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ISSN 1064–5624, Doklady Mathematics, 2006, Vol. 73, No. 1, pp. 100–101. © Pleiades Publishing, Inc., 2006.Original Russian Text © N.L. Zamarashkin, I.V. Oseledets, E.E. Tyrtyshnikov, 2006, published in Doklady Akademii Nauk, 2006, Vol. 406, No. 5, pp. 602–603.
1. INTRODUCTION
A matrix
T
=
is said to be Toeplitz if
a
ij
=
t
i
–
j
. It is natural to solve systems with such matrices byusing iterative methods. However, rapid convergencerequires preconditioning. In [1], circulant precondition-ers were suggested; they were studied in [2, 3] and inmany other works. All known proofs of the rapid(superlinear) convergence of iterative methods arebased on decompositions of the form [3]
T
=
C
+
R
+
E
,
where
C
is a circulant matrix,
||
E
||
≤
ε
, and
R
is amatrix of rank
r
=
r
(
ε
,
n
)
�
n
. The rank estimatesobtained in [3] and some other works have the form
r
(
ε
,
n
) =
�
(
ε
–
α
)
·
o
(
n
)
, where
α
> 0. In this paper, weprove that, in typical cases (including all examplesfrom papers concerning construction of superlinearpreconditioners), the circulant matrix can be chosen sothat
r
(
ε
,
n
) =
�
((ln
ε
–1
+ ln
n
)ln
ε
–1
)
.
2. TOEPLITZ MATRICESWITH RATIONAL SYMBOLS
We say that a matrix
T
is generated by a symbol
f
if
Lemma 1.
Let
T
be a lower triangular Toeplitzmatrix with first column
t
k
=
αρ
k
.
Then
,
T
=
C
+
R
,
where
C
is a circulant matrix and
R
is a matrix of rank
1.
aij[ ]ij 1=n
tk1
2π------ f t( )e itk– t.d
π–
π
∫=
Proof.
It is sufficient to consider the rank-1 Toeplitzmatrix
R
= [
r
i
–
j
],
r
k
=
k
= –
n
+ 1, –
n
+ 2, …,
n
– 1
as
R
. It can be verified directly that the matrix
C
=
T
–
R
is circulant.
Corollary 1.
If a Toeplitz matrix
T
is generated bythe symbol
f
(
z
) =
z
=
e
it
,
|ρ|
≠ 1,
then
T
=
C
+
R
,
where
C
is a circulant matrix and
R
isa matrix of rank
1.
Lemma 2.
Let
T
be a lower triangular Toeplitzmatrix with first column
t
k
=
kq, where q is a positiveinteger.
Then, T = C + R, where C is a circulant matrix andrankR ≤ q + 2.
Proof. For R we take the Toeplitz matrix
R = [ri – j], rk = p(k),
where p is a polynomial of degree q + 1 satisfying theequality
p(k) – p(k – n) = kq.
Obviously, the rank of the matrix R does not exceed q + 2,and the matrix C ≡ T – R is circulant, because
ck – ck – n = tk – tk – n – rk + rk – n = 0.
Theorem 1. Suppose that a Toeplitz matrix T is gen-erated by a rational trigonometric symbol
f(z) = P(z)+ , z = eit,
where P, Q, and L are polynomials, L has no roots onthe unit circle, the degree of Q is smaller than that of L,and the polynomials have no common roots. Then,
T = C + R,
αρk
1 1
ρn-----–
--------------,
11 ρz–--------------,
Q z( )L z( )-----------
MATHEMATICS
Approximation of Toeplitz Matrices by Sums of Circulantsand Small-Rank Matrices
N. L. Zamarashkin, I. V. Oseledets, and E. E. TyrtyshnikovPresented by Academician V.V. Voevodin June 7, 2005
Received July 20, 2005
DOI: 10.1134/S1064562406010273
Institute of Numerical Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russiae-mail: [email protected], [email protected], [email protected]
DOKLADY MATHEMATICS Vol. 73 No. 1 2006
APPROXIMATION OF TOEPLITZ MATRICES 101
where C is a circulant matrix, and rank(R) ≤ degP +degL + 1.
Proof. It is sufficient to represent as a sum of
simple fractions and apply Corollary 1.
3. TOEPLITZ MATRICES GENERATEDBY SYMBOLS WITH LOGARITHMIC
SINGULARITIES
Lemma 3. Let T be a lower triangular Toeplitzmatrix with first column
Then, for any ε, there exists a circulant matrix C and amatrix R of rank r such that
|(T – C – R)ij | ≤ |Tij |εand
r ≤ lnε –1[c0 + c1lnε–1 + c2lnn],
where c0, c1, and c2 depend only on α.Proof. For any ε, there exist fm and wm such that [4]
It remains to apply Lemma 1.Corollary 3. Suppose that a Toeplitz matrix T is
generated by the symbol
f(z) = ln(z – ζ), z = eit, |ζ| = 1.
Then, for any ε, there exists a circulant matrix C and amatrix R of rank r such that
|(T – C – R)ij | ≤ |T ij |and
r ≤ lnε–1[c0 + c1lnε–1 + c2lnn].
Corollary 4. Suppose that a Toeplitz matrix T isgenerated by the symbol
f = (z – ζ)αln(z – ζ), z = eit, |ζ| = 1, α ∈ �.
Then, for any ε, there exists a circulant matrix C and amatrix R of rank r such that
|(T – C – R)ij | ≤ |Tij|ε,
and
r ≤ lnε–1[c0 + c1lnε–1+ c2lnn] + 2α.
The results of this section are gathered in the follow-ing theorem.
Theorem 2. Suppose that a Toeplitz matrix T is gen-erated by a piecewise-analytic symbol of the form
where g is an analytic function on an annulus contain-ing |z | = 1.
Then, for any ε, there exists a circulant matrix C anda matrix R such that
|(T – C – R)ij| ≤ | Tij |εand
rank(R) ≤ lnε –1[c0 + c1lnε–1 + c2lnn] + c3,
where c0, c1, c2, and c3 do not depend on n and ε.
ACKNOWLEDGMENTSThis work was supported by the Russian Foundation
for Basic Research (project no. 05-01-00721).
REFERENCES1. G. Strang, Stud. Appl. Math. 84, 171–176 (1986).2. R. Chan, Linear Algebra Appl. 10, 542–550 (1989).3. E. E. Tyrtyshnikov, Linear Algebra Appl. 232, 1–43
(1996).4. N. Yarvin and V. Rokhlin, SIAM J. Sci. Comput. 20 (2),
699–718 (1999).
Q z( )L z( )-----------
tk
0, k 0,=
ρkk α– , k 1, …, n 1, α 0.>–=⎩⎨⎧
=
k α– wmef mk–
m 1=
r
∑– k α– ε,≤
r ε 1– c0 c1 ε 1–ln c2 nln+ +[ ].ln≤
f g Aka z ζk–( )α z ζk–( ),lnk 0=
m
∑α 0=
l
∑+=
z eit, ζk 1,= =
