ETNA
Electronic Transactions on Numerical Analysis.
Volume 29, pp. 31-45, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
PRECONDITIONING BLOCK TOEPLITZ MATRICES
THOMAS K. HUCKLE AND DIMITRIOS NOUTSOS
Abstract. We investigate the spectral behavior of preconditioned block Toeplitz matrices with small nonToeplitz blocks. These matrices have a quite different behavior than scalar or mulitlevel Toeplitz matrices. Based on
the connection between Toeplitz and Hankel matrices we derive some negative results on eigenvalue clustering for
ill-conditioned block Toeplitz matrices. Furthermore, we identify Block Toeplitz matrices that are easy to solve by
the preconditioned conjugate gradient method. We derive some useful inequalities that give information on the location of the spectrum of the preconditioned systems. The described analysis also gives information on preconditioning
ill-conditioned Toeplitz Schur complement matrices and Toeplitz normal equations.
Key words. Toeplitz, block Toeplitz, Schur complement, preconditioning, conjugate gradient method
AMS subject classifications. 65F10, 65F15
1. Introduction. We consider ill-conditioned symmetric positive definite (spd) block
Toeplitz (BT) matrices of the form
..
.
.
.
.
.
.
(1.1)
..
..
..
..
..
.
.
.
..
.
.
. . ! $#%'&)(***(+&,%-# , fixed and independent of & .
with small general matrices , "
/. , because for this case we can display already all
Mainly, we are interested in the case 0#
the problems and techniques that are different from the scalar case with / 13254 &6. 7#8(9.:(9;<(***
Furthermore, we assume that the family of BT matrices
is related to a generating => matrix function
,
,
213?@4 B1 A 9C D 13?@4+4F+CE DHG JIK LNM:O 1 %QP ?R4 I6 I6 LNM:O 1 P ?@4 IS
.YX periodic in the interval Z %QX)(+XR[ , which is
that may be T , TVU , TVW , or continuous
and
+C
D
A
\
1
@
?
4
(
]
#8(***N( , is T , TVU , T^W , or continuous and
equivalent
," ._X -periodictoinwrite
%Z QX)(+that
XR[ asevery
standard scalar-valued functions.
into the => block form
By a simple permutation we can transform the matrix
a C b a C E ` ..
(1.2)
...
a E C a E . C E
Yd (+e]f#g(***(
+C
D
&
&
a
/
a
c
+C
D ,"
, related to the sameYgenerating
with Toeplitz matrices
d
+C
D
2
\
1
R
?
4
A
\
1
@
?
4
h
a
c
9C
D
matrix function
. Then, each
is the scalar generating function to
.
i Received October 31, 2006. Accepted for publication November 12, 2007. Published online on December 17,
2007.
Recommended by M. Hochbruck.
Fakultät
für Informatik, Technische Universität München, Boltzmannstr. 3, D-85748 Garching, Germany
(huckle@in.tum.de).
Department of Mathematics, University of Ioannina, GR–45110, Greece (dnoutsos@uoi.gr). This research
was co-funded by the European Union - European Social Fund (ESF) & National Sources, in the framework of
the program “Pythagoras I” of the “Operational Program for Education and Initial Vocational Training” of the 3rd
Community Support Framework of the Hellenic Ministery of Education.
31
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32
T. K. HUCKLE AND D. NOUTSOS
of
Note, that in contrast to multilevel Toeplitz matrices we consider in (1.1) blocks
fixed size that will be in general of non-Toeplitz structure. Hence, the analysis derived for the
multilevel case based on trigonometric polynomials [8, 9, 15] cannot be applied in this BT
case. In fact such negative results as in [8, 9, 15] do not hold in the block case; see [10, 12]
for related positive results concerning strong clustering with block circulant preconditioners
and spectral equivalence with band block Toeplitz preconditioners, respectively. So, the BT
case is quite different from the scalar case and the multilevel case.
Block Toeplitz matrices are closely related
to Schur complements
a ina Toeplitz matrices,
e.g., Toeplitz Schur complement
a %ja
a
am
Uk l k
to BT matrix
n aRmoa lU p
. So, for q.
the given block equations can be reduced to solving Toeplitz matrices and Schur complement
systems of Toeplitz matrices. In the following we will derive theoretical and numerical results on block Toeplitz systems and also on Toeplitz Schur complements. Note, that also the
Toeplitz normal equations are a special case of a Schur complement, and therefore are related
to block Toeplitz systems.
a related to a continuous generating funcFor
a well-conditioned spd Toeplitz matrix
r
A
\
1
@
?
4
there exist various preconditioning techniques that ensure fast convergence of the
tion
preconditioned conjugate gradient (pcg) method; see [2, 7] and the references therein. Fast
Transform preconditioners based on circulant matrices or other classes
of matrix algebras
a . Here
lead to a preconditioned system with clustered eigenvalues for s
computa1 &vuxwgy 1 & the
4F4 operations
tion of the preconditioner and each step in the pcg algorithm take t
based on fast algorithms for the trigonometric transforms. Hence, if the linear system can be
&
solved
in1 a 4Fbounded
number of iterations - independent of - then the whole procedure takes
1
4
z
&
x
u
g
w
y
&
t
operations [2, 7]. This case occurs if the symbol is strictly positive or if it is
nonnegative and polynomial.
Another important class of preconditioners for ill-conditioned matrices is based on banded
Ar13?@4 has only zeros of even order then we
Toeplitz matrices. If the scalar generating function
A
can define a trigonometric polynomial { that has the same zeros of the same order as . Then,
{ is related to a band Toeplitz matrix. Furthermore, the preconditioned systemAr| 1\?R 4+} a1\ ?R4is
{ ;
well-conditioned with eigenvalues contained in an interval given by the range of
this again leads to fast convergence of the pcg method
[3,
10].
In
[12]
the
author
considers
u~w8y 1 & 4 by choosing, e.g., the partial Fourier
also band preconditioners
with bandwidth up to
2
\
1
@
?
4
213?@4 . For well-conditioned matrices
polynomial
for the given Fourier expansion of
this approach leads to clustered spectrum of the preconditioned system. For ill-conditioned
21\?@4 the banded approximation given by 2)13?@4 may be indefinite and
matrices with singular
therefore in [11, 12] a generalization of the approach in [10] is considered in order to deal
with the ill-conditioned case; see Theorems 3.1, 3.6, and sections 4.1, 4.2 in [11] and sections 3 and
[12]. Here, we introduce a different banded approximation that can deal with
2134?@4 inalso
singular
allowing a broader bandwidth.
Aim of this paper is to generalize the positive preconditioning results for circulant and
band preconditioners for Block Toeplitz matrices, e.g., [6, 11, 12, 13, 14], and to derive further
results on the eigenvalue distribution and clustering of preconditioned block Toeplitz systems
and Toeplitz Schur complements (like the normal equations). This also leads to some negative
results on the eigenvalue clustering of preconditioned BT matrices. The paper is organized as
follows. In section 2 some results are presented concerning BT, Hankel Matrices and Schur
complements of BT matrices. In section 3 some spectral and convergence properties are
presented and proved for pcg method applied on BT systems and for preconditioning Toeplitz
Schur complements. General propositions for constructing efficient prconditioners for BT
systems and Schur complements are given in section 4. In section 5 numerical examples
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PRECONDITIONING BLOCK TOEPLTIZ MATRICES
33
are presented showing the validity of the theory. Finally, concluding remarks are given in
section 6.
2. Toeplitz, Block Toeplitz and multilevel Toeplitz matrices. In this section we present
some useful known results that are necessary to obtain the main results in the following sections.
T HEOREM 2.1 (Spectrum of preconditioned BT system, Theorem 4.1 in [6], see also [11,
a 1B54 and a 1B254 be two sequences of matrices generated by Lebesgue integrable
12]). Let
2 and  defined on ‚ Z %QX)(FXR[ , where  is
Hermitian € -matrix-valued functions
„
?
ƒ
positive definite for almost every
‚ , that is, the Lebesgue measure
† is zero with
a 1B54 aof‚h1325…g4 ‚‡lie
all
the
eigenvalues
of
in the (not
‚†7ˆ Š‰ ?0ƒ ‚oˆ 1\?R4Œ‹ {@<Ž .(9Then
[ , where
necessarily bounded) interval Z 
ƒ ” ˆ–•˜—™›š 1 œ ž_Ÿ ^4 -¡>¢ wg£¥¤Yux¦Œw8‘F§ L ¨8L £9© M ƒª ^† Ž
 q‘+’ O ‰J“ ,
² ¡>¢ wg£¥¤Yux¦Œw8‘F§ Lg¨ L £+© M ƒ ª †VŽ
«­¬x® ¢ J‰ “ ƒ>” –ˆ •¯—°± 1œ ž Ÿ 4^-
A F( ³ ƒ ^T W , then
1A4
91 ³ µ 4 | I ‚ 1 Aµ 4
1³ 4 ‚ a 1A ³ 4 z
% a 1BA4 a R1 ³ 4 | k
^
k
´
^
k
´
k
k
^
´
^
k
´
k
3
1
?
4
\
1
?
?
4
3
1
?
4
\
1
?
?
4
(
*
*
*
(
*
*
N
*
(
(
*
*
*
(
*
*
N
*
(
with |
and ‚
are matrices picking out
&
the first entries of an infinite vector, and the infinite Hankel matrix
m U
m
1BA4 ¶ mU ¶ ¶ *
´
¶ ¶
¶
C OROLLARY1\?R2.3
Toeplitz and Toeplitz matrix). For a real trigonomet4 and(Product
³ ƒ T^W ofitband
ric polynomial {
holds
a 1 { ³ 4 Sa R1 { 4 a 1 ³ 4 I  T HEOREM 2.2 (Product of Toeplitz matrices [19]). If
and
a 1 { 4 a 1 { ³ 4 Sa 1 ³ 4 I 
U
 , " ·#8(9. , matrices of low rank that depends only on { \1 ?@4
with
&
.
and not on the dimension
C OROLLARY 2.413?@(Product
BT matrix and BT matrix). For a real trigonometric
4 and ¸of
ƒ T^band
W with =, blocks it holds
matrix polynomial |
a R 1 | 54 Sa R 1 | 4 a 1354 I  and

,"
¹#8(9.
a 1 | 4 a 1 | 54 qa R1B54 I 
U
, matrices of low rank.
be the & & Hilbert matrix
T HEOREM 2.5 (Spectrum of Hilbert matrix [18, 7]). Let

1
A
4
}
¼
²
»
²
´
7
#
7
g
#
9
(
<
.
(
x
*
~
*
*
X
to
with
" ,"
, the number of eigenvalues of
´ with absolute
» . Then for any º
values exceeding is given by
with
´
¶
. uxwgy¥&
X
‘ L½¾ X » 1 # IK¿ 1 & 4F4 *
k
k
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34
T. K. HUCKLE AND D. NOUTSOS
»
Hence for every cluster
eigenvalues around zero with small radius the number of outliers
u~w8y 1 & 4 . of
is growing like
This behavior
¿ 1 & 4 . is denoted as weakly clustered because the number of
outliers is allowed to grow like
T HEOREM 2.6 (Negative results for multilevel Toeplitz matrices [8, 9, 15]). For illconditioned positive definite multilevel Toeplitz matrices there exist examples where for every
positive definite multilevel circulant preconditioner there holds:
1. If the spectrum
of the preconditioned
matrix is bounded ( •RÀ)ÁFÂÄÃÅ4 ²­Æ , Å indepenH
D
Ç
v
È
1
4
1
&
&
&
dent of ), then •
eigenvalues tend to zero, with Å
tending to infinity
º and Å
&
when tends to infinity.
º,
2. If the spectrum
of the
preconditioned
matrix
is bounded from below ( •RÀ)É Ê1 ˼
Ì ÈÐ
Q
D
Ï
Í
Î
Ð
È
Æ
1
4
–
4
Æ
&
&
&
Ì independent
of ), then •
and Ì
eigenvalues tend to infinity, with Ì
Ñ
È
Æ
&
for
.
. . The close
Now, let us turn to Schur complements related to BT matrices for relationship between BT matrices and Schur
complements is displayed by the equations
a n aÒ
U
º
a a m
º aRmU p Qn Óº U
!
n
Ô
p k º am p k n a m Ó aÒ p
Ó
a º
a U a Ó n aÒ Ó a º
m p nQÓ
Up
p n
Ó k a º %QÔ a k º a Ó
U
n Ô
ºm
º
(2.1)
p k n %Qa Ò a m U p k n º a m p
º
Ô a m %aÒ a U a
aÒ
S
a
=
%
a
a
­
m
m
with
for
U
U . Hence, an efficient preconditioner and
Ô
U and Ô
U
a BT matrix directly leads to an efficient preconditioner for both Schur complements
m . On the other side an efficient preconditioner for one of the Schur complements directly
Ô
Ôleads
to an efficient preconditioner for the whole BT matrix.
a
3. BT matrices and Toeplitz Schur Complements. First we present some useful results on the location of eigenvalues of Schur complements of Toeplitz matrices. These results
are based on the connection between Schur complements and block matrices.
T HEOREM
3.1 (Preconditioned normal equations).
the family of Toeplitz maa 1BA4 , &„/
#g(9.<(*x*~* with generating function A , realConsider
Lebesgue integrable. Then it holds
trices
a 1BA U 4 Ë a R1BA4 U *
Proof. We consider the BT matrix
2
to
21\?R4 ˆ n Ar13# @? 4ÕA Ar1\1\?R?@4 4 *
U p
º and • This
rank-1 symmetric positive semidefinite with eigenvalues •
1\?@4 U . Theis generating
# I Armatrix
function is positive semidefinite, therefore the BT matrix
is also positive semidefinite. Hence, both Schurcomplements
positive semidefinite and
a 1A U 4 %Öa U 1BA4 Ë­º and %Öa 1BA4 a 1A U 4 a 1A4 ˁareº (for
a R1BA U 4 nonsingular).
therefore
Ó
2
3
1
@
?
4
1B254 may be even positive
Note that if
positive
semidefinite,
Ë A º is symmetric
0
r
A
3
1
@
?
4
?
definite, depending on , e.g.,
a 1+× A)× U 4 Ë a Ò 1A4 a 1BA4 U .. Theorem 3.1 can be also shown for general
Toeplitz matrices:
³ 1\?R4 nonnegative with
3.2.
integrable
and
Ø a, 1³ ³ 4 Lebesgue
× Ø 1\?R4L× U EMMA
} ³ 13?@4^²q
Æ forConsider
?
+
1
×
×
}
4
a
³
Ø U nonsingular. Then it holds
all , and
and
a R1+× Ø × U } ³ 4 Ë a Ò 1 Ø 4 a 1 ³ 4 a 1 Ø 4
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35
PRECONDITIONING BLOCK TOEPLTIZ MATRICES
and
a R1 ³ 4 Ë a 1 Ø 4 a +1 × Ø × U } ³ 4 a Ò 1 Ø 4 *
Proof. Consider the BT matrix
to
µ
213?@4 n × Ø 1\?R4× 13U ?@} 4 ³ 13?@4 ³ Ø \1 \1 ?R?@44 *
p
Ø
A ³
T HEOREM 3.3 (Preconditioned Toeplitz Schur Complement). Let , , Ø Lebesgue inte× × } ³ ²ÙÆ , and the matrices a 1 ³ 4 , a R1+× Ø × U } ³ 4 , a 1BA % × Ø × U } ³ 4
³
grable and nonnegative, Ø U
all nonsingular.
A µ Then it holds for the Toeplitz Schur complement relative to the matrix
21\?R4 ˆ n
Ø
Ø ³ p
:
a 1BA % × Ø ³ × U 4 à a 1BA4 %za Ò 1 Ø 4 a 1 ³ 4 a 1 Ø 4 ˆ 1BA Ú( ³R( Ø 4 *
Ô
The eigenvalues of the Toeplitz preconditioned Toeplitz Schur complement
#
a R1BA % × Ø × U } ³ 4 1BA F( ³@( Ø 4
Ô
are greater or equal .
Proof. Direct Corollary of Lemma 3.2.
Ar1\?R4 equals zero on a whole
Note, that for Toeplitz matrices with generating function
B
1
A
4
Ar1\?R4 has only
a
interval, all matrices
will be singular. Furthermore, if in the scalar case
isolated zeros of polynomial order thenA the smallest eigenvalue tends to zero polynomially
depending on the order of the zeros of . The next theorem displays a basic difference between scalar Toeplitz matrices and BT matrices. To prove this Theorem we need the following
Lemma
A
._X periodic continuous function, then the eigenvalues of aÛ 1BA U 4 %
3.4. If is a
a 1BAL4 EMMA
U are clustered at zero and the minimal eigenvalue tends to zero exponentially.
Proof. The related Hankel matrix is a compact operator; see [1]. Therefore, the sum of
the eigenvalues of the Hankel matrix is bounded which guarantees the strong clustering, e.g.,
the infinite Hilbert matrix is a bounded operator [4, 17, 18]. Furthermore, Tyrtyshnikov
has
1BA4 tends to zero
shown in [17] that the smallest eigenvalue of the associated Hankel matrix
´
exponentially.
BT matrix). There exist examples with generating function
L 3.5
21\?R4 T HEOREM
1\?@4+4 º such
§ 1B2(Ill-posed
13254 are spd. For such a
Ù
Ë
º
but Ü
the related BT matrices
the condition number may grow that
&
exponentially with , resp. the smallest eigenvalue may
tend to zero exponentially.
Proof. As example we consider
1B254
to generating function
2 n A#
A
AUp
. Then
can be bounded from above by the smallest eigenvalue of the
the smallest eigenvalue of
R
B
1
A
4
a
U %a 1A4 U Ë­º which is exponetially converging to zero following
Schur complement
the previous Lemma 3.4. (To see the relationship between the spectrum of a natrix and the
spectrum of the Schur
the Rayleigh
examAr1\?Rcomplement
4 ? U . Thenwe2can
13?@4 isconsider
? quotient).
13254 isAsstillspecial
ple we can choose
singular for all , but
spd in view
of Theorem 2.2.
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36
T. K. HUCKLE AND D. NOUTSOS
Now we want to analyse the spectrum of different preconditioned BT matrices. The first
results show, that well-conditioned BT matrices behave similar to scalar Toeplitz matrices.
T HEOREM
BT, Theorems 8.4 and 9.3 in [13]). For well-conditioned
2133.6
?@4 ._(Well-conditioned
X -periodic continuous
spd BT with
the optimal block circulant preconditioner leads
to a strongly clustered spectrum.
A Ëqº be real .YX -periodic continuous with a finite number
T HEOREM 3.7.
Let
of zeros

1
A
4
a
of even order and
U # nonsingular. Then the spectrum of Ýa 1BA4 a 1A U 4 a 1BA4 is
strongly clustered around .
Ar13?@4 { 13?@4+Þß13?@4 with Þà1\?R4 Ë e º and trigonometric polyProof. We can write
3
1
@
?
4
B
1
A
4
a
$a R1 { ޘ4 $a R1 { 4 a 1BÞ@4I  « a R1Þ˜4 a @1 { 4 I  with
nomial {
. Then
 of low rank
a k R1BA U 4 Ùa 1 { 4 a k 1Þ U 4 a 1 { 4 U I m .
(Lemma
2.3).
In the
sameway,
á
Æ
ã
â
ß
Þ
3
1
@
?
4

1
˜
Þ
4
e
a
a 1Þ U 4 are well-conditioned and can be
Because
and
Ë
Ë
º,
Ȋ approximated by circulant matrices s such that it holds: For each
å!æ8(+å!çg(+å!è and äæg(+äçY(+äè for which é å é à » , " ¼ê<(+ë<(Ïì and  , " º ¼there
ê<(+ë<(Ïexist
ì are matrices
matrices
»
U
&
of low rank (the rank depends on and not on the dimension ) [2, 7] with
a B1 Þ U 4 a 1BÞ@4
k 1BÞ@4 k B1 Þ@4 a
kæ sI  æ 4 s 1 I å k ç
k Ó
I  l 1BÞ U 4 s B1 Þ@4 s B1 Þ@4 a 1BÞ@4 I  l I  ç 4 k 1 I å è I  è 4 I k  l I å I ¹*
k Ó
Ó
T HEOREM 3.8 (Well-conditioned Toeplitz Schur1BAcomplement).
well1 Ø 4 with real
A
ÖSa 4 %za Ò 1 Ø 4 a 1 For
³ 4 auniformly
conditioned spd Toeplitz
Schur
complements
í
²
×
\
1
@
?
4
×
}
3
1
@
?
Ö
4
ã
²
Æ
A
×
×
}
³
³
%
³
Y
.
X
Ô
Ø q
U a 1BAr1\?R4 % ,× 1\?R4× } Ø ³ U 13?@4F4 º , all -periodic continuous, the
and Ë«º with º
Toeplitz preconditioner |
Ø U
clustered spectrum
Æ leads to1\?Ra4 weakly
with eigenvalues uniformly bounded away from º and . If Ø
is real with zeros of even
order the spectrum of the preconditioned Schur complement is strongly clustered around
and | are uniformly well-conditioned and the spectrum of | 1. is
Proof. All
#8( â [ . With the following Theorem 3.9 this proves the first part
Ô
Ô of
bounded in an interval Z
the Theorem. Furthermore, it holds
| qa 1A % × Ø × U } ³ 4 1 a 1BA4 %va 1 Ø 4 Ò a 1 ³ 4 a 1 Ø 4+4 k Ô I 1 a 19× k × } ³ 4 %za Ò 1 4 a 1 ³ 4 a 1 4+4 *
Ø U
Ø
Ø
Ó | k
13?@4 has zeros, then ³ 1\?R4 has the same zeros of double order. If Ø has only zeros of even
If Ø
1\?@4 1\?R4FÞRî13?@4
{ Þ¯13ï ?@4 . e Furtherorder then
we can write Ø ³ 1\?R4 { 13?@4FÞ¯ï<1\?Rwith
a trigonometric
polynomial
4
@
Þ
î
H
î
³
e
{RU
Ë
º and Ë ïh º .
more, can be written as
with positive
Then, for real Ø it holds
a Ò 1 Ø 4 a 1 ³ 4 a 1 Ø 4 qa Ò 1 { Þ î 4 a 1 { U Þ ï 4 a 1 { Þ î 4 ­a 1BÞ î 4 a 1BÞ ï 4 a R1BÞ î 4 I 
a 1 ³ 4
in view of Corollary 2.3. So we have to analyse only the case of well-conditioned
äŒSa B1 Þ@4
Sa B1 Þ@4
1 I å
Ó
which can be treated like in the previous Theorem by using circulant approximations.
Next we consider ill-conditioned matrices. Here it turns out that for BT matrices the
block circulant preconditioner can only ensure a weak clustering but no strong clustering.
The proof of the weak clustering is based on the definition of approximating class of matrix
sequences:
BT matrices,
T HEOREM 3.9 (Weak clustering of BC preconditioner for
3.1
1BA (FTheorem
³@( Ø 4 it holds
in [16]). For the Hermitian Toeplitz Schur complement of T functions
a 1BÔ A % × ØÏU × } ³ 4 . Therethat the spectrum is distributed like the spectrum of the Toeplitz matrix
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PRECONDITIONING BLOCK TOEPLTIZ MATRICES
37
#
fore, the eigenvalues of the preconditioned Schur complement are weakly clustered around .
The same is true for related circulant preconditioners.
T HEOREM 3.10 (Counter example
strong clustering). For the Toeplitz Schur
Awith
ƒ THU ,nowhere
complement there exist examples, e.g.,
the above Toeplitz preconditioner from
Theorem 3.9 that leads to a weak clustering, does not lead to a strong clustering.
1 aR 1 { 4 I a 1BA U 4F4 %5a 1BA4 U with a trigonoD
Proof. Consider the 13Schur
complement
½
@
?
4
1
3
1
@
?
F
4
4
#<% wð‘
Ô and function A with coefficients 0# } " . Note
metric polynomial like {
A
that the sequence is in ñ3U and therefore the related Fourier series is in T^U . The Toeplitz
R
1
4
a
¶
preconditioner is given by
{ . Then it holds
(3.1)
¶
a 1 { 4 I a 1 { 4 1 a 1A U 4 %za R1BA4 U 4F4
k Ô Ó I a 1 4 k 1 1BA4 1BA4 I 1BA4 1BA4 <4 *
{ k | ´
|
‚ ´
‚
´
´
Ó
1A4 is the infinite Hilbert matrix. Because of Theorem 2.5 the eigenvalues, and
Here,
´
therefore
the singular values are not clustered, and therefore also the eigenvalues of
1 uxwgy 1 & 4+4 outliers.
a 1 { 4 also
are not clustered around 1 but each cluster leads to t
Ô
T HEOREM 3.11 (Counter example for Toeplitz Schur complement
A U 4 %Ka 1A4 a with
1BA4
I a 1Bpreconditioner
unbounded norm). For the Toeplitz Schur complement
Ô
with Toeplitz matrices
there exists an example where the Toeplitz peconditioner that leads
to a weak clustering
following
Theorem 3.9 has unbounded spectrum of the preconditioned
A„ƒ TVU .
system, e.g., for
, ¿_ò {@ñ P 1 .:(%#g( º (*x*~* º ((# 4 and Ar1\?R4 with
Proof. Define the skewcirculant matrix
Ý
ô
ô
coefficients
with
related
Hankel matrix coefficients . (Here we use the MATLAB_
¿
ò
3
1
@
?
4
?
f
a
P
¶
Ú
¶
ó
{@ñ
notation
to denote the symmetric Toeplitz matrix generated by
vector ).
¶
With the eigenvectors õ of
, relative to the standard eigendecomposition
on
Ò õ and õ Ò to õ . Itbased
¶
¶Úó
the Fourier matrix, we consider the Rayleigh quotients õ
can
be
´
1
}
4
Ò
#
&
Therefore, for the eigenvecshown by direct computation that it holds õ ´ õ
it holds õ Ò äö õ t t 1 # } & U 4 , .and
tor to small eigenvalues of
the Rayleigh quotient of
õ Ò 1 I ´ <4 õ } õ Ò õ is unbounded.
aß 1BA4 U the Toeplitz preconditioner a 1A U 4
T HEOREM 3.12. For the normal equations
A„ƒ TQU .
from Theorem 3.9 guarantees only a weakly clustered spectrum, e.g., for real
1BA4 U precondir
A
\
1
R
?
4
}
÷
#
a
÷
8
#
9
(
:
.
(
*
*
*
Proof. Consider
;
˸º with coefficients
" ,"
B
1
A
4
a
tioned by
U leads to
(3.2)
and t
%za B1 A U 4 a U 1A4 ­
a
k
Ó
a
­
1 uxwgy 1 & 4+4
¶
B1 A U 4 1 a 1A U 4 %za U 1A4F4 1BA4 k 1 | 1BA4 1BA4 | I ‚ B1 A4 B1 A4 ‚ 4
k
´
´
´
´
outliers in view of Theorem 2.5.
Finally, let us summarize the clustering results on preconditioning multilevel and block
Toeplitz matrices by multilevel or block circulant preconditioners:
e
equispaced grid, blocksize & , and total size ø Dc d3ù D matrices
D with
1. For -level
Toeplitz
1
4
& D there
q
&
be t ø
outliers (this includes
the scalar Toeplitz case
1 # 4 may
S& U and t 1 & 4 also
t 1ûú ø 4 outliers).
with t
outliers and the 2-level case with ø &
k , with fixed , there may be t 1 uxwgy 1 ø 4+4 2.
For
a BT matrix of size ø
1
1
+
4
4
x
u
g
w
y
&
t
outliers in the ill-conditioned case.
So, in the ill-conditioned case for scalar Toeplitz matrices we get strong clustering while
e
for BT and -level Toeplitz matrices we 1 get only
weak one. Nevertheless, the pcg method
u~w8y 1 ø 4F4 aoutliers)
e
D
for BT systems D converges
much faster ( t
than the one for -level Toeplitz
3
d
ù
1
4
c
systems ( t ø
outliers).
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38
T. K. HUCKLE AND D. NOUTSOS
4. Finding efficient preconditioners for BT matrices and Toeplitz Schur Complements. An important class of preconditioners for scalar and multilevel
Toeplitz matrices is given by trigonometric polynomials, resp. band Toeplitz matrices. A
polynomial BT matrix is defined as a matrix with generating function whose entries are
trigonometric polynomials. Unfortunately, given an ill-conditioned BT matrix it is not obvious whether one can find a spectral equivalent trigonometric polynomial BT matrix. So for
the three examples
æ
2 1\?R4 n ? ? U # I ? ?
( 2 1\?R4 n ?× ?rU × # I × ?Û?×
( 2 m 1\?R4 n ? U I6? ü × ?Û× # I ? ?
U
U p
U p
U p
(4.1)
ü º , the polynomial BT matrix
with small
. 1 #H% ½ wð‘ 13?@4F4+4
‘+¬x® 1\?R4
‘F¬x® 1\?R4
# I . 1 #Q% ½ wð‘ 13?@4F4+4 p Ë-º
(4.2)
| 1\?R4 n
2 2 m
2
is spectral equivalent
to and , but for we cannot find a spectral equivalent polynomial.
2
2
U
Note, that
and
have the same determinant, trace, and eigenvalues.
U
In the following, we present a recipe to define a trigonometric
BT matrix
213?@4 . For simplicity we assume that 213?@4 has ? º as polynomial
for given
only
singularity.
L § 13 213?@4+4 and assume that at ? º it allows the expansion Ü L § 13213?@First,
4F4 D
weD compute
Ü
ý ? I t 1+× ?Û× † 4 . We split the partial functions A Çþ 1\?R4 of 21\?R4 into D A Çÿ 13 ?@4 A  Çÿ 13?@4 I
A & Çþ 1\?@4 with A  Çÿ 13?@4 differentiable up to order e and A & Çþ 1\?R4 t 19× ?r× † 4 not necessarily
? Then, we can reduce the partial function
D A  Çÿ L 13§ ?@1B4 2to1\?@4+the
differentiable
?
4 . coefficients
A Çÿ we
#g( ? (***( ? atthat .give
a contribution
to
the
leading
term
in
In 
Ü
þ
Ç
?
(
*
*
*
(
ù
º
ñ by trigonometric polynomials of the form
can13?@
,
½ w8‘ 1\?@4+4 terms
4 or 1 .àthe%Ä. occuring
‘+¬~® replace
U
.
In
this
way
we
can
polynomial BT matrix
L § 1 | 13?@4F4 has the same singularity as Ü L define
§ 1B21\?@a4+4 trigonometric
where Ü
. If the resulting polynomial has no
other singularity and is positive definite it may lead to a spectral equivalent preconditioner.
We can prove the following
21\result.
?R4 be the generating matrix function for an spd ill-conditioned
T HEOREM 4.1. Let
? º . The
213?@4 at ? º is assumed to be
D
BT matrix with
singularity at D I
determinant
of
L
§ 1B21\?R4F4 ý ?
of the form Ü
t 19× ?r× † 4 . Suppose thatÇþ 1\we
4 2a13?@trigonometric
4 % | 13?@4 are
3
1
@
?
4
?R4 inhave
1\?Rfound
D
D
matrix polynomial |
such Çþthat all the
scalar
functions

ÿ
Ç
I
\
1
R
?
4
?
9
1
×
r
?
×
4
†
ý
1 4 | a 1\?@1B4 25is4
approximated at º such that 
. Furthermore assumethat
t
\
1
R
?
H
4
?
a
spd and |
of the preconditioned system
|
º for º . Then the spectrum
Æ .
1\?R4
D
is uniformly bounded away
from
and
º
L § 1 | 1\?R4F4 Ü L § 13213?@4+4 I t 1+× ?Û× † 4 near º . The entries
D
Proof. Note, that Ü
of
|
13213?@4 % | 13?@4F4 are also bounded near ? º because Ü L § 1 | 13?@4+4 ý ? I t 1+× ?Û× D † 4 k .
1\?R4Ú21\?R4 are bounded from
1\?@4Fof21\the
Therefore, the eigenvalues
preconditioned
system |
R
?
Q
4
?
above. Furthermore, |
all . 2 1\?R4 21\?R4+} L § 1321\?R4F4
213?@4 canº beforwritten
Ü
Note that the inverse of
as
With this
1\?@D 4 . all
D
|
notation the condition
in
Theorem
4.1
can
be
formulated
in
the
form
that
for
 213?@4 1\?@4 have to be of the form  Çÿ 1\?@4 -³ ? I t 19× ?r× † 4 .entries
of the matrix ˆ
Now let us return to the general problem of preconditining ill-conditioned BT matrices
or Toeplitz Schur complements. We are interested in BT matrices where the solution can be
reduced to the solution of scalar Toeplitz systems.
¸. . The solution
T HEOREM 4.2 (BT Schur complement I). Given a BT matrix with of the BT1 system
Toeplitz
1Abe4 %hreduced
1BA4 systems
4 aSchur
R1BA4 comple³ 4 ˆ qa can
a Ò 1 Ø 4 toa solving
a 1 Ø 4 or
Sa R1 if³ 4 one
%ha of1 Øthe
a Ò 1 Ø 4
1 ³ 4 scalar
ments
ˆ
1
k
4
k

1
A
4
k
k
³
Ô
Ô
is well-conditioned. We call
and
dual Schur complements.
Ô
Ô
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39
PRECONDITIONING BLOCK TOEPLTIZ MATRICES
Proof. This is a direct consequence that via (2.1) we can reduce the BT matrix to scalar
Toeplitz matrices and one of the Schur complements. Furthermore, in view of Theorem 3.8
we have efficient preconditinoers for well-conditioned Toeplitz Schur complements.
. . BT matrix one of the Schur complements may be
Note that for ill-conditioned ?
?} .
ill-conditioned and the other may be well-conditioned, e.g.,
21\?R4 n ?R} U .
# p
.
T HEOREM
and Schur complement
Given the pair of dual Schur comple1BA4 and4.3(BT
1 ³ 4 . Then
1BAII).
4 can
1 ³ 4 .
ments
the solution of
be reduced to the solution of

1
A
4
1
4
³
Ô
Ô
Ô
Ô
Therefore, for ill-conditioned
we can find an efficient preconditioner
if
is well1 ³ 4 .
Ô
Ô
conditioned or if there exists an efficient preconditioner for
Ô
Proof. Again, the result follows from equation (2.1) and Theorem 3.8.
One example is
213?@4 n A
{
{
{@U p
with trigonometric polynomial { . Then for the Schur complement it holds
aR 1A4 %va 1 { 4 a 1 { U 4 a 1 { 4 S
a 1BA4 % I ‡(
Ó
a 1 {@U 4 qa 1 { 4 a 1 { 4 I  .
in view of Corollary 2.3 with
T HEOREM 4.4. Given a BT matrix with generating matrix function
integrable that admits the factorization
21\?R4 T 1\?R4
k
13?@4
213?@4
Lebesgue
13?@4
13?@4
with T
a trigonometric
polynomial and
a triangular matrix or vice versa.
a 13254 matrix
can be reduced to solving scalar Toeplitz systems and band maThen the solution of
trices.
consider a BT matrix in the form (1.2). In view of Corollary 2.4 the matrix
4 can -We
a 1325Proof.
up to low rank - be transformed in the product of a band matrix and a triangular
matrix with Toeplitz matrices on the main diagonal blocks.
One example is
A
A % Ø º
21\?@4 n Ar1\13?R?@4 4 Ø 1331 ?@?@44 n # ##
Ø
Ø p
º p k n Ø
Ø p
with real , Ø)T functions and the above Schur complement factorization according to (2.1).
213?@4 Lebesgue
T HEOREM 4.5. Given a BT matrix with generating matrix function
integrable that admits the factorization
1\?R4
1\?@4
21\?@4 13?@4  13?@4
k
k
1\?@4
 1\?R4
with
and 13254 trigonometric matrix polynomials and
a triangular matrix. Then
a
the solution of
can be reduced to solving scalar Toeplitz systems and band matrices.
consider a BT matrix in the form (1.2). In view of Corollary 2.4 the matrix
4 can -We
a 1325Proof.
up to low rank - be transformed in the product of a band matrix and a triangular
matrix with Toeplitz matrices on the main diagonal blocks.
One example is
A I Ø
#
# #
21\?R4 n Ar131\?@?@4 4 ArØ 1\13?R?@4 4 1 # } . 4 n ## º%
A
%
#
n
n
p
p
Ø
º
Ø p #b%# p
A
with and Ø real T functions.
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etna@mcs.kent.edu
40
T. K. HUCKLE AND D. NOUTSOS
TABLE 5.1
Number of eigenvalues of the circulant preconditioned well-conditioned BT system outside
Size
outliers
2*10
9
outside
Number of eigenvalues of
Size
outliers
2*20
5
2*40
5
2*80
5
2*160
5
.
2*320
5
TABLE 5.2
for Toeplitz preconditioned Toeplitz normal equations.
10
2
20
2
40
2
80
2
160
2
320
2
640
2
5. Numerical Results. First, we present numerical examples related to the Theorems
from sections 3 and 4.
E XAMPLE 5.1. Table 5.1 presents an example for Theorem 3.6 with optimal circulant
preconditioner for BT matrix to
21\?R4 n # I × ?r×× ?Û} X × } X
× ?r× } X
# p
?ƒ Z Q% X)(FXR[ . The spectrum is clustered and the number of outliers is bounded.
E XAMPLE 5.2. Next we display Theorem 3.7. Table 5.2 shows theAr1\number
outliers
?R4 ? of
U
for a Toeplitz preconditioned Toeplitz normal equations. The
function
has
zero
ˆ a 1\? U 4 a 1\? l 4 aR 13? U 4 with
º
ø
of
even
order
at
and
we
consider
the
matrix
?÷ƒ Z %QX)(+XR[ . Note, that ø is spectral equivalent to matrix ak 13? U 4 U preconditioned
k
by
a 1\? l 4 .
with
E XAMPLE 5.3. Table 5.3 presents an example for Theorem 3.8 with strong clustering
for
the
well-conditioned
optimal
I a 1+× ?Ûcirculant
× 4 %'a R1+× preconditioned
?Û× 4 a R1+× ?Û× 4 and
× 4 Toeplitz
× ?r× 4 complements
%'a R19× ?rdual
a 1 # I Schur
a 1+× ?Û× 4 with
?„
k
k
k
Ô ƒ Z %QÓ X)(+XR[ .
Ô U Ó
E XAMPLE 5.4. Next we consider the strong clustering
4 %=a 13? U by
4 a Theorem
. I a 1\? U described
a 13? Ta 13? l 4 3.8.
ble 5.4 deals with the Toeplitz
Schur
complement
U4
I
R
1
?
4
k
k
,
?
ƒ
a
#
Q
%
)
X
F
(
R
X
[
Ó
preconditioned by
.
U with Z
3.10 with Toeplitz precondiE XAMPLE 5.5. Table 5.5 presents an example
4 %5Theorem
a 1 { 4 I a R1A for
a U 1BA4 ; the
tioner for Toeplitz Schur complement
by
U
a 1 { 4 with Ar13?@4 × ?Û× , ? ƒ Z %QX)(FXR[ , and { 13?@4 preconditioner
.«% . ½ w8‘ 13is?@4given
,
resp.
{ 13?@4 që% ½ w8‘ 1\?R4 I . ½ w8‘ 1 . ?@4 .
E XAMPLE
3.10 by Table 5.6. For
the Toeplitz
I a5.6.
Ar1\?@4 × Schur
?Û× withcom?zƒ
1BA U 4We%'adisplay
with
U 1A4 weTheorem
plement
choose
the
preconditioner
%Z QX)(FXR[ and 1 { 4 the skewcirculant matrix given by the polynomial { 1\?@4 «.h%. ½ wð‘ 1\?R4 ,
1\?R4 Së% ½ w8‘ 13?@4 I . ½ w8‘ 1 . ?R4 .
resp. {
E XAMPLE 5.7. Table 5.7 presents an example
polynomial BT
1\?R4 for the three BT matrices 2 1\?R4 , for
2 1\?@Theorem
4 , and 2 m 4.1
1\?R4 with
preconditioner |
from
(4.1) and (4.2).
U 2 2 m
We see, that | leads to bounded spectrum when applied on # or ? .
E XAMPLE 5.8. We consider the matrix function
2
matrix generated by
is:
2 n ?
1B254 n a Ò Ó \1 ?R4 ._aa R 1\1\?@? 4 4 *
U p
For the solution of the BT system
1B254 M Ø
.?U p
. The associated BT
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41
PRECONDITIONING BLOCK TOEPLTIZ MATRICES
TABLE 5.3
Number of eigenvalues outside as example for strong clustering for well-conditioned Toeplitz
Schur complement with circulant preconditioner.
Size
outliers !#"
outliers !%$
10
3
3
20
2
2
40
2
2
80
2
2
160
2
2
320
2
2
640
2
2
TABLE 5.4
Number of eigenvalues outside as example for strong clustering for well-conditioned Toeplitz
Schur complement with Toeplitz preconditioner.
Size
outliers
10
2
20
2
40
2
80
2
160
2
320
2
640
2
we choose as preconditioner the block band Toeplitz matrix:
a 1 ‘+¬~® 31 ?@4+4
n a R1 ‘+Ó ¬~® \1 ?@4+4 ._a R1 .ä%€. ½ w8‘ 13?@4+4 p *
The Schur complements of the above BT matrix are
‡
% 1 # } . 4 aR 13?@4 Ò a \1 ? U 4 a 1\?R4 (
k
k
k
Ô
Ó
which is well-conditioned in view of Theorem 3.1 and does not need any preconditioning,
and
Ô U
Ù._a \1 ? U 4 %va 1\?R4 Ò aR 13?@4 (
k
which is ill-conditioned and needs preconditioning. For this we choose the band Toeplitz
preconditioner
a 1 .ä%€. ½ wð‘ 1\?R4F4 *
&
In Table 5.8 we give the number of iterations of the pcg method for various values of and
for using various preconditioners. Thereby, we use the following abbreviations:
BT: cg for block Toeplitz System,
BBT: Block band Toeplitz preconditioning for block Toeplitz System,
CBT: Block circulant preconditioning for block Toeplitz System,
WS: cg for well-conditioned Schur complement,
IS: cg for ill-conditioned Schur complement,
BIS: Band Toeplitz preconditioning ill-conditioned Schur complement, and
CIS: Circulant preconditioning for ill-conditioned Schur complement.
We have to remark that for this example we get only a weak clustering and not a strong
clustering for the band Toeplitz preconditioned
ill-conditioned Schur complement, since the
?
Hankel matrix related to the function is similar to the Hilbert matrix (Theorem 3.10). Hence,
ì
the number of iterations in the th column shows a logarithmic growth.
Since the dual Schur complement is well-conditioned we get a strong clustering for the block
band Toeplitz preconditioner applied on the block Toeplitz system (Theorem 4.2). As a con;
sequence, the number of iterations in the rd column tends to be constant.
For the circulant preconditioned matrix for the Schur complement as well as for the block circulant preconditioned matrix for the BT system we get only a weak clustering (Theorem 3.9).
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42
T. K. HUCKLE AND D. NOUTSOS
Number of eigenvalues of outside
tioned Toeplitz Schur complement.
TABLE 5.5
as an example for no strong clustering for Toeplitz precondi-
10
2
2
Size
outliers &('*),+.-0/1&(24365789 :
outliers &('*)9+;-0/1&(2431<7>=4, :
20
2
2
40
2
3
80
2
4
160
3
4
320
3
6
640
4
6
1280
4
8
TABLE 5.6
Maximum eigenvalue as an example for unboundedness for Toeplitz preconditioned Toeplitz Schur complement.
?;@BA*C for &('*)9+;Size
-0/1&(2431578, :
? @BAC for &('D)9+;-/6&(2E36<7>=49 :
10
3.1
2.0E1
20
6.0
1.8E2
40
12.2
1.6E3
80
25.3
1.4E4
160
52.2
1.1E5
320
106.7
9.2E5
640
216.5
7.5E6
1280
437.1
6.0E7
Therefore, the number of iterations in the F th and th columns show logarithmic growth.
2 n ? . ? ? U l (Œa 13254 n a Ó 1\ ? 4 _. a a 1\13? ? U l 4 4 * We use as block
E XAMPLE 5.9.
p
p
U
U
#
band Toeplitz preconditioner for the BT system the matrix:
a 1 .ä%€. ½ wð‘ 1\?R4F4
an R1 .ä%6Ó . ½ wð‘ 1\?R4F4 .Ya 1F1 .ä%€. ½ w8‘ 13?@4+4 U 4 p
and as band Toeplitz preconditioner for the ill-conditioned Schur complement the matrix:
a 1F1 .ä%'. ½ w8‘ 1\?@4+4 U 4 *
In Table 5.9 we give the number of iterations of pcg method as we have done in Table 5.8.
The meaning of “*” is that the number of iterations exceeds the dimension of the considered
matrix.
We get the same remarks as in the previous example except the band Toeplitz preconditioned
Schur complement. Note, that the Hankel matrix related to the function
? U is a ill-conditioned
compact operator (Theorem 3.4, see also [1]). Consequently, the number of iterations
ì
?
in the th column tends to be constant.
?
a 1\? 4 a 1\? 4
2 n ? U ? U („a 1B254 n a 13? U 4 a 1\? U 4 * We use as
U U U p
U U
U p
U
U
block band Toeplitz preconditioner for the BT system the matrix:
aR 1 .%'. ½ w8‘ 1\?@4+4 aR 1 #Q% ½ w8‘ 1\?@4+4
n a 1 #Q% ½ wð‘ 13?@4F4 a 1 .%€. ½ w8‘ 13?@4F4 p *
m a 1\? 4
U , so we use as band Toeplitz
Both Schur complements are the same Toeplitz matrix: l
E XAMPLE 5.10.
preconditioner for the ill-conditioned Schur complement the matrix:
F
; a 1 .ä%'. ½ w8‘ 13?@4+4 *
This example is a special case of Theorem 4.5, so
2
m
21\?R4 1 # } . 4 n ## %# # n ? U
p Uº
can be written in the form
# # *
m º?
n
p #b%# p
U U
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43
PRECONDITIONING BLOCK TOEPLTIZ MATRICES
TABLE 5.7
Minimum and maximum eigenvalue of the preconditioned spectrum GI H
? V
@BAC
@ U ? Size
? @VU ;? @BAC
? @VU ?;@BAC
10
0.83,2.4
0.1,9.0
0.90,8.0
for NM"
for N $
for NXW
20
0.81,2.6
4E-2,26.9
0.84,8.9
40
0.80,2.7
1E-2,96.5
0.82,9.4
80
0.80,2.9
4E-3,4E2
0.81,9.8
" 36JK:%LMG 31N;OP: , QRS95,T
160
0.80,2.9
1E-3,1E3
0.80,9.9
320
0.80,2.9
4E-4,6E3
0.80,10.1
.
640
0.80,3.0
1E-4,2E4
0.80,10.1
TABLE 5.8
Iterations for BT system and for Schur Complements of example 5.8
&
4
8
16
32
64
128
256
512
1024
BT
8
15
31
64
128
202
350
619
1189
BBT
7
10
12
14
14
14
14
13
13
CBT
5
12
14
15
16
16
18
19
20
WS
2
3
3
3
3
3
3
4
4
IS
2
4
8
17
39
86
187
395
819
BIS
2
4
8
9
10
11
11
12
12
CIS
2
4
5
5
6
6
6
9
10
Theorem 4.5 is applied, which means that the problem is reduced to solve the scalar Toeplitz
a 13? U 4 . This system is efficiently solved by band Toeplitz precondisystem of the matrix
2
tioned method to get superlinear convergence. Since could be written also as the product:
213?@4 n ? U ? º n #
º
U p U
#U p (
where the first factor is a diagonal matrix function with roots of even order and the second one
is a constant positive definite matrix function, we can use efficiently the block band diagonal
Toeplitz preconditioner
. %'. ½ w8‘ 1\?R4F4
13?@4 n a @ 1 ä
a 1 .ä%€º. ½ w8‘ 13?@4+4 p *
º
In Table 5.10 we give the associated numbers of iterations for all the methods mentioned
above, where the column named by BBDT corresponds to the block band diagonal Toeplitz
preconditioner. We get superlinearity in all preconditioned methods which confirms the validity of Theorem 4.5.
It should be mentioned here that the block band diagonal Toeplitz preconditioner for this
example could be obtained also by Theorem 3.1 of [12], as an analogous preconditioner has
been obtained in the Example 4.2 of the same paper.
6. Conclusions. We have characterized different methods to reduce the solution of BT
matrices or Toeplitz Schur complements to the solution of scalar Toeplitz matrices. Furthermore, we have developed different preconditioning methods for block Toeplitz matrices
and Toeplitz Schur complement matrices. In ill-conditioned cases we have to reckon with
t 1 uxwgy 1 & 4+4 outliers and growing iteration count with increasing & . Furthermore, we have
developed results on the spectrum of preconditioned BT matrices and Toeplitz Schur complements. We have shown that for some cases neither circulant preconditioner nor banded
ETNA
Kent State University
etna@mcs.kent.edu
44
T. K. HUCKLE AND D. NOUTSOS
TABLE 5.9
Iterations for BT system and for Schur Complements of example 5.9
&
BT
4
11
25
*
*
*
*
*
*
4
8
16
32
64
128
256
512
1024
BBT
4
7
9
12
16
18
20
20
20
CBT
3
6
9
12
11
15
17
25
24
WS
2
2
2
2
2
2
2
2
2
IS
2
4
9
28
*
*
*
*
*
BIS
2
4
8
10
12
13
14
14
14
CIS
2
4
6
8
11
10
14
15
24
TABLE 5.10
Iterations for BT system and for Schur Complements of example 5.10
&
4
8
16
32
64
128
256
512
1024
BT
2
4
8
16
36
82
177
372
768
BBT
2
4
7
9
10
10
10
11
11
BBDT
2
4
7
9
10
10
10
11
11
IS
2
4
8
16
37
82
177
372
768
BIS
2
4
7
9
10
10
10
11
11
preconditioner lead to an efficient iterative method. In such cases as alternative method the
Multigrid approach for BT matrices [5] may be preferable, because these methods can also
deal with zeros of uneven order in the generating function.
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ETNA
Kent State University
etna@mcs.kent.edu
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45
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