase form is complete. If … †%† is singular, determine a rank
X
revealing factorization or symmetric Schur decomposition † " … †%† † ~ X X „ with † orthogonal and nonsingular.
Record the inertia 0ˆÛ ‡ÜÌ%c®3 of for use in Theorem 3.3 below and perform a further
pencil equivalence
o c " c
o c
c †
c c + …… "™%™ †™ …… †‡™ †† c †
q ™‡™ … q ™ † … q ™ TU
c“c TU
D8PR c c“c +gPR … q "™ †
c t
c c“c
q… "™ c c
Q
MC
!
R
TS
S
S
S
WV
U
S
S
YX
S
S
L
L
Z
C
Q
S
S
S
S
R
!
\[
]
C
L
^]
Determine a rank revealing factorization or singular value decomposition
]
with and _
]
" … q ™ c cc
^]`_-]
H
!
orthogonal and H nonsingular. Perform another pencil equivalence
]
q ™%™ … q ™ † … q ™ TU
c c UT "
c“c TU
c c TU
PR cc oc c
PR cc oc c
PR cc c“
c“cc PR …… qq ""™™ † c cc
c“c T
c T
%
™
™
™
™
‡
™
™
™
™
‰
‰
‰
†
†
P ‰ ™ " † ‰ †%† c c“c U P … "™ † … †%† … † c c U
·
" c c c“c + … "™ … †"
c c
(3.2)
c
" c c c c
R ‰ ™ cc cc cc c“
R
c“c
c c c c c
~ «ƒ« « « „ and ‰ ™ c . The ‰ ™ block may fill with nonzero entries later in
where
the process, so we do not distinguish it from other blocks that may be nonzero.
apply the even staircase reduction to the central subpencil
X
r~ Recursively
r
H
~
X X˜„
„ recording the inertias of the submatrices as they occur. This
corresponds to performing another pencil equivalence to (3.2) that modifies rows and columns
— and ‘ typically modifying ‰ ™ † , ‰ ™ , … ™ † and … ™ along with the central subpencil. At that
bca
]
^]
f
f
lnl
hg
g
i ml
ikj
f
f
^]
i
l>l
/o >l p
o j
]
f
f
ml
i lnl
o
H
]
g
]
f
i
]
g
f
0de
]
L
_ ]
C
^]
C
H
^]
]
hg
g
g
g
_ ]
L
]
l>p
L
X
^]
^]
point the pencil is in even staircase form.
R EMARK 3.2. It should be noted that the rank decisions in the recursive procedure described in the proof of Theorem 3.1 have to be carried out with great care. Ideally one would
need a structured version of the procedure for general pencils in [16, 17]. The development
of such a procedure is currently under investigation.
The recursive construction of the even staircase form also generates a sequence of inertias
of certain ephemeral symmetric submatrices that appear briefly during the construction. The
following theorem shows that the characteristic quantities describing the singular part and the
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A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
11
+O
ú » û » ™ ú : » û » ™
ú ÛÌ»{û » í ™ ™ úÜ ´» û » í ™ ™ ú » Û» EHS »{û » í ™ ™
rO
ú ÛÌ»´û úÜ »û ú » ÛÌ» E[Ü »{û
+
•
™ : » Ý» í ™ 0 » í ™ » 3
p wtztwtÔ
ë †»
†
í ™ ·c
•
p ztwtztÔ E p
» » šš™ ™
ë † » š™
•
Û
»
Ì
Û
»
p
Ü » 6Ü » ê™
p
MÜ X X [c
Û
X
c «  , «1 Oc «  , «
— wtwtztÔ
Ž » š™
í ™ . í ™ O í ™ . » í ™ : »
+O p
are determined by the integer sequences 9 rq ,
eigenvalue infinity of
rq ,
`s
,
and
.
rq
rq
rq
T HEOREM 3.3. Suppose that an even pencil
has been reduced to the condensed form (3.1) by Algorithm 1 with integer sequences
,
, and `s
.
Then
has the following block structures associated with the singular part and the
eigenvalue in the even Kronecker canonical form (2.2) of Theorem 2.3.
s
vs +w
1. For every @
A , the pencil has ut
9
blocks
corresponding to the eigenvalue . (Here we set 9
).
2. For every @
, the pencil has s xs
odd-sized blocks
correA
sponding to the eigenvalue ,among which
blocks have sign index and
vs
blocks have sign index
. (Here we set
.)
3. The pencil has a singular block <
.
<
A , the pencil has 9
4. For every @
singular blocks
.
5. The subpencil
is a regular pencil of index at most . It
contains the Jordan structure associated with all finite eigenvalues of
.
Proof. See Appendix B
E XAMPLE 3.4. This example demonstrates the effect of rank decisions and the ability
of the even staircase algorithm to determine a nearby even pencil with non-generic structure.
Our experimental M ATLAB implementation of the even staircase algorithm makes rank
, i.e. singular values of magnitude less
decisions using a singular value drop tolerance y
than an absolute threshold y
are taken to be zero. In this experiment, the threshold y
varied from
to
. (The unit round is approximately
.) Oversimplifying
slightly, the algorithm searches for a “most non-generic” even pencil in the cloud of pencils
that lie within a distance of roughly y of then nominal input pencil.
We constructed even pencils
where
× c
c
cp š™2› p c š™ [×
+O
c p c
P p c“
c
c”c“c
c“c
R c”
c”c“c
c“c p c
c
P c c”
p c“
“
c
p c”c c
R c“
c“c”c“cp
— t —Ø— 7 p c š™2›
c”c T
c”c U "
c”c E º f
c
h
p cp h
c T
c U "
c E º f
c h
h
ü
ê™î› to p c š™ , is a random real orthogonal
where º is a positive real number varying from p c
h3
matrix generated as described in [40], and
and are skew-symmetric
and symmetric
Ì
c
matrices, respectively, whose nontrivial entries are normal 0 p random variables.
If º [c , then these unperturbed pencils have simple finite eigenvalues 4— and an index 2
infinite eigenvalue. If º4× c , then the perturbed pencils typically lie at a distance of roughly
º from the º Dc unperturbed even pencil.
In this experiment, for each value of the singular value drop tolerance , we chose ten
random equivalence matrices . For each and , we varied the perturbation magnitude º
š™2› , h p c ê™îœ , p c ê™îœ , . .h . , p c ê™ and recorded the smallest value of
logarithmically as º p c
the selected º ’s for which the algorithm did not find an even pencil with an index 2 infinite
eigenvalue (and two finite eigenvalues) with a distance of roughly of the test pencil. We
plotted the recorded points 0 º 3 in Figure 3.1. As expected the points fall near the line º .
g
g
g
f
g
f
f
f
C
hg
g
g
f
g
f
f
f
C
C
C
z
{
y
}| ~
y
}| 
y
my
y
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12
R. BYERS, V. MEHRMANN AND H. XU
0
10
−2
perturbation magnitude ε
10
−4
10
−6
10
−8
10
−10
10
−12
10
−14
10
−16
10
−16
10
−14
10
−12
−10
10
−8
10
10
−6
10
−4
10
singular value drop tolerance τ
−2
10
0
10
F IG . 3.1. Figure from Example 3.4. For each selected value of the singular value drop tolerance € and each of
ten random pencil equivalences, the graph plots the smallest of the selected perturbation magnitudes  for which the
even pencil staircase algorithm did not find an index 3 pencil. As expected, the points line near the line  ‚€ . In
every test case, for which  was significantly smaller than € , the experimental staircase algorithm successfully found
a nearby index 3 even pencil.
-
º
In every test case with perturbation magnitude significantly smaller than the singular value
drop tolerance y , the staircase algorithm successfully located a nearby index 3 even pencil.
<
~ X ‚ XX؄
If the skew-symmetric matrix in the pencil (1.5) is of the special form as in
applications from linear quadratic optimal control or
control, then from a perturbation
theory point of view it is advisable to preserve this structure as much as possible, i.e., we
would like to compute a staircase form, where the middle block associated with the finite
eigenvalues and the infinite-eigenvalue-index- part is again of the same form as the original
pencil with a (possibly smaller) skew-symmetric part /ƒ
. An algorithm to compute a
p
~ X XX؄
X
variant even staircase form while preserving the ~ X X „ structure of the skew-symmetric part
„ ƒ
has been presented in [10].
p
4. The regular, index one case. It remains to determine the finite eigenvalues and index
infinite eigenvalues contained in the central block of the even staircase form (3.1). To
avoid the hazards of introducing asymmetric rounding errors demonstrated above, a structure
preserving numerical method is necessary. In this section we outline how to modify a skewHamiltonian/Hamiltonian structure preserving algorithm from [3] for regular even pencils of
index at most . For ease of notation, in this section we assume that the even pencil is regular
of index at most .
p
p
In order to use the skew-Hamiltonian/Hamiltonian algorithm, we must transform the
skew-symmetric/symmetric pencil into skew-Hamiltonian/Hamiltonian form. For this we
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A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
13
c ‹† c“c TU
™‡™ ™ † ™ ™ TU
c
" P ‹c † cc c“
(4.1) áD R
c“c + PR '™™" † #†%† " † † † c c c“c
'™" #† " '" % ( * . . For a general
where † is positive diagonal, † % ™‡™ % †‡† ( * ´. , and even pencil this may be achieved, for example, by computing and reordering the real Schur
form of ™%™ .
The pencil (4.1) has even size. If the size of the original pencil is odd, then add one more
index p eigenvalue infinity by appending one row and column as
c cc + c c where is a nonzero scalar. The extended pencil is simply a direct sum of the original pencil
and the scalar pencil cN . So the eigen-structure of the original pencil will not be affected
by this adding.
Let k8 ¢^¤£{¥ 0 ‡cÌ #%cØ3 and : ‹E . By interchanging the — nd and ‘ rd columns
and rows and then multiplying with , " from the right, the pencil (4.1) is equivalent to the
skew-Hamiltonian/Hamiltonian pencil
™ † ™ ! ™%™ ! ™ TU
n·-k , k " , " + PR ††%† † !!##™™"" !!'" t

#† " !#™" † ! †
We then have the following structured Schur form.
k
" , " , and let (Œ*-†%,. †%, be HamT
4.1. Let kŸD¢^¤£{¥ 0 %cd ‡cØ3 , ,
iltonian. Then there exist orthogonal matrices ™
h h † and orthogonal symplectic matrices
™ † such that
T
‡
™
™
™
™
™
†
"™ † PR cc c †‡† † c † U h h
c c
c k ™ † c k ™ TU
"™ k ™ PR cc k c †‡† cc k c † h
c c c k
c c c c TU
Pc
(4.2)
† " h † R cc cc †%† c † c † ‡k
" " (Œ* {. are upper
where ™%™ g" (Œ* . are upper triangular †%† Ýk †‡† †‡† k
triangular, and
is lower quasi-triangular. Furthermore, †‡† Ýk †‡† are nonsingular.
+ , and + in (4.1) as well, are the same as the
The finite eigenvalues of k
c
finite eigenvalues of the index pencil
C·+OG · k †‡† c k
" " c †%† + c " c †‡† t
assume that the even pencil already has the form
hg
g
f
f
,
f
^]
†
]
†
f
^]
]
†
†
E E
Q
]Y]
R
!
]Y]
]^†
]r†
†r†
2‡ ‡
†Y†
ˆ
ˆ
‰ˆ
r
‹Š
Qs
†
f
OŒ
]
f
]r†
HEOREM
r
]Y]
†
]r†
]
†Y†

Œ
Œ
f
Œ
f
Œ
Œ
Œ
Œ
Œ
]
Œ
]r]
Œ
†r†
g
g
†
f
g
†
Œ
†Y]
g
†
]
Œ
g
]
†
†
hg
^]
†
f
†Y†
hg
g
f

f

]

Œ
 †/]
YŒ
]Y]
Œ
‡ ‡
Œ
†

 †r†
/Œ

†r†
†Y†

ŽE E
†Y†  †r†
†r† 
OŒ
 †Y†
†Y†
 †r† 
Œ
Œ
†Y†
hg
g
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14
R. BYERS, V. MEHRMANN AND H. XU
Proof. The proof appears in Appendix C, where (4.2) is proved constructively by Algorithm 4.
5. Application to optimal control. Consider the linear quadratic control problem described by (1.2)–(1.5). It is well known that if the pencil is regular then the boundary value
problem is uniquely solvable [8]. We therefore assume that the pencil (1.5) is regular. (For
the singular case, see [9].)
The following proposition shows how the even staircase form (3.1) characterizes consistency of boundary conditions for the special problem (1.3). (The general theory of linear
differential-algebraic equations of [25, 26] uses a normal form to characterize consistent initial conditions.)
T HEOREM 5.1. Consider the boundary value problem (1.3) with a regular matrix pencil. Transform the boundary value problem to the staircase form (3.1). With as in (3.1),





partition 
UT
A
" PR SI ™" tztwt " " í ™ " í † twtzt †%" í ™ "
(5.1)
? ?
w
z
t
w
t
z
t
c
, and ™ wtwtztÔ Mc . The solution of the boundary value
conformally. Then ™
problem
?
?
í ™ . , í ™ í ™ c cc í ™ · í ™ . í ™ í ™ "™‡™ ™ †‡™ †† í ™ †
(which is of index at most p ) uniquely determines the remaining components, í † , tztwt ,
™.
†% í Proof.
Because the pencil is regular, we have : » » and » . †‡ í †  » » , p ztwtztÔ . Hence,
w
z
t
w
t
Ô
t
p , we obtain recursively that
for
í
? í
í » í ™ š™  » í ™ í »  » í ™ . í »  » í ™ . t
™
™

F



I


C



DL
L
L
L
H
9
A

@

A
’
H
‘

“
@

’
“
“q


“
“m”
“q
í † wtwtztw †‡ í ™
í ™ †%šo † ™ "™ † a  a o c í ™ 0ˆV X 3¶
ê™ 0 ™‡™ ™ † †%š† ™ "™ † 3 , and (5.1) with » âc for p ztwtwtw
with l”
observation was made about more general pencils in [9].
The consistency of the boundary conditions in (1.3) may be checked by using the recur

sion formulas for
, the explicit solution representation
›š^œ
L
C
L
L
L
L
L
–•{—™˜
F 
!I
@
A
. A similar
In this way we may reduce the general linear differential-algebraic boundary value problem (1.3) in an even structured way to a smaller linear differential-algebraic boundary value
problem of index at most , to which appropriate methods may be applied. See for example,
[2, 1, 27].
p
6. Conclusion. Even pencils have paired eigenvalues and a structured Kronecker-like
canonical form with paired blocks. Even otherwise numerically stable numerical methods
that allow asymmetric rounding errors can return computed “eigenvalues” that are unrealistic
in the sense that they do not have proper pairing and, hence, are not eigenvalues of an even
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A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
15
pencil. Numerical procedures including asymmetric staircase forms for determining Kronecker indices do not calculate the sign indices of the even Kronecker-like form and, if they
allow asymmetric rounding errors, can return unrealistic results.
This paper presents an even staircase form for even pencils that displays the structure
and characteristic indices of the singular and infinite eigenvalue structure of even Kroneckerlike canonical form. Using only orthogonal matrix multiplications and rank decisions, the
accompanying numerically stable numerical method preserves even structure throughout and
introduces only even rounding errors.
The use of the even staircase form is illustrated using an application to boundary value
problems arising from optimal control of differential-algebraic systems. As outlined in Section 4, the even staircase form may be the first step in a method for calculating eigenvalues of
an even pencil.
REFERENCES
[1] U. A SCHER , R. M ATTHEIJ , AND R. R USSELL , Numerical Solution of Boundary Value Problems for Ordinary
Differential Equations, SIAM, Philadelphia, PA, second ed., 1995.
[2] U. M. A SCHER AND L. R. P ETZOLD , Computer Methods for Ordinary Differential and DifferentialAlgebraic Equations, SIAM, Philadelphia, PA, 1998.
[3]
, Numerical computation of deflating subspaces of skew Hamiltonian/Hamiltonian pencils, SIAM J.
Matrix Anal. Appl., 24 (2002), pp. 165–190.
[4] P. B ENNER , R. B YERS , V. M EHRMANN , AND H. X U , A robust numerical method for the  -iteration in
-control, to appear in: Linear Algebra Appl., (2007).
[5] P. B ENNER , V. M EHRMANN , AND H. X U , A new method for computing the stable invariant subspace of a
real Hamiltonian matrix, J. Comput. Appl. Math., 86 (1997), pp. 17–43.
[6] C. B ISCHOF AND G. Q UINTANA -O RT Í , Algorithm 782: codes for rank-revealing QR factorizations of dense
matrices, ACM Trans. Math. Software, 24 (1998), pp. 254–257.
[7] A. B OJANCZYK , G. G OLUB , AND P. VAN D OOREN , The periodic Schur decomposition; algorithms and
applications, in Proc. SPIE Conference, vol. 1770, 1992, pp. 31–42.
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url:
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, A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratificationenhanced staircase algorithm., SIAM J. Matrix Anal. Appl., 20 (1999), pp. 667–699.
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Kent State University
etna@mcs.kent.edu
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Appendix A. Proof of Theorem 3.1: Algorithm 1. We prove Theorem 3.1 constructively.
The proof is provided by the following algorithm. Note that in the algorithm, some blocks in
the partitioned matrices may be void, i.e., they may have zero rows or zero columns or both.
A LGORITHM 1. Staircase algorithm for even pencils.
For
this algorithm computes an orthogonal matrix such that , are in the form of (3.1). In addition, the algorithm produces a
sequence of inertias
of nonsingular, symmetric submatrices that will be used in
Theorem 3.3.
4
©¨
! #O" "$% 8 " # "@(*-,/. ,
0ˆÛÌ» ‡ Ü » ‡cØ3
À
À
Set
flag
, §
3
5
(+*Z,/. ,
,
3
,
ETNA
Kent State University
etna@mcs.kent.edu
17
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
ª
ª
Ê
))«7
Ê
6
¬
DO WHILE
À
flag
Ð
!©­
ª
with ´
))«!‰",³‘´
)m·¸% )m·
®‘¯
.
ª
Àð
"
"
³
"‰
¶µ
Set
­
À
±
Àð
"
³
­
À
¬
±
ª
µ
IF ¼½
Set
THEN
Á
flag
À
¿
­
(Here ¹
±
)) ®»¯µ °
­¿
ELSE
ð
'1À À À 'W¿
À
­/Á
À
Á
§ Â
b
.
Set §
Perform the Schur decomposition of
%
® ¯‰Ä Ä
k
where Ã
¨„&7¶ÊË
rank
Set
¬
)) ! )
¹
³kÃ
)m·¸%
)(T³
)«T³
¬
Å
°
¬
">)
µ¹
¬ ¹ ))
¬
)m·
Ê
µ
).
« ÀÀ ð Ä Ð
¬
'À À À ' Á
))
¹
,
À ÀÀ
Ê
)
µ
&
is nonsingular with inertia index ÅÇÆ
&
&
Æ
Â
.
½
È
ª
­„)m·
À À
Ì)
ª
µ
­ )m·
À À
Ì)
À À
Ì)
À À
µ
Á
г
">)
¬ ÎÃ
¬ Î
µ
½ À « ð À zð « « À À Ê Â
¿
Ä
À
À
ð
«
«
ÀÀ
À
À
‰"
­
­ ¿
³
2Ñ
±
'À À À '*¿ 2Ñ
­„)m·
)
¾
µ
µ
¹
&
ÄÊ
À
À
½ ÀÀ ÀÀ ÀÀ
Â
¿
ÄÊ
À ÀÀ
T¬ Î ">)
¬ Î "mÏ
Ž)
¬
Ê ÊÀ
È
Â
´
""
µ
­ )m·
"ͳ
partitioned analogously.
ÊË 4
IF
¼½ THEN
¾
Set flag
and
¾¹
¿
­„)m·
"ͳ
,
Â
2"
ÀÀ
° ²±
Ê
À ÀÀ
""
³º¹
¬ ¹ ">)
µ
¬
½ «
¾
and
³‘´
±
"
¬
partitioned analogously.
4
­
³
µ
ð
À
À ð
À
"
³
%
))®k¯‰° \±
Ê
"
À ÀÀ
ð ð
ª
Perform a rank revealing factorization of
­ Á
'À À À ' Á 2Ñ
"mÏ
¬ Î
and
¸É
ETNA
Kent State University
etna@mcs.kent.edu
18
R. BYERS, V. MEHRMANN AND H. XU
ELSE
Perform a rank revealing factorization or SVD
&
"mÏ0B1ÏJ³»Ò
where Ò
½ Å Å
¿ À
½¾¾¾ ÀÀ À
¾Å
¿ À
ÀÀ
4
1Ï
ª
½
¿
¬
‰"
­¿
­
and
¾
END IF
END IF
Form
6
6
¼^½
Ê
4
¾
Å
¼½
,
Å
×
END WHILE
Õ
Ó ))
±
¼^½
Ï
Â
³
Ê
¾
.
­
Å
À
Ó
2Ñ
1Ï
­„)m·
³
­
Ì)
¾
Ä
Ï
µ
Ó
­ Á
'1À À À ' Á 2Ñ
ÕÖÊ
))
Ï
¬ Ã
Ê
¬
Õ
¼^½
Õ
¼½
Ê
³
))
¬
)+Ï
¬
µ
¾
Ä
&
Ò
)+Ï
¬
)+Ï
Ï
¬
"mÏ
)) Ó
Ä
1Ï
'À À À 'W¿ 2Ñ µ
ª
­
Ó
¹
Õ
Â
1Ï
)
Ä
¬
Set
ª
">)
ª
¬
½ À«
¿
ª
">)
À
¿
)
Ï
>" )
¬
¬ Ò &
³
¹
""
">)
"mÏ
¬
¿
¾
­
½¾¾¾ ÀÀ
¾
¾
ª
ÌÏ
¬
½
ÀÀ Ä
Ä
""
ª
Â
and
Ä
À
À
ÀÀ Â ÃÃ À À
ÀÀ ÀÀ ÀÀ ÃÃ
ÀÀ ÀÀ ÀÀ ÀÀ Ä Ê
À À À À
 ½
Â
À ÀÀ Ä ¿ À À ÀÀ Ä
À
À Â ÃÃÃÃ À
ÀÀ ÀÀÀ Ä=Ê
ÀÀ ÀÀ ÀÀ ÀÀ ½
Â
¿ À À ÄÊ
ðÀ «
À
ÀÀ Â À À
¬ð « À
ÄÐ
À
À
ð
«
«
ÀÀ
À
À
Å
Å
Ê
ÀÀ Ê
­
Ï8
Ï8
Ê
¼^½
Ê
µ«Ó
is nonsingular.
&!©Õ
&
, 5
Set 3
ª
À ÀÀ
¬ Î
%
&K®»¯2Ô Ô
Ï
)+Ï
¬ Ã
%
®‘¯ ° °
µ
¾
, and
†‡†
!
.
Algorithm 1 will stop after finitely many steps, because at each recursive call, the order
of the even pencil decreases. At some stage S
must be either nonsingular or void.
Our experimental M ATLAB implementation of the even staircase algorithm 1 makes rank
decisions using a singular value drop tolerance y
, i.e. singular values of magnitude less
than an absolute threshold y
are set to be zero. Ordinarily, y should be slightly larger than
B× c
× c
ETNA
Kent State University
etna@mcs.kent.edu
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
S
19
the magnitude of errors or uncertainties in the data. For example, if the data are perturbed
only by rounding errors and is the unit round, then it would not be unreasonable to use
8Ø !؄Ù
8Ø BØ`Ù
y
for rank decisions on submatrices of and y
for rank decisions on
submatrices of .
S w
S Ô
Appendix B. Proof of Theorem 3.3. We prove Theorem 3.3 constructively using another
staircase algorithm to obtain a more condensed even staircase form followed by a further
reduction closer to the even Kronecker-like canonical form of Theorem 2.3. In contrast to
Algorithm 1 these reductions use extra non-orthogonal transformations, so they are theoretical
in nature and may not be well suited to finite precision computation. The extra work displays
a relationship between successive values of the inertias
.
A LGORITHM 2. Given
, this algorithm computes a real
Ú ,Ú
Ú are in the even staircase form (3.1).
nonsingular matrix Ú
such that Ú
Set
flag
ª
ÝÜ %c®3
!'"Z‡ ' "D(@*0ˆÛ ,» . , »
(]*
,/. ,
" O" À
À
Ê
Ê
Ê
À ÀÀ
3
, §
ª
)) ª
Û³
©¨
5
4
,
))
))
± š
¬
®µ ¯ °›Ü ± š
)+Ï »
ª
%
®k¯ ± š ± š
À
,
))
Û³
6
.
and
ª
.
³
³
"‰
¬
Å
ð
À
À
ð
À
À
­
ª
µ
2"
­
±
µ
IF ¼½
4
Set
¨„&
THEN
flag
and
Ý
½ «
Á
¿
±
µ
® ¯ °
)) »
(Here ¹
partitioned analogously.
­¿
Ä Ê
ð
""
¹
¬ ¹ ">)
¬ ¹ "mÏ
¬ )m·
²±
µ
­
¬
ÀÀ Ä Ê
Â
À
±
2"
³
">)
¹
¬ ¹ ) )
¬ ¹ )+Ï
% ¬ )m·
°
²±
¬
'1À À À 'W¿
Á
ð
6
¹
¹
¬ Ã
"mÏ
)+Ï
).
Â
À
« ÀÀ ð Ä Ð
"
ÀÀ
ELSE
,
Â
´
­
±
° ²±
"
‰"
³
%
))®k¯ ° \±
µ
Ê
!©­
ð ð
ª
Ê
À ÀÀ ½
ð ¿ À
À
½ ÀÀ ÀÀ
À
ð ¿
À
À
ð
Set
2"
"
µ
are void.
¬
))«!‰",³‘´
)m·¸% )m·
®‘¯
with ´
Ê
¬ Ã
¬
Perform a rank revealing factorization of
ª
)+Ï
)+Ï
¬
¬
Ã
where Ã
, and
¨
Since
, the initial last row and¬ column of
DO WHILE flag
À
3
'1À À À ' Á
­/Á
À
Set §
§ Â
b
.
Perform a congruence transformation with
½
Þß
¿
­
ÀÀ Å À
)+Ï
¬
"›à 7Þ
ª
" Þá
ª
"
Ê
"›à ©Þ
¬
"
Ã
to annihilate the blocks ¹
ª
Â
­
¬
and
­
¹
)+Ï
¬
¹
)+Ï
" Þß
¬
ÀÀ Ä
½
¿
in
"
¬ ""
¬
¹
¬ â ">)
¬ ¹ "mÏ
.
Set
¬
â >" )
¬ â ))
À
6
Ã
Â
ÄÐ
À
"mÏ
¹
ETNA
Kent State University
etna@mcs.kent.edu
20
R. BYERS, V. MEHRMANN AND H. XU
Perform the Schur decomposition
&
â ))0!Ì)J³‘Ã
where
Let Å Æ
Set Î
ª
¿
¿
­„)m·
À
À
½ À À
Ì)
"mä
"mä
¬
½
Â
¸É
­„)m·
Ê
µ
is nonsingular.
&
be the inertia index of à .
Ê ÊÀ
½
ÈÎ
À ÀÀ
¬
%
&K®k¯ãÄ ãÄ
Ã
&
&
)
±
ð ÀÀ « Ä
Â
Ì)
­
ª
¿
"݈
2Ñ
­
ÀÀ ÀÀ
ð ÀÀ « Ä
±
Ž)
¿
"݈
2Ñ
À
À
½ À À
¬
­
 ¾½¾
ð ÀÀ « Ä ¿ À
 ½¾¾ ÀÀ
ð ÀÀ « Ä ¿
­)m·
´
2Ñ
±
­„)m·
""
¹
¬ å
ÀÀ ÀÀ
Ì)
­
±
¬
"mÏ
¹
ÀÀ Ä Ê
å
Ã
å
(Ñ
ÀÀ ÀÀ
ÀÀ ÀÀ
 ÃÃ
ÀÀ
å
"mÏ
¹
 ÃÃ
ÄÊ
À
À
ÀÀ ÀÀ À
&
¬
Ã
partitioned analogously.
Let æ be the permutation that interchanges the last two columns
Ï
and rows of
. Set
½¾¾
¬
ª
)k
ª
æ
"mä
´
 Ãà ½
ÀÀ ÀÀ Ä ¿ À À Ä Ê
ÀÀ ÀÀ
 Ãà ÀÀ ÀÀ ÀÀ
À ÀÄ
 ÀÀ À ÀÀ
ÄÐ
ÀÀ
¿ À ÀÀ
½¾¾ ÀÀ ÀÀ
¿
æ
""
)
¬
¨„&7¨„&
Set
4
¨ &
Set
Å
"mä
½
¬
¿
;ç
(B.1)
IF
æ
"
¬ Îå
æ
¹
¬ å
""
">)
Ê
¬ Î ">)
¬ Î "mÏ
¼^½
Á
flag
Ý
¹
г
Ý
¬ Î
À
.
THEN
Î
Â
å
"mÏ
Ã
&
¬
"mÏ
¹
å
Ã
"mÏ
¬ ÎÃ
¬ Î
½
and
Â
ðÀ « ¿ À
½ À « ¬ð « ÀÀ À
¿
À « À ð
ÀÀ
À
À
2"
­
­¿
Â
´
;ç
±
2Ñ
'1À À À 'W¿ 2Ñ µ
­ )m·
Ì)
Þ
­
Ý
¹
­ Á
'À À À ' Á 2Ñ
±
ð ÀÀ « Ä Ê
Â
«ÄÐ
2Ñ
æ
ELSE
Perform a rank revealing factorization or SVD
"mÏ0B1ÏJ³»Ò
where Ò
&
%
®»¯2Ô Ô
¬ Î
&
À ÀÀ
is nonsingular.
Ï
µ«Ó
Ê
ETNA
Kent State University
etna@mcs.kent.edu
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
Å Â Å ½
¿ Àð À Ä ¿ Àð
½¾¾¾ ÀÀ À À
ÀÀ À
Â
Ã
Ã
¾Å
ÀÀ ÀÀ ÀÀ ÃÃ
¿ À À À À ÀÄÊ
½ ÀÀ ÀÀ ÀÀ Â ÀÀ ÀÀ ½
¿ Àð À Ä ¿ Àð
ÀÀ Â ÃÃ À
½¾¾¾ ÀÀ À À
ÃÃ
¾
À
ÀÀ ÀÀ Ä Ê
¿
À ð ÀÀ « ÀÀ ½¿ ÀÀ ÀÀ
À ð ÀÀ «
À
À
À
½ « Àð «À
¿
À «
ÀÀ
À
Å
Å
Å
Ê
À
Å À À
½
4
&7Õ
&!
, 5
Set 3
ª
ª
">)
ª
)) Ó
""
ª
>" )
ÌÏ
Ï
­
¬
¬
¬
Ý
¹
Û³
±
Ï
""
">)
">)
"mÏ
))
¬
¬
¬ Ò &
"mÏ
END IF
Form
6
4
­
6
­
µ
¼^½
Ï
Ó
 ½
Â
ð
Ä ¿ À À ÄÐ
ÀÀ À À
Â
ÀÀ ð « Ä Ð
Å
Ê
³
Õ
á
­/Á
¬
Ý
±
Ï
'À À À ' Á (Ñ
Õ
¼^½
¨„&
))
³
)+Ï
¬
¬
µ
.
­
Ó
Õ
¼^½
¨„&
1Ï
æ
2Ñ
Ý
'À À À 'W¿ 2Ñ
))
¨„&
Â
±
Õ訄&
ª
Ý
,
ÀÀ Ä
±
)
Þ
2Ñ
±
Õ
¼½
Ý
Â
­„)m·
‰"
¼^½
¨„&
and
END IF
Ý
Ó
¬ Ã
¬
¹
ª
Ï
&
Ò
)+Ï
Ó ¬
)+Ï
­
¬
­ ¿
Set
±
1Ï
)
¾
Set
­
)
Ï
ª
ÀÀ Ä
Ï
ª
±
Â
, and
Ï
­
Ï8
¨„&
¼½
21
)+Ï
¬ Ã
%
®‘¯ ° °
µ
Ý
, é
.
é
END WHILE
We now show that the subpencils generated by two algorithms are equivalent. For this
we need the following lemma.
vs
and U WV
. If
L EMMA B.1. Suppose that
Cg(Œ*
¡. , £˜Ú 1CD
㠙" C ™ c ™ cc 㠆 " C † c † cc where ™ † ('* . , 㠙 㠆 (*- j. , and ™ † (+*-,/. , are nonsingular, then there exist
nonsingular matrices
k6 kk †™ k c (Œ* j.  †™ c (]* ,/. ,
  
where k ™ ™ (* . , such that

㠙 㠆 k- ™ †  ™ Mk ™" †  ™ t
L
Ú
L
DL
‡ ‡
Ú
]
DL
Ú
!
Ú
]
‡ ‡
Ú
Ú
L
L
ETNA
Kent State University
etna@mcs.kent.edu
22
R. BYERS, V. MEHRMANN AND H. XU
C iCj" or C ! Cj" and 㠙 ™ , 㠆 † , then k&

kŸ 㠆 ê™ ã ™ and † ê™ ™ . Then
 c
c
c ™ c Dk " c † c  t
"
 ™ † ™
In particular, if
.
L
Proof. Let
Ú
Ú
and
Ú
L
™
L
Ú
L
The result follows directly by comparing the blocks on both sides.
To show the relationship between Algorithms 2 and 1 we denote the blocks in Algorithm 2 by a and in Algorithm 1 by a S .
Assume that at the beginning of the A th reduction
q
‰ q + … q " 0 ‰ + … 3
†™%™™ c †‡† , where ™‡™ (*  ð .  ð , %† † (
for some nonsingular matrix
* ð . ð . Then for ‰ q †%† and ‰ †%† in ‰ q and ‰ , respectively, we have
‰ q †%† ™‡" ™ ‰ †%† ™‡™ t
bê
(B.2)
ê
‡
‡
S
ê
ê
ê
S
ê
‡
N
‡
N
ê
S
Qê
Let
^ê
S
ê
S
q ™ " ‰ q †%† q ™ cq cc ™ " ‰ †‡† ™ c cc t
C
S
S
S
CS
By Lemma B.1,
q ™ ‡™ š™ ™ ™ †%™ ™™ c %† † is nonsingular and ™‡™ (r*-† . † . Then a simple calculation
where
yields
@‰ q ™ +Ÿ… q ™ q " 0 ‰ ™ + … ™ 3|q where
TU
c
c
"
‡
™
™
q
c
c
ßq c ™ o ð
c ™ o ð PR †¶™™ †‡†† c
†‡† . Clearly, then
with
q… †%† … q † †‡† c " PR … †%† … † TU †‡† c t
(B.3)
†
†
q "† q
"†
In Algorithm 2 we then determine a nonsingular matrix q such that

q
q
q " … q †‡" † … q † q cq cc  …†

vê
Œ
Œ
Œ
Œ
Œ
Œ
S
Œ
E E
S
Œ
S
Œ
S
Œ
Œ
‡
ê
‡
Œ
Œ
Œ
Œ
]
Œ
]
]Y]
bê
]r]
ë
]
S
S
S
]
L
Œ
ì
Œ
Œ
]
SS
]r]
SS
SS
]
]
LS
ë
]
S
S
S
]
L
ì
L
Œ
Œ
]
Œ
]r]
ETNA
Kent State University
etna@mcs.kent.edu
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
23
q [¢^¤£{¥ 0 Ì q 3 and in Algorithm 1 we determine an orthogonal matrix such that

TU
c
" PR … †%† … † c c t
 … "†

By (B.3), the new q and must have the same inertia and the same size í ™ 7 í ™ .
Moreover, by Lemma B.1,
q †%†† c š™ Ñ 

ð
c
«. ð «.
Ñ †‡†
where Ñ is nonsingular and Ñ †%† (*
Ñ † Ñ
We then have
@‰ q † +Ÿ… q † áÑ q " 0 ‰ † + … † 3Ñ q (B.4)
where
c c TU
š

™
%
™
™
Ñ
o
c
o
c
†
†
q c q PR Ñ † ™ Ñ †%† c Ñq c
Ñ Ý™ Ñ † Ñ


™‡™ .
with Ñ ™%™
Let … q ™ be the block of … q † in (B.1) and … ™ be the corresponding block in … † . By
comparing the blocks in (B.4) we have
q ™ Ñ "™‡™ … ™ Ñ t
Let
q " … q ™ q q c cc " … ™ c cc
be the computed rank revealing factorizations. Then q and must have the same size
7 . Again by Lemma B.1
q Ñ ™‡ê™ ™ k- q Ñ ê™ where
k+ kk †™‡™™ k c †‡† †™‡™™ c †‡† and k ™‡™ ™‡™ (Œ* z. . Then
)‰ q Ÿ… q k q " 0 ‰ … 3škq where
cð c TU ê™ q ð c c TU
k6q PR c o « c Ñ q PR c o « c
c c q
c c
k ™%™ c c c c T
Pk k c c c
k †Ý™™ k †%†† k c c U R k ¶™ k † k k c
k œÝ™ k œ † k œ k œ k œ‡œ
where Lví
L
S
L
SS
S
SS
SS
]
S
LS
LS
s
LS
L
]Y]
Œ
Œ
]
]r]
S
=
‡
E
S
E
Δ
=
‡
S
7s
Œ
]
]
S
]
]
]r]
ïŒ
]
S ]
^]
S ^]
]r]
H
^] _
S
H
S ^] _ S
S
y
H
H
S
S
y

_
S
]Y] _ S 




‰ð ð
]
]
S
S ]
=
‡
S ]
‡
_S
_
hg
g
g
f
g
f
f
f
]
†
†
]
]r]
†Y]
†Y†
r]
r†
=
ETNA
Kent State University
etna@mcs.kent.edu
24
R. BYERS, V. MEHRMANN AND H. XU
k Ñ †‡†
k ‡™ ™ , k œ † ™ , and k œ‡œ †‡† . It is then evident that the newly
)‰ q +Ÿ… q and ‰ + … satisfy
)‰ q +Ÿ… q kk †‡†† k c " 0 ‰ … 3 kk †‡†† k c and
with ]Y]
generated subpencils
†r†
r†



S
S
]
]Y]
S
J
S
]
]Y]
which is the same as (B.2). Since both algorithms start with
‰ q ‰ [ … q … D
it follows by induction that they generate the same integers :š» , ¶» , » . The inertia indices
satisfy ÛÌq » ÛÌ» Û» ê™ and Ü q » MÜ » 6Ü » š™ .
S
S
s
9
In the following we will show that by carrying out some further block Gauß elimination steps, the staircase form computed by Algorithm 2 can be reduced close to the even
Kronecker-like form. In Algorithm 2 it is not necessary to move all the blocks L toward the
center. So the permutation with in (B.1) does not necessarily have to be carried out. The
staircase form has the following block structure.
s
½¾¾¾
ß
¹
é
¾¾¾
#""
¾¾¾
¾¾¾ Å
¾¾¾ Å
¾ Å
..
.
..
.
¾¾
¹
жÐÝÐ
.
..
" % &
" % )+&
""
é
..
.
..
.
.
..
.
..
.
" % &('1"
6
" % &('-)
..
.
..
.
ÐÐÝÐÝЇЇÐÐ
ÐÝÐÝÝÐ ÐÝÐÝÐÝÐ
ÐÐÝЇÐ
..
" % )+&('1"
..
6
.
..
.
..
.
,&0% &
Å
6
..
.
..
.
&0% &
6
&0
% &2'-)
..
.
..
.
"% &2'-)
..
.
&
À
&0% &2'1"
..
.
..
.
..
.
6
.
 ÃÃà À
ÃÃÃ
ÃÃÃ
ÃÃÃ
ÃÃÃ
ÃÃÃ
"% )+&('"
.
&('*)r% &('*)
6
.
..
.
q » » » š™ ¶q » Ý» E q » , ( p
), and
» . †‡ í ™Ý » (6* , ¹ . ¹ « £{Ú  » . †‡ í ™¶ » í ™ . í ™ c c ðc « (]* † ´. †
:»
q
:
»
»
»
» . †‡ í †  » : » » » c p
s
9
#?©@Ë?BA
>=
WV
U
DC
=
‡ò
s
H
E E
9
C
I
.
6
)+&('1"% )+&('1"
Ä
9
s
$?©@?KA
#?7@?BA
..
.
..
.
3 &
4
5 &
..
.
..
.
Î
5 "
Î
..
.
..
.
3 &
4
5 &
..
.
..
.
Î
5 "
— ÖE q í ™ . The blocks have
p
q» í ™ p
Š
ñ
3 "
3 "
Î
(B.5)
F .
.
..
ó< ò
À ÃÃÃÃ
ÃÃÃ
ÃÃÃ
ÃÃÃ
ÃÃ
Ä
&('"% &('"
6
.
where for s Qs ¶s
,9
the following properties.
..
..
..
&0% &('*)
6
..
ÐÝжРÐÝÐÝÐ
"% &2'-)
6
 ÃÃÃ
#"% )+&
À
&('1"% &('1"
..
.
..
.
6
&0
% &2'1"
..
.
..
.
"% &('"
6
6
$"% &('-)
À
"% &
ЇÐÝÐ
#"% &('"
,&0% &2'1"
0
& % &('1"
..
.
..
6
.
"% &2'-)
&
ÐÝÐÝÐÐÝЇÐ
..
" % &
#"% &
жжЇЇÐÐ ¶Ð¶Ð ÐÝÐÝÐÐ
жЇРŠ..
.
u6
6
6
6
..
" % &
¿ Å
½¾¾¾ À
¾¾¾
¾¾¾
¾¾¾
¾¾¾
¾¾¾
¿
é
жЇÐ
" % &('"
6
é
Ê
Ê
ETNA
Kent State University
etna@mcs.kent.edu
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
25
q » ¶c »
†‡ í †  » . †‡ í †  » ¶q »» c » c p
í ™ . í ™ "™%™ ™ ™ í † ™ ™%™ (* † ´. † í ™ (Œ* ð « . ð « †
where all the blocks
» » are nonsingular. Without loss of generality, we may assume
that and » D¢^¤£{¥ 0 o ¹ w!o ¹ 3 .
†
The property that » . †‡ í ™Ý » has full column rank can be shown as follows.
After the first step of reduction we have
c”c“c T
‡
™
™
™
‰
‰
†
P "
c”c“c
f ‰ c ™ † ‰ c †‡† c”c“c U t
R c c c”c“c
c c c”c“c
In the next step, after having compressed ‰ †‡† , is changed to
c“c”c T
%
™
™
™
™
‰
‰
‰
†
P ‰ ™" †
c c“c”c
‰ ™ " c c c“c”c U t
c”c
R cc cc cc c“
c“c”c
c c c c“c”c
‰ ‰ ™‡™ " ‰‰ ™ † and are nonsingular, ‰ ™ has to be of full column rank. It is easily
Since ™ † †‡†
seen that ‰ ™ is equivalent to ™ . †‡ . So ™ . †‡ has full column rank. By induction, it follows
that the other blocks » . †% í ™¶ » have full column rank as well.
We now begin further reductions on the pencil (B.5). The reduction process is described
in the following algorithm.
" and "O be given as in (B.5).
A
3. Let s
s
9
L
DL
L
L
C
E
C
$?©@Ë?!A
M
9
ôL
E E
L
=
‡ò
L
=
õ‡ ò
H
L
÷ò
öò
L
g
g
g
f
g
f
f
f
S
f
f
S
f
S ^]
f
f
hg
g
g
g
g
g
S ]
S
C
f
S ^]
C
S ]
LGORITHM
Ú
í
Ú
Ú
í
">)
">)
Ú
&('1"
Å
&('1"% &('1"
6
Annihilate the blocks Ã
and Ã
with pivot block Ã
in
.
0
,&0% &('"
& % &('"
(
) above and to the left of
Annihilate the blocks in
´
in
&('"% &('"
with the pivot block ´ . Then
Å
³
,&0% &
0
& % &('"
¿
µ
Because by the reduction procedure
ø &
Ï
ðÀ « Ð
has to be of full column rank.
ÞÎ
³
such that ù
ù
À À
­
µ
³
Å
ø
Û³
Î
,&0% &
0
& % &('1"
­
±ã
½
,&0% &('1"
&('"% &('"
=
&0% &
ÅÀ
Î
³
,&('1"% &('1"
µ
Î
is nonsingular,
So we can determine µ a nonsingular matrix
½¾¾
ù
À À ÄÐ
À À
&8% &('"
& '"% &('"
(
0
& % &('"
Then
µ
,&0% &('"
´
ø &
,&0% Î &
Å
³
Â
ø &
À À
¿Å
­
""
µ
ú
">)
ú
ÀÀ
">)
))
ÀÀ
Å Àð «
À
­ú
ú
Î
±
=
´
­
±
Î
ð À « Â ÃÃ
ÄÐ
ÀÀ
=
ETNA
Kent State University
etna@mcs.kent.edu
26
R. BYERS, V. MEHRMANN AND H. XU
))
Ê
Å ð «
">)
">)
, and
We then annihilate
­
transformation with pivot blocks ± ã
of
Å
³
,&0% &
,&0% &('1"
ú
&('"% &('"
8
& % &('"
ú
ú
""
,
Å ð «
ðð «
by performing another block Gauß congruence
­
=
±ã
and
. Again by the nonsingularity
=
has to be nonsingular.
&Qç Rûýü ¿
±ã
So it can be compressed further
to
ú
µ
more congruence transformation. Eventually,
&0% &
Å
³
½¾¾
by performing one
&
Å ¿ À ÀÀ ð « ÀÀ
ÀÀ Å À À
Ï
Ê Ê
&0% &('1"
0
& % &('"
ú
= ^þóÿ )
,&('"% &('"
ú
µ
­
±ã
´
=
±ã
­
ð À « Â ÃÃ
ÄÐ
ÀÀ
Ï
=
6
Applying the same sequence of congruence transformations to
, it is easy
to check that the block structures and ranks will not change.
­ ¿
6
Ò
"
We now proceed by working on
. First we simplify
and Ò
to
­
­
6 &8% &('*)
6 &0
6 &8% &('*)
% &2'-)
Ò
in
and
by post-multiplying
to
and pre-multiplying
Å
6 &0
% &('*)
its transpose to
. Note that this transformation
does not affect
É
6 &('*)r% &('*)
and the blocks in
.
&
6 &0% &('*)
6 &0
6 &0% &2'1" 6 0
&
% &('-)
& % &('" 6 &0% &
Second, we annihilate , in
,
and
­¿
6 &0% &('*)
6 &8
% &('*)
with pivot block
in
and
.
Å
ð
Å ÝÊ Ð¶ÐÝÐÝÊ
Á Á
Ï
ʶÐÝÐÝÐ‡Ê Ï
Ï Å
ð
Ï
Ê
Ï ÏÏ Ï
ÏÏ Å Ï Ï
Å
ÊÝÐÝÐÝÐ‡Ê Å
ÊÝÐÝЇжÊ
¶Ê ÐÝЇÐÝÊ Å
Ï
Ï Å Ï ÏÏ Ï
Å ¹À «
¹À «
¹« Å ¹«
Ï
Ï Ï
ÏÏ Ï
ÀÀ Ï Ð
À
Ï
¹ ¹ «
Ï Ï Ï Ï
Ï Ï ¹
Ï
Ï
Ï
Ï
Ï
Ï
Ï Ï
Ï Ï Ê‡ÐÝжÐÝÊ Ï Ï Ï Ï ÊÝÐÝÐÝÐ‡Ê Ï Ï¹
Ï Ï
Ï Ï
With this further reduction, the matrices and are transformed as
 ÃÃÃ
À ÃÃÃ
ÃÃÃ
ÃÃÃ
Å
ÃÃÃ Ê
À
Å ÃÃ
À
À
Ä
Å
À
À
FOR §
ÏÏ
Ê
% '"
a) Annihilate the blocks
in
with the nonsingular blocks in
&0% &('"
'"% )+&
as pivots.
Simplify
% )+&2'1"
and
% )+&('"
% )+&
³
to
­ Á n=
ã
%
,
%
ú
ú
6
, Ò
Annihilate the blocks
6
, Æ
in
as well as
6 % )+&('-)
6 % )+&('*)
from
and
­ Á >=
ã
% )+&
,&('"% &('"
%
6
,
% )+&('*)
in
6
µ
, respectively.
­ Á >=
ã
and µ
from
and
must be nonsingular and thus
% )+ú&('*)
% '1"
6
6
and
% )+&('1"
% )+&('*)
6
% )+&('*)
6
to
6
% '"
­ ¿
.
% )+&('"
,
,
­ ¿
with block pivot
.
END FOR
½¾¾¾
Þ
k޺
¾¾¾
¾¾¾
#""
¾¾¾
..
.
¾¾¾
..
&
.
¾¾
,&0% &
¿
" % )+&
..
.
..
.
&
8
& % &('"
,&0% &('1"
"% &('1"
..
.
..
.
"% )
3 "
.
..
.
..
.
..
3 &
4
,&('"% &('"
5 &
"% &('1"
..
.
..
,
­ Á >=
ã
г
With the same argument as before,
û ü¿
Á n= þ ÿ )
ã
we reduce
to
.
b) Reduce the blocks Ò
³
and
with pivot blocks
Annihilate the blocks in
µ
% )+&('1"
% )+&('"
and
to get
% '"
0
& % &('"
as well as
'"% )+&
..
.
..
.
Î
.
..
.
5 "
Î
ETNA
Kent State University
etna@mcs.kent.edu
½¾¾¾
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
Þ
¾¾¾ À
¾¾
..
.
6
6
..
" % )+&('"
À
&2'1"% &2'1"
6
&('*)r% &('*)
6
0
& % &('*)
.
..
3 "
..
.
ÃÃÃ
.
..
&0% &('-)
6
ÃÃÃ
"% )+&('"
6
¾¾¾
¿
 ÃÃÃ
Þß
u6
.
6
)+&('"% )+&('"
Ä
27
ÃÃ
3 &
4
5 &
..
.
Î
5 "
q q
Ï w» Ý» » )
Â
Ï
ÀÀ Ä Ê Á Å Á Ê
Ï Ï
Ê
Ê
Ê
ÀÀ
Á
À
Å
ÀÀ À Ê
ðÀ « Ê
ÏÀ Ï Ï Å Ï Ê
À
Ï Ï Ï Ê
À ÏÏ Ï À Ê Á
Ï
Ï Ï Ï
Ê
Ê
À
À
À Á Ð
À Ê
E p are signature
Note that all » ·¢^¤£{¥ 0 o À ¹ w!o ¹ 3 , p
matrices.
By performing a congruence transformation to the pencil with ã " ã ã "š ã in (B.6)
|O
(B.6)
ã
Ï
½
Ï ÏÏ Ï Å Ï ¿ À
ÏÀ Å
Ï
Ï
ÏÏ Ï Å
for some nonsingular matrix
% )+&('"
Î
3
¨
5
Î
, where (note that 9
¨
' "
'"
'"
'1"
5
Î
3
%
5
'"
5
³
Î
Î
&
& '"
¨ (
3
,&0% &2'1"2
¨„&('"
Î
Î
&2'1"% &2'1"2гk´
±ã
¨
% )+&('-)
6
Î
)+&2'-)
6
% )+&('*)
5
¨
¨„&2'1"
Î
Î
§
Bû
µ `·
3
§
³ ÃÎ
§
Î
Ã
&('"
÷ò
öò
¨„&('"
­
´
µ
µ
5 &('"2
5
&('"% &('" г
L
Î
=
5
'1"
5
Î
Î
­
3
'"
5
û
³
3
¨
6
¼^½
§
­
3
'1"
5
­
Î
'1"
s
9
µ
®»¯
)m·¸% )m·
µ ?7@?KA
J
with an appropriate permutation, we obtain the structured Kronecker form (2.2) of
This leads to the conclusion in Theorem 3.3.
Let us illustrate this complicated process by an example.
E XAMPLE B.2. Let
c c
P c c c
OfD p c c p
c p c
c c
R cc cc
c c
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
C
c c”c“c T
pc c “
c c”c“c
p “
“
c
c
p
c
c“c
c“c
c“c”c“c U
c“c”c“c
c“c”c“c
c“c”c“c
hg
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
.
ETNA
Kent State University
etna@mcs.kent.edu
28
R. BYERS, V. MEHRMANN AND H. XU
c c
”
P c”c c
c c c p c“c T
c c c c p c
c p c
c
c c
c c
p c”c“c U
c c”c“c
c c”c“c
c c”c“c
hg
g
g
f
g
f
f
f
g
c
f
f
g
g
ˆ
g
g
c
c”c cp
c”c
R cp cp
c”c
is nonsingular. We have ‘ , and
: ™ — : † D cÌp : D[cÌc ™ · cd‘ t † — q™ p q†
q
q
f
f
f
f
f
f
f
f
f
f
where C
g
g
g
g
g
g
g
A
]
s
Then
s
s
]
9 ]
9
†
p
: ™ p † 3‡Ù : † D cÌp ; : p (for
ŽšD» šc̙ )
‡
3
Ù
q
q
Ü q0W:™ ™ p † q † † [cd— q Dc̆ q 0W: †[cÌ (for— ë † » ê™ ) (for ë † » )
By Theorem 3.3, we conclude that the pencil has one singular block ¦ ™ cF@Oc ; one
block Ž ™ ; one block Ž † ; and one block ë ™ with ù & p . The canonical structure of 8=O
.
associated with the finite eigenvalues is the same as that of In fact, if we rearrange the columns and rows of and in the order ü , , p , ‘ , , , ,
— , , p c , p˜p and let s be the corresponding permutation matrix, then
T
P c
c c c p c
c c p c“c
c”c“
c“cc
c p c p c”
s " 0 O3îs8[
c c c”c“c
c p c U
c“c
R
c p c“c
c
T
P p
c c c“c p
c c c p c
c c c“c”c
c c“c”c
p c p c“c”c
c c p U
c c”c
R
p c”c
c
s
9
™; ™
q™ s
9
9 ]
9
]
9
s
9 ]
s
s
]
s
s
]
†
C
hg
g
C
f
2ˆ
g
g
f
g
f
g
f
g
f
g
f
g
f
g
f
g
f
g
f
g
f
g
f
g
f
g
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
ˆ
hg
g
g
g
g
g
g
g
g
g
g
g
g
g
g
g
ETNA
Kent State University
etna@mcs.kent.edu
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
29
which is in the structured Kronecker form.
p
Appendix C. A structured algorithm for computing the structured Schur form of
regular skew-symmetric/symmetric pencil of index at most .. In this appendix we present
an algorithm for computing the structured Schur form (4.2). We call a matrix . In this algorithm we will deorthogonal-symplectic if and note by m @ a Givens rotation operating in rows or columns and @ . If “
“ “ is orthogonal-symplectic and also then m @ í
is
orthogonal-symplectic. Finally in the algorithm we use the orthogonal-symplectic permutaw
tion matrix
.
•
•
•
•
•
t•
OŒ
A LGORITHM 4. For a regular skew-Hamiltonian/Hamiltonian pencil
of index
Y
OE E
at most with
and
, where
is
positive diagonal, this algorithm computes orthogonal matrices
and orthogonal symŒ
, such that
,
are in the form (4.2).
plectic matrices ()*-†‡,/. †‡,
" o ‡† , " , ,
= 0 3 3 ¢^_£˜¥ 3
» *
†‡( ,/. †‡*, ,. ,
#
3
&
(
´0
0 » »
=0W: —´:
sg , ™ ztwtztÔ , ê™ †‡, , í ™ wtztwtÔ †‡, ê™
‹
á
k
k
"
"
D
(
*
‡
†
/
,
.
‡
†
,
k
¢
¤
^
{
£
¥
‡
Ì
c
%
®
c
3
p 
0 ™  (M* .
, ,
"™ † "™ k ™ † " 0 , k
" h , " 3 h ††
™ †
h h h
h
Let
ß gsu ™ ™ ·o †‡, † † [su
[s " 0 , k h " , " 32sgD¢^¤£{¥ 0 cÌ h %c ,  ê™ %cd ‡c ,  š™ 3Ôt
‡k
to a form that is as (4.2) with the exception
Step 1. Reduce
that
is lower Hessenberg.
FOR ;F p ztwtztÔ :
% Annihilate , í 9¬. 9 ztwtztw †‡, š™ . 9
FOR : E ;wtztwtz —´: p
Determine =0 E p 3 to annihilate B» 9 . Set
n " 0 E p 3 kŸ " 0 E p 32k
™ ™ =0 E p 3Ôt
h h í
E
3
Determine 0
: : p to annihilate ùÔ» . » ™ . Set
kŸDk 0 : : E p 3Ô ™ ™ ´0 : : E p 3¶t
Determine =0 : : E p 3 to annihilate ùÔ»  , í ™ . »  , . Set
n " 0 : : E p 3 k6 " 0 : : E p 32k
E 3¶t
h ™ h ™ =0 : : p
END FOR
% Annihilate †%,/. 9
Determine =0W: —´: 3 to annihilate †‡,/. 9 . Set
n " 0ˆ: —´: 3 k6 " 0W: —´: 3%k- ™ ™ =0W: —´: 3Ôt
h h
3
Determine another =0ˆ: —´:
to annihilate ù †‡,/. , . Set
k6Mk =0ˆ: —´: 3¶ ™ ™ =0ˆ: —´: 3Ôt
% Annihilate ,/. 9 wtztwtÔ 9 í ™ . 9
FOR : wtwtztԇ; E p
Œ
bŒ

Œ
Œ
Y
r
E
E

†r†
A
A
@
@
Œ
@
@
A
^Œ
@
@
@
@
@
@
@
@
Œ
@
rŒ
@
ó@
ó@
@
A
A
^Œ
A
A
@
@
@
@
Œ
ó@
@
@
ETNA
Kent State University
etna@mcs.kent.edu
30
R. BYERS, V. MEHRMANN AND H. XU
=0 p 3 to annihilate » . 9 . Set
ß " 0 p 3 kŸ " 0 p 32k- ™ ™ =0 p 3¶t
hí h
3
p to annihilate ù » . » ™ . Set
Determine ´0
kŸ[k 0 p 3Ô ™ ™ 0 p 3¶t
p : E 3 to annihilate ù , í » š™ . , í » . Set
Determine =0ˆ: E
ß " 0ˆ: E p : E 3 kŸ " 0ˆ: E p : E 3%k-
E p : E 3Ôt
h ™ h ™ =0ˆ:
END FOR
% Annihilate , í 9¬. 9 í ™ wtwtztÔ , í 9. , š™
FOR M; E p wtwtztÔ : p
Determine =0 E p 3 to annihilate , í 9¬. » . Set
ß =0 E p 3¶ =0 E p 3¶ † † =0 E p 3¶t
hí h
E
3
Determine 0
p to annihilate Vƒ» ™ . » . Set
" 0 E p 3 † † ´0 E p 3¶t
E p 3 to annihilate V , í » . , í » í ™ . Set
Determine =0ˆ: E : E
ß =0ˆ: E : E E p 3¶ =0W: E : E E p 3¶
E E E p 3¶t
h † h † =0W: :
END FOR
% Annihilate , í 9¬. ,
Determine =0W: —{: 3 to annihilate , í 9¬. , . Set
ß =0W: —´: 3Ô =0ˆ: —´: 3Ô † † =0ˆ: —{: 3¶t
h h
Determine another =0ˆ: —´: 3 to annihilate V †‡,/. , . Set
" 0ˆ: —{: 3 † † =0ˆ: —´: 3¶t
% Annihilate , í 9¬. †‡, ztwtwtz , í 9¬. , í 9 í †
FOR —´: ztwtwtw : Er; E —
Determine =0 p 3 to annihilate , í 9¬. » . Set
n =0 p 3¶ =0 p 3Ô † † =0 p 3Ôt
h h
3
Determine 0
: p : to annihilate V§» š™ . » . Set
" 0 : p : 3 † † 0 : p : 3Ôt
Determine =0 : p : 3 to annihilate V§»  ,. »  , ê™ . Set
n =0 : p : 3¶ =0 : p : 3¶
3¶t
h † h † =0 : p :
Determine Œ
@
@
ó@
A
^Œ
@
@
ó@
@
@
@
ó@
@
@
Œ
@
@
@
@
rŒ
@
@
A
@
@
@
@
A
@
@
Œ
ïŒ
@
@
A
@

ó@
@
@
ó@
@
@

ó@
@

@
Œ

@
@
ïŒ
@
@
@
@


@
@
@
A
A
Œ
bŒ



A

A
@
@
Œ
ïŒ
@
@
A
@

ó@

@
@
vŒ
ó@
@
@
@
@
@
Œ


@
@
@
@
@
ó@

@

ó@
@
ETNA
Kent State University
etna@mcs.kent.edu
31
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
END FOR
END FOR
(Note that the (3,3) block of  now is zero.)
% Annihilate A
A
E
E
Š
FOR
Determine to annihilate A
.
í í ztwtztw í í í
;F : E p , wtwtz™ tÔ. , : † E , ´. , ™
=0 ;‡;ÖE p 3
9¬. 9 í ™ Set
ß =0 ;š‡; E p 3¶ =0 ;‡; E p 3Ô † † =0 ;Ý;ÖE p 3¶t
h h
Œ
ïŒ
END FOR
Step 2. Reduce

Œ

to lower quasi-triangular form.
†r†
‡k-
p ztwtzhJtÔ 
†

Partition the matrices Œ
as in (4.2). Apply the periodic
-algorithm,
see e.g. ([7, 22, 23]) to the formal product
Œ
Œ

 †r†
, @
†r† to determine orthogonal matrices
†Y†
†Y†
such that
and
 †Y†
are lower triangular,
and
]
Œ
Œ
† ,
† are upper triangular, and
†Y†

is lower
]
quasi-triangular.
Let
k  " k †‡š† ™ ‡† † †%š† ™  " g"
gl " k l
lg" †‡† l lg™ œ " %† † l †
lgœ " l ›
l» l8" k l
l › ‡† †
l8™ "
™ [[ ¢¢^_^_£˜£˜¥¥ 0 oo ‡‡ ll ‡‡oo ‡‡ll ™ 3Ô3Ô † DD¢¢^_^¤£˜£{¥ ¥ 0 o o ‡‡l l ‡‡o o ‡‡l l › 3¶3Ôt ™
0 † † †
0 œ œ
Set
n "™ † kŸ "™ k ™ † " † ™h h ™ ™ h † h † † ™ ™ ™ † † † t
Once the form (4.2) has been obtained, we introduce
k6 "™ k ™ † " † and n "™ † t
(C.1)
h
h
h h
Because , k
" , " , ß8 , , " , ™ , , ™ , and † , , † , we have
™ " J0 ," h 3 " ™ k , " 3Z , k "" , "" 0, † , † , ,
0 , h † , " 3 " h 0 , h "™ , 3 " 8 , " , "
(C.2)
and from Mk , we have

"™ 0 , ™ , " 3$ "™ k ™ ™ " J0 , ™ , " 3Z k , k " , " h
(C.3)
0 , h  † , " 3 h " jh † h0 , h † , " 3%k † h† " h † , , " t
+ are exactly the finite eigenvalues
It was shown in [3] that the finite eigenvalues of 
of
ñ +! ·  c c + c c 
(with doubled algebraic multiplicity). Let ™ ·¢^¤£{¥ 0 ™ , † , " Ý3 , † D¢^¤£{¥ 0 , ™ , " † 3 .
h h
h h
If follows from (C.1), (C.2), (C.3), and (4.2) that
"™ 0 ñ +! f3 † [ k , k
c " , " c " + gc " " c
, ,
, ,
]
E
E
E
Œ
E
Œ
E

S
S
Œ
Œ


Œ
S
S
S
Œ
Œ

E
E

†
E
Œ
S


S 
S
#
S S
OŒ
Œ
Q
S S
#
S B S
Œ
Œ
Q
Œ
S
S
ETNA
Kent State University
etna@mcs.kent.edu
32
c
c k %† † k
" " "
P“
c c c c
D c c " k k
‡†" † c " "
c " ‡† † "
R
c c c
c c "
c ™%™
P
c
c
g" "
"
"
c g" "
"
R c
c g™‡" ™ c
c
c
"
g†‡" †
f
T
R. BYERS, V. MEHRMANN AND H. XU
"
"
hg
g
g
g
g
g
g
g
g
g
"
†Y†
f
f
f
f
†r†
f
f
f
f
 †r† 
f
"
Œ
c †%†
Œ
f
f
f
f
f
Œ
f
f
]Y]
Œ
f
†r†
c
"
 †Y†
c
f
f

%† " †
U
"
Œ
"
c
"
"
]Y]
Œ
T
g
"
g
g
"
g
g
g
†Y†
U
g
g
g
g
t
Œ
Œ
p — ‘ ü
Rearranging the rows and columns by a block permutation in the order , , , , , , , ,
the pencil is equivalent to the pencil
õ " " " TU
U
#
"
"
"
"
Pc
PR c$
c c c c R c ä c u õê" c c c "%# "
c c " ä"
where the asterisks indicate (possibly) nonzero blocks and
k †‡† k " c ä c †%† Qõ# c ™‡™ t
#
c " †‡†
g" c
g" c
are exactly those of & #g6 ä and &#|"]EH ä " . It
The finite eigenvalues of ñ +!
is easily verified that
o c 0 &#DEŸ ä 3 o c [&#g+ ä c !o
c !o
#â· & # " E ä " are the same. Therefore, · and
the eigenvalues of &
#g ä have the same finiteä and
&
eigenvalues with the same algebraic multiplicity.
k
Ý
k
Since †‡†
†‡† are nonsingular, the pencil ' #g+ ä is equivalent to
c
k  " k †‡š† ™ †%† t
# š™ ä š

™
†‡† " g"
c
š™ ä are the square roots of the eigenvalues of the
Then, obviously, the eigenvalues of #
matrix
 š™ ê™ 
0 # š™ ä 3 † 8 k " k †‡† †‡† c †%† " g" †‡š† ™  " g" c k  " k †‡ê† ™ †‡† t
Note that the two diagonal blocks have the same eigenvalues. Using the triangular forms of
š™ ä 32† can be computed from the diagonal blocks of the
these blocks, the eigenvalues of 0 #
upper quasi-triangular matrix
¡k  " k †%š† ™ †%† †%š† ™  " " t
c
f
"
"
T
hg
"
hg
g
g
f
f
f
†Y†
 †r† 
Œ
Œ
†Y†
Œ
Œ
]r]
‰Œ
†r†
 †Y†


†Y†
Œ
 †Y†

Œ
†Y†
Œ
†Y†
Œ
 †r†
†r†

†Y†
Œ

 †Y†
Œ
 †Y†
†r†
Œ
†Y†
†r†
Œ
ETNA
Kent State University
etna@mcs.kent.edu
A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS
6O
g+ ä
6 
g+ ä .
This then allows to compute the eigenvalues of &#
OŒ
Because
is equivalent to
'
#
be obtained from
, as well.
, the finite eigenvalues of
+
33
can