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Electronic Transactions on Numerical Analysis.
Volume 30, pp. 187-202, 2008.
Copyright  2008, Kent State University.
ISSN 1068-9613.
Kent State University
http://etna.math.kent.edu
LOW-RANK ITERATIVE METHODS FOR PROJECTED GENERALIZED
LYAPUNOV EQUATIONS
TATJANA STYKEL
Abstract. We generalize an alternating direction implicit method and the Smith method for large-scale projected
generalized Lyapunov equations. Such equations arise in model reduction of descriptor systems. Low-rank versions
of these methods are also presented, which can be used to compute low-rank approximations to the solution of projected generalized Lyapunov equations with low-rank symmetric, positive semidefinite right-hand side. Numerical
examples are presented.
Key words. projected generalized Lyapunov equations, alternating direction implicit method, Smith method,
low-rank approximation
AMS subject classifications. 65F10, 65F30, 15A22, 15A24,
1. Introduction. Consider a linear time-invariant descriptor system
!
where , #"$&%(' % , )"*$%(' + , #"*$,-' % , ."/$% is the state vector, 0"*$&+ is
the
input and "1$, is the output.
The
matrix may be singular, but the pencil
2 4control
2
3 is assumed
9
3
:<;=> . Descriptor systems arise in many
to be regular, i.e., 5768 (1.1)
different applications including electrical circuit simulation, multibody dynamics and spatial
discretization of partial differential equations, e.g., [6, 7, 8, 40]. Stability analysis and some
control problems for (1.1) are strongly related to the projected generalized continuous-time
algebraic Lyapunov equations (GCALEs)
? A @B C? @C 3 DE FG@ D E @ @ ? @ C? 3D H @ @ D H (1.2)
(1.3)
?
?
DH?CD H @ D E @ ?CD E
and the projected generalized discrete-time algebraic Lyapunov equations (GDALEs)
(1.4)
(1.5)
2 L3 where
DE
DH
? @ 13 <?C @ *I E G @ I E@ @ ? 3K @ ?C *I H@ @ <I H ? JI H ? I H@ ? JI E@ ? I E I E NM 3/D E
I H OM 3/D H
and
are the spectral projectors onto the left and right deflating subspaces of
corresponding to the finite eigenvalues,
and
are the
spectral projectors onto the left and right deflating subspaces corresponding to the eigenvalue
at infinity. Let
be in Weierstrass canonical form
2 3 >
T >
QPSR MU)
(1.6)
> VJW X and J*PSRZY> M %([ T W\X P and X are nonsingular, Y corresponds to the finite eigenvalues of
where
2 ]3 theandmatrices
V being nilpotent
corresponds to the eigenvalue at infinity. The index ^ of
_
Received March 10, 2006. Accepted for publication March 21, 2008. Published online on August 4, 2008.
Recommended by P. Van Dooren. This work was supported by the DFG Research Center M ATHEON ”Mathematics
for key technologies” in Berlin.
Institut für Mathematik, MA 4-5, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
(stykel@math.tu-berlin.de).
187
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T. STYKEL
V
nilpotency of is the index of the pencil
be represented as
2 3 T >
D E JP R M )
> > W P [a`
(1.7)
. Using (1.6) the projectors
and
2 d3 D E and D H
can
T >
D H X [` R M b
> > W Xc
It has been shown in [34] that if the pencil
is stable, i.e., all its finite eigenvalues
have negative real part, then the projected GCALEs (1.2) and (1.3) have the unique symmetric, positive semidefinite solutions which define the proper controllability and observability
Gramians of the descriptor system (1.1). Furthermore, the projected GDALEs (1.4) and (1.5)
have the unique symmetric, positive semidefinite solutions which are the improper controllability and observability Gramians of (1.1). The Gramians play a central role in analysis
and control design problems for descriptor systems, such as the characterization of controllability and observability properties, computing
or Hankel norm, minimal and balanced
realizations as well as balanced truncation model order reduction [4, 34, 35, 36].
has been the topic
The numerical solution of standard Lyapunov equations with
of numerous publications [2, 3, 12, 13, 14, 17, 18, 22, 24, 29]. A variety of direct and iterative methods has been proposed there for computing the solutions of such equations, their
Cholesky factors or low-rank approximations. The case of nonsingular has been considered in [5, 9, 15, 16, 21]. Until now, only direct methods have been extended to projected
Lyapunov equations [33]. The solutions of (1.2) – (1.5) can be computed by the generalized
Schur-Bartels-Stewart or the generalized Schur-Hammarling methods that are based on the
preliminary reduction of the pencil
to the generalized Schur form, solution of the
generalized Sylvester and Lyapunov equations and back transformation. Since these methods cost
operations and require
memory location, they can be used only for
problems of small or medium size.
Due to the practical importance of the numerical solution of large-scale projected generalized Lyapunov equations that occur in balanced truncation model reduction of descriptor
systems [35], the development of iterative methods for such equations is a challenging problem. In this paper we generalize the alternating direction implicit (ADI) method [17, 18, 22]
and the Smith method [22, 29] to the projected generalized Lyapunov equations (1.2) and
(1.4) with large sparse matrix coefficients. The dual equations (1.3) and (1.5) can be solved
in a similar way. Low-rank versions of the ADI and Smith methods are also presented, which
can be used to compute low-rank approximations to the solutions of (1.2) and (1.4) with
a low-rank right-hand side. Such a problem arises, for example, in model reduction. Note
that the number
of columns of the matrix in (1.2) and (1.4) relates to the number of
inputs of the underlying descriptor system (1.1) and it is usually small compared to the state
space dimension of the problem.
A major difficulty in the numerical solution of projected Lyapunov equations is that we
need to compute the spectral projectors ,
or ,
. Fortunately, in many applications
such as computational fluid dynamics and constrained structural mechanics, the matrices
and have some special block structure. As the following examples show, this structure can
be used to construct the projectors and
in explicit form.
E XAMPLE 1.1. Consider the descriptor system (1.1) with
egf
4M
2 h3 i j f
i jakl
m
j
DE DH IE IH
(1.8)
DE
`o`
DH
R > `n` > ` f W Q R n` ` ` f W f ` of f
where
is nonsingular. Such a system arises, for example, after linearization and spatial
discretization of the Euler equation that describes the flow of a fluid through a supersonic
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ITERATIVE METHODS FOR PROJECTED LYAPUNOV EQUATIONS
189
f ` D `na[ `E ` ` f D 3 H fnf is nonsingular, then the pencil 2 p3 in (1.8)
and
are given by
D E )R M> q ` f 3 `n` `n[a` ` ` f > f ` `n[a` ` ` f 3 fnf ![` W D H R M 3 `n[a ` ` ` f q[ ` f ` `n[a3` ` ` f 3 [ ` fnf r[a`! f ` 3 Ms`n[a` g` q ` f f [a` ` `o[` ` 3 ` f 3 [afnf ` ![ `! fof W c
f ` `n` ` f fof f `
f ` `n` ` f fof fnf
E
1.2. Consider the descriptor system (1.1), where and have the form
R > `n` >> W Q R `n` > ` f W c
(1.9)
f`
Such systems arise in spatial discretization of the instationary incompressible Stokes equation
[`
[6, 7] and
the convection equation [19]. If `n` and f ` `n` ` f are nonsingular,
then the
2
t
3
in (1.9) is of index 2. In this case the spectral projectors D E and D H have
the
pencil
following form
D E uRwv > E 3 v E `o` `n[` ` ` f > q f ` `n[a` ` ` f r[a` W D H 4R 3 q [` r[v aH` [` H >> W f ` `o` ` f f ` `o` `n` v
E )M 3 ` f f ` `n[` ` ` f ![` f ` `n[a` ` is a projector onto the kernel of f ` `n[` `
where v
H /M 31 `n[a` ` ` f f ` `n[` ` ` f r[`x f ` `n[` ` v E `o` .
along the image of ` f and v
diffuser [40]. If the matrix
is of index 1, and the projectors
XAMPLE
E XAMPLE 1.3. The motion of multibody systems with holonomic constraints can be
described by nonlinear differential-algebraic equations of the first order [8, 28]. Linearization
of these equations around an equilibrium state leads to the descriptor system (1.1) with
> ‚
M > >
M 3ƒ >
€
}
>
|
>
zy
{
Q
}y
{
(1.10)
ƒ > >}@ ~ > > >~
€

ƒ
where | is a nonsingular mass matrix, is a stiffness matrix, 2 is a damping matrix and
ƒ
4
3
is of index „ , and
is a matrix of constraints.
If H has full row rank, then the pencil
E
D
D
the spectral projectors and
can be computed as
3
Bƒ
D E y{ 3 v @  v M 3 v v > @ 3 v @ € v |  [v ` | [a` `  ƒ ` ~
>
>
>
>
>
3 v | [` v  M 3 v
D H y{
> c
ƒ @` € v 31 v | [a`  qM 3 v
ƒ @` v  v
>~
ƒ | [` ƒ @ ƒ | [a` ƒ @ r[a` and v QM 3 | [` ƒ @ ƒ | [a` ƒ @ r[` ƒ *M 31ƒ ` ƒ is
Here `
ƒ
ƒ
a projector onto the kernel of along the image of | [a` E @ ,H see E [28] forH details.
D
D
In the following we will assume that the projectors , , I and I are given. Clearly,
we do not compute these projectors explicitly. Instead, we use matrix-vector multiplication
and linear system solvers if an inverse is required. In Section 2, we present a generalization
of the ADI method and its low-rank version for the projected GCALE (1.2). In Section 3,
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T. STYKEL
we discuss the numerical solution of the projected Lyapunov equations (1.2) and (1.4) via the
(cyclic) Smith method. Section 4 contains some results of numerical experiments.
Throughout the paper the open left half-plane is denoted by
. We will denote by
and
the spaces of
real and complex matrices, respectively. The real
part of a complex number is denoted by Re . The matrix
stands for the transpose of
,
denotes the complex conjugate and transpose of
, and
. An identity matrix of order is denoted by or simply . We will denote
by
the spectral matrix norm and by
the Frobenius matrix norm of
.
$%(' +
%7' +
ˆ
" $ %(' + <[ @ 9q<[a`U @
‰ ‰f
‡
jJ† m
‡
j
‰ ‹‰ Š
[
@
M%
"
M
%(' +
" %(' +
2. Alternating direction implicit method. The ADI method was originally proposed
for linear systems [20] and then used in [17, 18, 22, 38] to solve standard continuous-time
Lyapunov equations. The case of nonsingular has been considered in [16]. In this section,
we present a generalization of the ADI method for the projected GCALE (1.2).
Assume that the pencil
is stable. Then the matrix is nonsingular, and the
projected GCALE (1.2) is equivalent to the projected standard Lyapunov equation
2 Œ3 (2.1)
[a` ? ? q [a` @ 3DH [` GF@ [ @ D H @ ?
DH?CD H @ c
In this case an approximate solution of (1.2) can be computed by the ADI method applied to
(2.1). The ADI iteration is given by
?.Ž tq<[a` p Ž M(r[a`- [` 3  Ž M ?.Ž [` <[a` 3  Ž M( @ [`  Ž M(r[ @
(2.2)
3A‘ Re  Ž [` ’ Ž M [` DH [` F @ [ @ D H @ [`  Ž M( [ @
?.“ > and the shift parameters  ` cUc‹c o Ž " [ . It follows from
with an initial matrix
D H <[a` 3  Ž M(”9q<[` 3  Ž M D H D H [` ’ Ž M![`•t <[a` p Ž M(r[a` D H
?–Ž D H ?–Ž-D H @ , i.e., the second equation in (1.2) and (2.1) is satisfied exactly. The
that
iteration (2.2) can also be written as
(2.3)
? Ž h ’  Ž :![`Z 3  Ž : ? Ž [` 3  Ž : @
3A‘ Re  Ž p Ž r[a` D E F @ D E @ 2 t3 ?
 Ž :r[ @
 Ž :![ @ c
 ` cUc‹c  Ž " [
The following propositions give the convergence results for the ADI iteration.
P ROPOSITION 2.1. If the pencil
is stable and
, then the ADI
iteration (2.3) converges to the solution of the projected GCALE (1.2).
Proof. Let be a solution of the projected GCALE (1.2). The error matrix
can
be computed from (2.3) recursively as
?
(2.4)
where
(2.5)
?L3—? Ž
?˜3?BŽ Q™ Ž ? ™ Ž ™ Ž D H p Ž [` 3  Ž š cUc‹c š› ’  ` : [` 3  ` c
Using the Weierstrass canonical form (1.6) and representation (1.7), we find that
(2.6)
™ Ž X [` R Y > Ž >> W X
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ITERATIVE METHODS FOR PROJECTED LYAPUNOV EQUATIONS
191
2 h3
and
3  ` (Y Ÿ Y . ,If t¡› Uc c‹c nis¢ ,stable
lie
inside
?
the unit circle, and, hence,
converges toward the solution .
2 £3 P
2.2. Consider the projected GCALE (1.2). Assume that the pencil
?
Ž
¢
is in Weierstrass canonical form (1.6), where Y is diagonal. Then the -th iterate
of the
Ž N MGœ Ž Y r[`Z M 3  Ž Y š cUc‹c šž MGœ ` Y r[a`-qM 3
Y
 ` cUc‹c  Ž " [ , then all the?BŽ eigenvalues of MŸ Y r[`Z M
with
ROPOSITION
ADI method satisfies the estimate
‰ ?¤3K?–Ž ‰Uf:¥/¦ f X ¨§ f ™ Ž ‰ ? ‰f where ¦ X © ‰ X ‰ f ‰ X [a` ‰ f is the spectral condition number of the right transformation
Ž is the spectral radius of the matrix ™ Ž given in (2.5).
matrix X in (1.6) and §ª ™
(2.7)
Proof. Estimate (2.7) immediately follows from (2.4) and (2.6).
™ Ž
2.1. Computing the shift parameters. As Proposition 2.2 shows, the convergence rate
of the ADI iteration is determined by the spectral radius of the matrix
as in (2.5) and
depends strongly on the choice of shift parameters. The minimization of this spectral radius
with respect to the parameters
leads to the generalized ADI minimax problem
 ` c‹cUc o Ž
¡ 3  ` š c‹cUc šÇ¡ 3  Ž (
Ž
¬

o

¯
­
n
®
²
°
0
±
µ
³
´
±
.
¯
­
¾
« ` ‹c cUc
(2.8)
¶o· '¹¸¹¸¹¸ ' ¶ º‹»¯¼-½ ¿ » wÀ Áà' ÄÅ&Æ Ç¡\’ ` š c‹cUc šÇ¡p Ž Æ
Æ
2 p3 . TheÆ computation
where SpÈ o: denotes the set of finite eigenvalues of the pencil
of the optimal shift parameters is a difficult problem, since the finite eigenvalues of the pencil
2 L3 are, in general, unknown and expensive to compute. This problem is solved for
ÉM and symmetric , e.g., [39], while the case of
standard Lyapunov equations with
complex eigenvalues of is still not completely understood; see [18, 27, 32, 39] for some
Sp
contributions. To compute the suboptimal ADI shift parameters for the standard problem,
a heuristic algorithm has been proposed in [22]. This algorithm is based on Arnoldi iterations
[25] applied to the matrices and
. It can also be extended to the generalized problem
(2.8). Due to the nonsingularity of this problem is equivalent to
F[`
3  ` &š cUc‹c š› 3  Ž Ž
¬

o

¯
­
n
®
&
°
0
±
Ã
³
´
±
.
­
¾
“
Ì
(2.9)
« ` Uc c‹c
¶ · '¹¸¹¸¹¸ ' ¶ º »¯¼ ½ ¿ » Á Ä ½ ·  qÅ ÊrË Æ p ` &š cUc‹c š› ap Ž Æ
Æ
denotes the spectrum of the matrix F[Æ ` . Thus, the suboptimal
where Sp q[`
ADI shift
 ` cUc‹c  Ž can be determined by the heuristic procedure
parameters
[22, Algorithm 5.1] from a set of largest and smallest (in modulus) non-zero approximate
eigenvalues of [` . A conventional approach for computing largest and smallest eigenvalues of a matrix is to apply an Arnoldi process to this matrix and its inverse, respectively.
However, if is singular, then the inverse of F[` does not exist. Observe that the recip
rocals of the smallest non-zero eigenvalues of G[` are the largest finite eigenvalues of the
2
4
3
pencil
D , where . The latter can be determined by an Arnoldi procedure applied to the matrix
T > P [` D D H D H 3 I H [` t D E 3 I E : [` D E X [` R M >)
>W
and X , P are
the transformation
as in (1.6), see [30] for details. Similarly to the
D H and
D E , the matrix D matrices
projectors
can be obtained in explicit form using the special block
structure of the matrices and .
Sp
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T. STYKEL
2.2. Low-rank version of the generalized ADI method. Recently, an efficient modification of the ADI method has been proposed to compute low-rank approximations to the
solutions of standard Lyapunov equations with large-scale matrix coefficients [17, 22]. This
is the low-rank alternating direction implicit (LR-ADI) method. It was observed that the
eigenvalues of the symmetric solutions of Lyapunov equations with low-rank right-hand side
generally decay very rapidly, and such solutions may be well approximated by low-rank matrices [1, 10, 23, 31]. A similar result holds for projected generalized Lyapunov equations. In
other words, it is possible to find a matrix with a small number of columns such that
is an approximate solution of the projected GCALE (1.2). The matrix is referred to as the
low-rank Cholesky factor of the solution of (1.2).
A low-rank version of the generalized ADI iteration (2.3) can be derived analogously to
the standard case [17, 22]. First of all note that the matrix
in (2.3) is Hermitian, positive
semidefinite, and the Cholesky factor
of
has the form
?
Í
Í
Í:Í @
?BŽ
ŽÍ ?–Ž Í Ž Í Ž
p Ž r[`Z 3  Ž Í Ž [`sÐ
Í Ž hÎUÏ 3A‘ ReE  Ž U p Ž :![` D E Bˆ
hÎ Ñ Ž¯ÒaŽ-D –ÓÑ Ž [` ÒaŽ-ÔՎ¯ÒaŽ [` D E B cUc‹c ÓÑ ` ÒaŽlÔՎ šUš‹š Ô f Ò ` D E Ð 3A‘ Re rŸl , Ò Ÿ:4 KrŸwr[` and Ô ŸA Q3 Ÿ‹ . Taking into account that
where ÑaŸ: Ï
Ò Ž Ò Ÿ: Ò Ÿ‹ Ò Ž Ô Ž [` Ô Ÿ Ô Ÿ‹ [a` Ô Ž Ò Ž Ô Ÿ:Q [` Ô Ÿ Ò Ž ‘
Ž
for ¢ , 4¡Z c‹cUc , the matrix Í can be rewritten as
(2.10)
Í Ž tÎ “ nÖ Ž [` “ £Ö Ž [ f Ö Ž [` “ cUc‹c £Ö ` Ö f šUšUšnÖ Ž [` “ Ð “ 4Ï 3A‘ Re  Ž ‹ ’ Ž :![` D E ttÏ 3A‘ Re  Ž D H p Ž :![`U and
where Ö Ÿ t× Re r Ÿo Ø Ÿ Ò Ÿ Ô ŸoØ ` 9× Re ! Ÿo Ø Ÿ 1Ù M 3  Ÿ  ŸoØ ` U p Ÿ : [` :Ú c
Re
Re
`
`
If we reenumerate the shift parameters in reverse order, then we obtain the following algorithm for computing the low-rank Cholesky factor of the solution of (1.2).
A LGORITHM 2.1. The generalized LR-ADI method for the projected GCALE.
D E "g$%(' % Û"£$%7' +
 ` ?O
cUc‹à c o ŽrÜÞÝqß " [ .
ŽÍ
Ž Í Ž of (1.2).
Í
E
A
3
‘
D
Í Á ` Å 9Ï¢. ‘  ` p ` [a` Í ` Í Á ` Å
„ c‹cUc
ŽÁ Å t× Re  Ž M 3  Ž p Ž U p Ž : [` Ú Á Ž [` Å Í
Ž
Í
(2.11)
[a`
Re  [` <Ù
Ž
Í Ž 9Î Í Ž [a` Í Á Å Ð c
I NPUT: , ,
,
, shift parameters
of the solution
O UTPUT: A low-rank Cholesky factor
1.
Re
,
.
2. FOR
END FOR
The ADI iteration can be stopped as soon as a normalized residual norm given by
(2.12)
á Í Ž ” ‰ Í Ž Í Ž @ D E Í FŽ Í @Ž D E @ @ D E F @ D E @ ‰ Š
‰
‰Š
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ITERATIVE METHODS FOR PROJECTED LYAPUNOV EQUATIONS
á Í Ž ¥ Çâ-ã
Çâ-ã
193
with a user-defined tolerance
or a stagnation of norsatisfies the condition
malized residual norms is observed. If the number of shift parameters is smaller than the
number of iterations required to attain a prescribed tolerance, then we reuse these parameters
in a cyclic manner. Note that computing the normalized residuals
even via the efficient
methods proposed in [22, 27] can still be quite expensive for large-scale problems. It should
also be noted that for ill-conditioned problems, the small residual norm does not imply that
the error in the computed solution is also small; see [33].
If
converges to the solution of (1.2), then
á Í Ž
?–Ž Í Ž Í Ž
Ž Ž
Ž ä å³Ã± æ Í Á Å Í Á Å Žä å³µ± æ ?–Ž:3?–Ž [a` ² > c
Therefore, just as in the standard
Ž case [17], theŽ stoppingŽ criterion in Algorithm 2.1 can also
be based on the condition ‰wÍ Á Å ‰:¥ Çâ-ã or ‰‹Í Á Å ‰w瞉‹Í ‰¥ Çâ-ã with some matrix norm ‰ š ‰ .
Ž
R
2.3. The matrices J r[a` in Algorithm 2.1 do not have Eqto
be com Ž
D è either
puted explicitly. Instead, we solve linear systems of the form œ ¨p
by
EMARK
computing (sparse) LU factorizations and forward/backward substitutions or by using iterative Krylov subspace methods [26]. In the latter case the generalized LR-ADI method has the
memory complexity
and costs
flops, where
is the number
of outer ADI iterations and
is the number of inner linear solver iterations. This method becomes efficient for large-scale sparse Lyapunov equations only if
is much smaller
than .
R EMARK 2.4. In exact arithmetic the matrices
satisfy
and, hence, the
second equation in (1.2) is fulfilled for the low-rank approximation
. However, in finite
precision arithmetic a drift-off effect may occur. In this case we need to project
onto the
image of
by pre-multiplication with . In order to limit the additional computation cost
we can do this, for example, at every second or third iteration step.
R EMARK 2.5. If is nonsingular, but ill-conditioned with respect to inversion, then
Algorithm 2.1 may provide a better result than the classical LR-ADI method [16, 17, 22]
applied to the matrices
and
.
Note that if at least one of the shift parameters is complex, then the low-rank Cholesky
factor
may be complex although the solution of (1.2) is real. As in the standard case
[17, 22], in order to keep the real factors, we can take the complex shift parameters in complex
conjugate pairs
with
and compute the iterates
as follows
i é ¢ Ä ê²Z¢ ë ì ím j
j
i q ¢Zì ír¢ Ä ê²ë m j
͎
DH
͎
(2.13)
DH
[`
«Ž ŽØ `¬
Í Ž 9Î Í
[`U
ŽØ `
Ž [a` Ž
ÁÍ Ž Å Ð
¢ Äê²ë
¢(ì ín¢ Äê²ë m
Í Ž Ž D Ž@ H Í Ž
Í Í
Ž
Í ÁÅ
?
if
Ž
͎
is real
ŽÍ 4Î Í Ž [a` Í ` Á ŽŽ Å Ð
Ž
(2.14)
î Í Ž Ø ` 4Î Í Ž Í f Á Å Ð if ï  Ž Ø is` complex
Ž c
Ž Å 4ð ‘  Ž  Ž Ø r[`U Á Ž Å Á Ž Å tð ‘  Ž Ø r[` Á Ž Å
Ž
Á
Í ,Í f
Í and Í Á Å is
Here Í `
`
`
as in (2.11). One Æ canÆ show that (2.13) and the double step (2.14) provide the real Cholesky
factors of the solution of the projected GCALE (1.2).
2 h3 is stable
P
2.6. Let , ñ"$&%(' % and u"$%(' + . Assume that
Ž Ž  Ž¬ .
and that the complex shift parameters
appear in complex conjugate pairs «   Ø `
Ž
Ž
Then the sequence of the matrices Í in (2.13), (2.14) is real. Furthermore, for complex  ,
we have
Ž Ž
Ž Ž
Í Ž Ø ` Í Ž @ Ø ` tÎ Í Ž [` Í Á Å Í Á Ø ` Å Ð Î Í Ž [a` Í Á Å Í Á Ø `ÇÅ Ð c
and
ROPOSITION
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T. STYKEL
¢ `
Proof. We prove the first part of this proposition by induction on . If
is real, then
Í ` Í Á ` Å ð 3A‘  ` p  ` [a` D E  ` , then we have
is real. If  ` is complex and  f
Á ` Å ‘ Ï 3 Re  `  `  ` :![`F  ` :![` D E B
Í
Í
`
`
(2.15)
Í&f hÎ Í ` ‘ Ï 3 Re  ` Æ Æ  ` :![` p ` :r[a` D E Ð c
3 ’ž [a` and ž: [` ž [`
Next we show that the matrices Þ: [` F
2
t
is stable, the matrix is nonsingular.
are real for any complex  . Since the pencil
Therefore,
ò:r[a`G pž:![` Ù ’žÇ<[a`¯ ò: Ú [`
h [` ‘ Re Þ  f r[` c
Æ Æ
Clearly, the inverse of a real matrix is also real. Furthermore, we have that
ž [` ’ž [` * [` Þ: [` F pž: [`
is real. Thus, Í ` and Í f in (2.15) are both real.
Ž
Assume now that the matrix Í is real, where ¢ is the index of a real parameter or of the
Ž Ž  Ž [a` ¬ . Then the matrix
second element in a complex conjugate pair «  [` 
Ž
Ž
Í ó Á Ŕh 3  Ž : Í Á Å
hÏ 3A‘ Re  Ž ‹ 3  Ž p Ž r[a`\š c‹cUc š 3  ` p ` :![` D E Ž
is real. For real  Ø ` , we obtain that
Ž
Ž
 Ž Ø
Í Á Ø `ÇÅ × ReRe  Ž ` p Ž Ø ` : [` Í ó Á Å
Ž
Ž ŽØ
Ž
is real, and, hence, Í Ø ` 4Î Í Í Á ` Å Ð is also real. If  Ø ` is complex, then
ÁÍ ` Ž Ø ` Å  Ž Ø ` × ‘ Re   Ž Ž Ø `  Ž Ø ` : [` F ’ Ž Ø ` [a` Í ó Á Ž Å ŽÁ Ø ` Å ×Æ ‘ ReÆ  Ž Ø Re`  Ž Ø : [a` ’ Ž Ø : [a` ó Á Ž Å
Íf
Ž
`
` Í
Re 
Ž
Ž Á Ž Ø ` ŽÅ Ø Ð and Í Ž Ø f 4Î Í Ž Ø ` Í f Á Ž Ø `ÇÅ Ð are also real.
are real. Therefore, Í Ø ` tÎ Í Í `
3’‘  Ž Ø ` ’ Ž Ø ` :r[a`! Ú Í Á Ž Å and
Ž
Further, for complex  , we have Í Á ` Ų Ù M
ŽÍ Ø ` Í Ž@ Ø ` Í Ž [a` Í Ž@ [` Í ` Á Ž Å Ù Í ` Á Ž Å Ú @ Í f Á Ž Å Ù Í f Á Ž Å Ú @
Ž Ž
Í Ž [a` Í Ž@ [` ‘ Í Á År Í Á Åo Ž Ž
3A‘  Ž Ø ` G Ž Ø ` :![`Ã Í Á Å! Í Á Å 3F‘  Ž Ø ` Í Á Ž Å Í Á Ž ÅÇ @  Ž Ø ` r[ @
:ô  Ž Ø ` f G Ž Ž Ø ` Ž [` Í Á Ž Å Ž Í Á Ž Å Ž @  Ž Ø ` [ @
Í Ž [aÆ ` Í Ž@ [Æ ` Ž Í Á Å!Ž ÍØ Á Åo Í Á Ø ` Å!Ž Í Á Ø Ž` ØÅ
dÎ Í Ž [` Í Á År Í Á ` Å Ð Î Í Ž [a` Í Á År Í Á `ÇÅ Ð c
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Observe that (2.13) and (2.14) involve, in general, complex matrix operations. Complex
arithmetic can be avoided if we rewrite (2.13) and (2.14) as
ÁÍ Ž ŝ × Re  Ž Ž ’ Ž : [` Í ó Á Ž [` Å Re [`
ŽÍ dÎ Í Ž [` Í Á Ž Å Ð Í ó Á Ž Å&4 3  Ž Í Á Ž Å
 Ž and
õÕù öµ÷!ø hú û ÷ úxü ý Reû þ û ÷-ÿ þ ù ý Re þ û ÷ ÿ ú û ÷ ú ÿ ù õ öµ÷ ù ø Re þ ÷ ù ÿ
õÕ öµ÷!ø ü ý Reû þ û ÷¯ÿ ù þ ù ý Re þ û ÷ ÿ ú û ÷ ú ÿ ù õ öµ÷ ù ø ú û ú ù FõÕù öµ÷!ø ÷
Re þ ÷ ù ÿ
ŽÍ Ø ` dÎ Í Ž [` Í ` Á Ž Å Í f Á Ž Å Ð õ öµ÷ ù ø Re ! #"$ Re %&('*) +) ,-!./ ! ('0$ Re 123'*) +) ,4!. 6795 8 &:
for real
Re for a complex conjugate pair ; < =?>@<=ACBED <F=G . A drawback of this procedure is that the
B
inverse of HIJ HLKNM Re O <=%P2HQKQR <=+R SI is required. Solving linear systems with this matrix
is usually more expensive than with HTKU<V=1I .
3. Smith method. For any parameter <#W#X J , the projected GCALE (1.2) is equivalent
to the projected discrete-time Lyapunov equation
(3.1)
Y ZNY0[]\EZ^D_\a`bdc ca[e` g
fb >
ZhD
`ib/ZN` 0
fb >
where
B
YjDjO HTKQ<+I.PJ
crDjs
O Hk\
<lI.PmD
\tM Re O <lPuO HKQ<+I.PJ
n \UM Re O <lP/OoHpKq<+I.PJ
Bv3w
B
I>
Note that if the pencil xlHy\qI is stable, then ` b is the spectral projector onto the invariant
subspace of the matrix Y corresponding to the eigenvalues inside the unit circle. In this case
the Smith iteration
(3.2)
Z{z.D` b cmc [ ` fb >
Z = DT` b c c [ ` fb
KqY Z =
J
B Y [
can be used to compute an approximate solution of (3.1); see [29]. The number of iterations required for a desired accuracy in Z = depends on the parameter < . Note that the
Smith method (3.2) is, in fact, the generalized ADI iteration (2.3) with a single parameter
www
<(D|< BqD
D}<= .
A modification of the Smith method has been proposed in [22] for computing a low-rank
Cholesky factor of the solution of standard Lyapunov equations with a low-rank right-hand
side. This version of the Smith method is based on the LR-ADI iteration with ~ shift parameters applied in a cyclic manner and referred to as the low-rank cyclic Smith (LR-Smith O~P )
method. It can be generalized for the projected GCALE (1.2) as follows: first one determines
€i
www 
using the generalized LR-ADI method with the shift parameters < B >
>2< and then solve
the discrete-time Lyapunov equation
Y

€  € 

ZNY [ \EZ‚Dƒ\
[ >
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™
¢.
T. STYKEL
. In summary, we have the following algorithm to compute
where „ is as in (2.5) with
the low-rank Cholesky factor of the solution of the projected GCALE (1.2).
‹ method for the projected GCALE.
D E D H "£$%(' % , Û"£$%(' + , shift parameters  ` c‹cUc o „ " [ .
I
: , , Ž
?Oà Í Ž „ Í Ž „ of (1.2).
O
: A low rank Cholesky factor Í „ of the„ solution
1. Compute Í „ using Algorithm 2.1 and set Í Á `r' Å Í „ .
‘ „ Ž cUc‹c „
2. FOR Ž ¢.“
Í Á ' Å ¡Í ›Á [a`r'& Å ,
FOR
ŽÁ ' Ÿ Å cUc‹c M 3’‘ rŸwU prŸ‹ [` :Ú Á Ž ' Ÿ [` Å Í
Í
(3.3)
Ù Re
END FOR
Ž
Í Ž „ tÎ Í Á Ž [` Å „ D H Í Á ' „ Å Ð .
A LGORITHM 3.1.The generalized LR-Smith 
NPUT
UTPUT
END FOR
Í
͎
Note that if „ is real and the complex shift parameters appear in complex conjugate
pairs, then the matrices „ are also real.
R EMARK 3.1. The generalized LR-Smith( ) method is equivalent to the generalized LRADI method with cyclically repeated shift parameters.
However, comparing (2.11) and (3.3),
„
where
and
, respectively, one can see that Algorithm 2.1 is
more efficient than Algorithm 3.1.
R EMARK 3.2. At every iteration step in the generalized LR-ADI and LR-Smith methods the number of columns of the approximate solution factors
and „ grows by and
, respectively. To keep the low-rank structure in the Cholesky factors in case of large
and/or slow convergence, we can replace the iterate by its low-rank approximation computed
via the updated singular value decomposition; see [11] for details.
Consider now the projected GDALE (1.4). If the matrix is nonsingular, then equation
(1.4) is equivalent to the projected discrete-time Lyapunov equation
Ž
Í Á [` Ŕ" %(' +
Ž
Í Á ' Ÿ [ ` ŝ" %(' +
͎
m
͎
qãq
m
m
?˜3 q [a` ? q [` @ QI H [` F @ [ @ I @H ? JI H!? I @H c
H is the spectral projector onto the invariant subspace of the matrix G[a` corNote that I
H t<[a` I H is nilpotent with the
responding to the zero eigenvalues. In this case I [`
2 3 . The unique solution of
index of nilpotency ^ that is equal to the index of the pencil
(3.4)
the projected Lyapunov equation (3.4) is given by
a[ ` q [` Ž I H [` F @ [ @ I
Ž “
? ‰‰ @
Thus, the Cholesky factor ‰ of the solution
GDALE (1.4) has the form
H [` –² [` I H [` B cUc‹c ‰ 4ÎoI ?
T†
‡
ˆ
H@ q [` @ Ž c
of (3.4) and also of the projected
[` † [` I H [ ` Ð c
It can be computed by the following algorithm that is a generalization of the Smith method
for the projected GDALE (1.4).
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A LGORITHM 3.2. The generalized Smith method for the projected GDALE.
I H "£$%(' % Û"£$%(' +
? ‰ ‰@
H
Á ` ÅQI ‘ [`! ` Á `ÇÅ
† †
¢0 Ž „ cUc‹c ^ Ž
Ž
Á Ų/[` Á [a`ÇÅ Ž 9Î Ž [` ‰ Á Å Ð .
I NPUT: , ,
and
.
O UTPUT: A Cholesky factor ‰ of the solution
1. ‰
, ‰
;
‰†
2. FOR
, ‰
‰
‰
‰
END FOR
^
of (1.4).
ŽÁ Å 2 Û3Çâ-ã Á Ž Å Ž Çâ-ã
‰ ‰ ¥
‰ w‰ 瞉 ‰G¥
IH
ÁŽ Å
Note that if the index of the pencil
is unknown, then the iteration in Algorithm 3.2 can be stopped as soon as /‰
or /‰
‰
with some matrix
norm
and a tolerance . If we want to compute the solution of (1.4) as accurate as
possible, we should set
to the machine precision. To avoid the drift-off of the columns
of ‰ from the image of
, the matrices ‰
should be pre-multiplied with
after some
†
iteration
steps.
In the case of singular , we can consider the Lyapunov equation
‰š‰
Çâ-ã H
I
Çâ-ã
? 3 I < ?C I@ QI<GF@I<@
ñ
H D H I H ![`¤ D E I E :![`‹I E . This equation has the same solution
with ILhI as the projected GDALE (1.4), see [41]. Therefore, the Cholesky factor ‰ of the solution
? ‰ ‰ @ of (1.4) can also be computed by applying the Smith method to† equation (3.5).
H and I E , the matrix I can be constructed in explicit form using
† to
† the projectors I
Similarly
the special block structure of and .
(3.5)
4. Numerical examples. In this section we present some results of numerical experiments. Computations were done on IBM RS 6000 44P Modell 270 with machine precision
Š
2‹
using MATLAB 7. In all examples the suboptimal ADI shift parameters
were computed as described in Section 2.1.
E XAMPLE 4.1. Consider the 2D instationary Stokes equation that describes the flow of
an incompressible fluid in a domain. The spatial discretization of this equation by the finite
difference method on a uniform staggered grid leads to the descriptor system (1.1) with the
matrices and as in (1.9), where
,
and
. In our experiments
the state space dimension of the problem is
, and the matrix
is chosen
at random. Note that and are symmetric and is positive semidefinite. In this case the
finite eigenvalues of
are real.
In
Figure
4.1
we
present
the normalized residual norm
as in (2.12) and the ratio
Œ
for the generalized LR-ADI method with
real shift parameters. The spectral radius of the matrix
as in (2.5) is
. One can see that
the generalized LR-ADI method converges fast, and the solution of the projected GCALE
(1.2) can be approximated quite accurately by a matrix of rank 30. The normalized residual
normŒ
stagnates on a relatively small level, which is caused by round-off errors. Note
that
does not decrease monotonically and more iteration steps are required to achieve
Œ
than
.
Figure 4.2 shows the convergence history of the normalized residual norm for the generalized LR-ADI method and the generalized LR-Smith(10) method versus the iteration number. As expected, both methods give similar results.
Furthermore, we computed the full rank Cholesky factor ‰
of the solution
of the projected GDALE (1.4) using Algorithm 3.2. The Frobenius norms of the
‰ ‰
àh‘ c ‘›‘ †p¡ > [`
2 3 Í Ž ² ‹‰ Í Á Ž Å ‰UŠ”瞉‹Í Ž ‰Š
?
á ͎ Ž
Í Ž Í ¥ Çâ-ã
f f@
`n` *M n` ` Q @`n`
j’L¡Z ¡ > ô >
™ `“
f ` Q @` f
"1$ %7' `
ñ
á Í Ž A¡ >
§sq™ ` “ : > c > ¡
á Í Ž ¥ Çâ-ã
af "˜$%(' f
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T. STYKEL
Normalized residual norms and relative updates
PSfrag replacements
ºº
Generalized LR−ADI iterations, l=10
0
10
Ž  o‘
’ Ž ‘
−2
10
−4
10
−6
10
−8
10
−10
10
0
10
20
30
40
50
60
F IG . 4.1. Example 4.1: convergence history for the generalized LR-ADI method.
Generalized LR−ADI and LR−Smith(10) iterations
0
10
LR−ADI, l=10
LR−Smith(10)
−2
Normalized residual norm
10
Iteration k
−4
10
−6
10
−8
10
−10
10
−12
10
0
5
10
15
20
25
Iteration k
30
35
40
F IG . 4.2. Example 4.1: normalized residuals for the generalized LR-ADI and LR-Smith(10) methods.
‰ Á ` Å ‰ Š „ c 2 ôՆB3 ¡ > • , ‰/‰ Á f Å ‰ Š 9¡‘ c ‘%–— †©¡ > f
is of index .
update matrices are ‰
“?”
is not surprising because the pencil
‰ Ák Å ‰ Š ¥
and /‰
Š
. This
E XAMPLE 4.2. Consider the descriptor system (1.1) with non-symmetric matrices and
as in (1.9) that has been obtained by the finite element discretization of a convection problem with a boundary control, see [19] for details. The problem has the state space dimension
– –
, and the matrix
results from the boundary control.
In Figure 4.3 we present the convergence history in terms of the normalized residual
norms for the generalized LR-ADI method and the generalized LR-Smith(11) method. One
can see that the solution of the projected GCALE (1.2) can be approximated by a matrix of
rank 35. The spectral radius of
is
“ .
The solution of the projected GDALE (1.4) has been computed in factored form
with the full rank Cholesky factor ‰
. The Frobenius norms of the update
‰ ‰
–
—
Š
matrices are ‰
and ‰
.
” , /‰
”
j ‘ >
f f@
d9Î > ž f@ Ð @ "—$%(' `
™ `n` s§ q™ n` ` ² > c > ¡
ªf "9$†K%(' f¡ > [ k
‘
f
Ç
`
Å
Å
9¡
Á
Á
‰ ‰ Š c„ ‰ ‰ Š c„
?
‰ Ák Å ‰ Š ¥
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ITERATIVE METHODS FOR PROJECTED LYAPUNOV EQUATIONS
Generalized LR−ADI and LR−Smith(11) iterations
0
10
LR−ADI, l=11
LR−Smith(11)
−2
Normalized residual norm
10
−4
10
−6
10
−8
10
−10
10
−12
10
0
5
10
15
20
25
Iteration k
30
35
40
45
F IG . 4.3. Example 4.2: normalized residuals for the generalized LR-ADI and LR-Smith(11) methods.
*¡w
E XAMPLE 4.3.
Consider a damped mass-spring system with ˜ masses, see
[37, Section 3.9]. The ™ -th mass of weight ›š is connected to the ™
-st mass by a spring
and a damper with constants š and œš , respectively, and also to the ground by another spring
and damper with constants  š and Fš , respectively. Additionally, we assume that the first mass
is connected to the last one by a rigid bar and it can be influenced by a control. The vibration
of this system is described
by the descriptor system (1.1) with the matrices and as in
—
(1.10). For ˜
, we obtain a problem of the state space dimension
with
Ÿž
. The system parameters are
and
¢
¦
m
j #¡ >›>Z> ¡
Q
>›>Z>
m ` ‹c cUc m 9¡ >Z>
¢ ` cUcUc Q¢ ž [ ` Q¢0 ‘ ¦ ` cUcUc ¦ ž ¦ ôò
œ `
cUcUc œ ž [` œ „  ` c‹cUc  ž  ”(c
Í Ž and the relative updates Œ Í Ž
á
Figure 4.4 shows the normalized residual norms
‘
for the generalized LR-ADI method with :4¡ complex ADI shift parameters. We see that
Ž
the low-rank CholeskyŒ factor Í of the solution of the projected GCALE (1.2) computed with
Ž
the stopping criterion Í Ž ¥ Çâ-ã has about twice more columns than those computed with
the stopping criterion á Í
¥ Çâ-ã . Since the spectral radius §sq™ ` f > c >Z> „ is very small,
the generalized LR-ADI iteration converges very fast.
> c „ and  > ec ” that resultsŽ
Furthermore, we change the damping constants to œ
>
ª
§
™
¡
in
residual norms á Í
` f c ” “ . Figure 4.5 shows that in this case the normalized
s
§
q
™
”
decay quite slowly compared to the previous experiment with
` f > c >Z> „ . This example
Û"g$%(' `
demonstrates that the convergence rate of the generalized LR-ADI method strongly depends
on the spectral radius of the iteration matrix in (2.5).
Finally, we investigate the impact of the growing number of columns in
on
the convergence of the LR-ADI method. Assuming that the first
masses are influenced
2¡ ž
¡ ž
by controls, we obtain
, where ¡ denotes the -th column of ž .
Figure 4.6 shows the normalized
residual norms for the generalized LR-ADI method applied
—
to the problems with
, and inputs. One can see that for different the convergence
is about the same.
¤]Î Ø ` c‹cUc Ø + Ð
¡>
m ¡
‹Ÿ
m
m
" $”%(' +
Mf Ø `
5. Conclusion. In this paper we have discussed the numerical solution of large-scale
projected generalized Lyapunov equations that arise, for example, in model reduction of descriptor systems. We have presented the generalized low-rank alternating direction implicit
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T. STYKEL
Normalized residual norms and relative updates
PSfrag replacements
ºº
Generalized LR−ADI iterations, l=12
0
10
¢Ž  ‘
’ Ž  ‘
−5
10
−10
10
−15
10
0
10
20
30
40
Iteration k
50
60
70
F IG . 4.4. Example 4.3: convergence history for the generalized LR-ADI method.
Generalized LR−ADI iterations, l=12
0
Normalized residual norms
10
PSfrag replacements
d = 3, δ = 7
d = 0.3, δ = 0.7
−5
10
−10
10
−15
10
0
50
100
150
Iteration k
200
250
F IG . 4.5. Example 4.3: convergence history for the generalized LR-ADI method for different sets of damping
parameters.
method and the generalized low-rank cyclic Smith method for computing low-rank approximations to the solutions of these equations. The efficiency of these methods has been demonstrated by numerical experiments.
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ITERATIVE METHODS FOR PROJECTED LYAPUNOV EQUATIONS
Generalized LR−ADI iterations, l=12
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Normalized residual norms
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