The pseudospectra of banded finite dimensional Toeplitz matrices rapidly converge to the pseudospectra of the corresponding infinite dimensional operator. This exponential convergence makes a compelling case for analyzing pseudospectra of such Toeplitz matrices, not just eigenvalues. But what if the matrix is dense and its symbol has a jump discontinuity? The pseudospectra of the finite matrices still converge, but it is shown here that the rate is no longer exponential in the matrix dimension, only algebraic.\ud
\ud
The second and third authors were supported by UK Engineering and Physical Sciences Research Council Grant GR/M12414

Boettcher, A.

Embree, Mark

Trefethen, Lloyd N.

Czech

Mathematical Institute Eprints Archive

PIECEWISE CONTINUOUS TOEPLITZ MATRICES AND
OPERATORS: SLOW APPROACH TO INFINITY
ALBRECHT B

OTTCHER

, MARK EMBREE
y
, AND LLOYD N. TREFETHEN
y
Abstrat. The pseudospetra of banded nite dimensional Toeplitz matries rapidly onverge to
the pseudospetra of the orresponding innite dimensional operator. This exponential onvergene
makes a ompelling ase for analyzing pseudospetra of suh Toeplitz matries, not just eigenvalues.
But what if the matrix is dense and its symbol has a jump disontinuity? The pseudospetra of the
nite matries still onverge, but it is shown here that the rate is no longer exponential in the matrix
dimension, only algebrai.
Key words. Toeplitz matrix, pieewise ontinuous symbol, pseudospetra
AMS subjet lassiations. 47B35, 15A60
Let T be a Toeplitz operator (singly innite matrix) on `
2
(N) with symbol a 2
L
1
(T), where T is the unit irle. If a is ontinuous, then the spetrum spT is
the urve a(T) together with all the points this urve enloses with nonzero winding
number [6℄. This result generalizes to pieewise ontinuous a: If a
#
(T) is the urve
onsisting of the omponents of a(T) onneted by straight segments at points of
disontinuity, then spT is a
#
(T) together with all the points it enloses with nonzero
winding number; see [5, x1.8℄.
A long-reognized anomaly is that the spetra of Toeplitz matries T
N
of nite
dimension N look very dierent, typially onsisting of points distributed along urves
rather than aross regions, even asN !1 [1, 5, 10, 11, 15℄. Some kind of resolution of
this anomaly was obtained with the disovery that although the spetra of the matrix
and the operator do not agree, the "-pseudospetra may agree very losely [8, 9℄.
(The "-pseudospetrum sp
"
A of a matrix or operator A is the set of points z 2 C
satisfying k(zI A)
 1
k  "
 1
, where we write k(zI A)
 1
k =1 when z 2 spA; see,
e.g., [12, 13℄.) In partiular, if T
N
is banded, then for eah point z enlosed by a(T)
with nonzero winding number, k(zI   T
N
)
 1
k grows exponentially as N !1 [3, 9℄;
the ondition number kV
N
kkV
 1
N
k of any matrix V
N
of eigenvetors of T
N
is likewise
exponentially large. As illustrated by numerial examples in [9℄, the result is that for
small ", the "-pseudospetra of T
N
typially look muh like the spetrum of T for
values of N on the order of hundreds.
A more general onvergene result for sp
"
T
N
has been proved in [2℄. If a 2 L
1
(T)
is pieewise ontinuous, then for eah " > 0, sp
"
T
N
onverges to sp
"
T as N ! 1.
The question arises: If a is disontinuous, is the onvergene still fast enough to be
ompelling for modest values of N?
We have found that the answer is no. If the symbol is disontinuous, the rate at
whih k(zI T
N
)
 1
k and kV
N
kkV
 1
N
k inrease as N !1 may drop from exponential
to algebrai, hanging the qualitative nature of the pseudospetra strikingly.
We onsider the following simple example. Take a suh that a(T) is the right half
of the unit irle, speially, a(e
i
) = ie
 i=2
for  2 [0; 2). Then spT is the losed
right half of the unit disk, and T
N
is a dense Toeplitz matrix whose entries are given

Fakultat fur Mathematik, TU Chemnitz, 09107 Chemnitz, Germany (albreht.boetther
mathematik.tu-hemnitz.de).
y
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD,
UK (embreeomlab.ox.a.uk, LNTomlab.ox.a.uk). Supported by UK Engineering and Physial
Sienes Researh Counil Grant GR/M12414.
1
2 A. B

OTTCHER, M. EMBREE, AND L. N. TREFETHEN
N = 10
2
N = 10
3
N = 10
4
Fig. 1. Eigenvalues and "-pseudospetra for the Toeplitz matries T
N
given by (1) for three
values of N with " = 10
 1
, 10
 2
, and 10
 3
(from the outside in). The ross (
+
) marks the origin.
Exept in the rst image, the eigenvalues are so numerous that they appear fused into a urve. The
thikness of this urve is atually due to the boundaries of the 10
 2
- and 10
 3
-pseudospetra; the
boundary of the 10
 3
-pseudospetrum also aets the thikness of the middle eigenvalues in the rst
plot. We believe these images are orret to plotting auray.
by the Fourier oeÆients of the symbol,
(T
N
)
jk
:=
1
(j   k +
1
2
)
; j; k = 1; : : : ; N:(1)
Figure 1 shows numerially omputed "-pseudospetra of T
N
for N = 100, 1000, and
10 000 with " = 10
 1
, 10
 2
and 10
 3
. Note how far they are from spT for the
smaller values of ", and how the interior ars approximate irles passing through
i. Figure 2 shows resolvent norms as a funtion of N for points on the real axis.
For z =
1
2
, the bound k(zI   T
N
)
 1
k grows roughly like 3:8N
0:30
. At this rate, the
resolvent norm will not exeed 10
5
until N  10
15
. For z = 0, k(zI   T
N
)
 1
k grows
roughly like 0:4 logN+1:5; it will not exeed 10
5
until N  10
108572
. This behavior is
related to the \Moler phenomenon," the observation that the norm of the matrix (1)
approahes 1 spetaularly fast as N ! 1 while the smallest singular value deays
to 0 very slowly [5, x4.5℄, [14℄.
Here is a mathematial foundation for these observations. Let a be a pieewise
C
2
funtion with at most one jump disontinuity, say at e
i
0
2 T. For z outside a(T),
let arg(a   z) be any ontinuous argument of a   z on T n fe
i
0
g. Dene 
z
, the
Cauhy index of a with respet to z, by

z
=
1
2

arg

a(e
i(
0
+2 0)
)  z

  arg

a(e
i(
0
+0)
)  z

;
and put 
z
= j
z
j. If 
z
<
1
2
, then zI   T
N
is invertible for all suÆiently large N ,
and it is well known that k(zI   T
N
)
 1
k = O(1) in this ase [7℄. If 
z
 1, then
k(zI   T
N
)
 1
k may grow exponentially, as trigonometri polynomials (i.e., banded
matries) with nonzero winding number about z show. The following result tells us
that for
1
2
 
z
< 1, we have just algebrai growth at a known rate.
PIECEWISE CONTINUOUS TOEPLITZ MATRICES 3
PSfrag replaements
z =  
1
4
z = 0
z =
1
4
z =
1
2
z =
3
4
k(zI   T
N
)
 1
k
N
10
0
10
1
10
2
10
3
10
1
10
2
10
3
10
4
Fig. 2. The resolvent norm as a funtion of N for the lass of matries (1). The growth is
algebrai for z =
1
4
;
1
2
and
3
4
and logarithmi for z = 0. For z =  
1
4
, k(zI   T
N
)
 1
k is bounded
by 4 (see Theorem 3:19 of [5℄).
Theorem. If
1
2
 
z
< 1, then for every Æ > 0 there exist positive onstants C
z
and D
z;Æ
suh that
C
z
N
2
z
 1
 k(zI   T
N
)
 1
k  D
z;Æ
N
2
z
 1+Æ
(2)
for all suÆiently large N .
In the example (1), we have 
z
<
1
2
for all z outside spT and 
z
=
1
2
for z 2 ( i; i).
For z in the interior of spT , we have

z
= 1 
1

artan
1
x
;(3)
where x 2 (0; 1) is the point at whih the irular ar through  i; z; i intersets
the real line. In partiular,
1
2
< 
z
<
3
4
, and hene, by our theorem, the resolvent
norm inreases at most like O(N
1=2
) for z in the interior of spT , explaining the slow
onvergene seen in Figure 1. Moreover, formula (3) also reveals why the interior ars
of Figure 1 are lose to irles passing through  i and i. Finally, our theorem explains
Figure 2. For z =
1
2
, for example, we have 2
z
  1 = 0:295 : : :, in good agreement
with the growth 3:8N
0:30
estimated numerially.
Sketh of the proof of the theorem. The proof of the upper bound in (2) an be
based on the argument used to prove Theorem 6.1() of [4℄: A theorem by Verbitsky
and Krupnik (see, e.g., Theorem 7.20 of [5℄) states that the resolvent norm is uniformly
bounded on ertain weighted `
p
spaes, and appropriate hoie of these spaes together
with Holder's inequality gives the `
2
estimate O(N
2
z
 1+Æ
). To prove the lower bound
in (2), assume that
1
2
 
z
< 1. (The ase  1 < 
z
  
1
2
an be redued to this ase
by passing to adjoints.) We an write a   z = 
z
'

z
, where 
z
is a ontinuous and
pieewise C
2
funtion with no zeros on T and with zero winding number and where
4 A. B

OTTCHER, M. EMBREE, AND L. N. TREFETHEN
Fig. 3. Slow growth for the symbol a(e
i
) = e
i
. On the left, omputed eigenvalues and "-
pseudospetra for the Toeplitz matrix of dimension N = 1000 for " = 10
 1
, 10
 2
, 10
 3
, 10
 4
. (The
eigenvalues appear fused into a urve near the essential range of a.) The shaded region shows the
spetrum of the orresponding innite dimensional operator. On the right, ontour lines of onstant

z
for 
z
= 0:5; 0:55; : : : ; 1:45 (lokwise from right).
'

z
is a ertain anonial pieewise ontinuous funtion with a single jump (see, e.g.,
pp. 170{171 and 182 of [5℄). Here 
z
is a omplex number whose real part equals 
z
.
By Cramer's rule, the (N; 1) entry of (zI   T
N
)
 1
is ( 1)
N+1
times the quotient of
two Toeplitz determinants,

(zI   T
N
)
 1

N;1
= ( 1)
N+1
D
N 1
(
z
'

z
 1
)
D
N
(
z
'

z
)
;
and sine jRe 
z
j < 1 and jRe 
z
 1j < 1, we an invoke Renement 5.46 of [5℄ (whih
proves an important speial ase of the Fisher{Hartwig onjeture) to onlude that
the absolute value of [(zI   T
N
)
 1
℄
N;1
is asymptotially equal to a nonzero onstant
times





N
 (
z
 1)
2
N
 
2
z





=


N
2
z
 1


= N
2Re 
z
 1
= N
2
z
 1
:
As the norm of (zI   T
N
)
 1
is greater than the modulus of its (N; 1) entry, we arrive
at the lower bound of (2).
For z = 0, the estimate (2) asserts that C  kT
 1
N
k  D
Æ
N
Æ
for arbitrary Æ > 0.
Tyrtyshnikov [14℄ showed that in this ase we atually have
C logN  kT
 1
N
k  D logN:
We may summarize our observations as follows. Sine the pseudospetra, or
resolvent norms, onverge, T
N
must \behave" as if spT
N
= spT for suÆiently large
N . However, it is worth bearing in mind that a typial marosopi system has on
the order of 10
8
or 10
10
atoms or moleules in eah diretion (on the order of the ube
root of Avogadro's number or somewhat more). Thus for T
N
to behave like T , the
dimension N will have to be larger than the numbers that usually pass for innity in
the physis of gases, liquids, and solids.
Figure 3 presents a further example, the Toeplitz matries assoiated with the
symbol a(e
i
) = e
i
. The eigenvalues of these nite Toeplitz matries have been
PIECEWISE CONTINUOUS TOEPLITZ MATRICES 5
studied by Basor and Morrison [1℄. Our theorem provides us with the growth rate of
the resolvent norm as N !1 in the regions where 
z
< 1. Computational evidene
suggests that the same rate is valid throughout the interior of the spetrum, although
the values of 
z
range up to
3
2
.
One ould attempt to generalize our theorem and to raise onjetures suggested
by our omputations, but we will not pursue this here as our purpose is to point out
the slow onvergene phenomenon as briey as possible.
Aknowledgements. Some of our alulations were performed on the SGI Cray
Origin2000 at the Oxford Superomputing Center; the third plot of Figure 1 involved
omputation of the minimum singular value of 10
4
dense square matries eah of di-
mension 10
4
. We thank Rihard Brent and Walter Gander for suggestions onerning
fast Toeplitz algorithms, and Nik Birkett, Jeremy Martin, and TomWright for advie
onerning implementation and exeution.
REFERENCES
[1℄ E. L. Basor and K. E. Morrison, The Fisher-Hartwig onjeture and Toeplitz eigenvalues,
Linear Algebra Appl., 202 (1994), pp. 129{142.
[2℄ A. B

otther, Pseudospetra and singular values of large onvolution operators, J. Integral
Equations Appl., 6 (1994), pp. 267{301.
[3℄ A. B

otther and S. Grudsky, Toeplitz band matries with exponentially growing ondition
numbers, Eletron. J. Linear Algebra, 5 (1999), pp. 104{125.
[4℄ A. B

otther and B. Silbermann, Toeplitz operators and determinants generated by symbols
with one Fisher{Hartwig singularity, Math. Nahr., 127 (1986), pp. 95{124.
[5℄ , Introdution to Large Trunated Toeplitz Matries, Springer-Verlag, New York, 1999.
[6℄ I. Gohberg, On the appliation of the theory of normed rings to singular integral equations,
Uspekhi Mat. Nauk, 7 (1952), pp. 149{156. In Russian.
[7℄ , Toeplitz matries omposed of the Fourier oeÆients of pieewise ontinuous funtions,
Funkts. Anal. Prilozh., 1 (1967), pp. 91{92. In Russian.
[8℄ H. J. Landau, On Szeg}o's eigenvalue distribution theory and non-Hermitian kernels,
J. d'Analyse Math., 28 (1975), pp. 335{357.
[9℄ L. Reihel and L. N. Trefethen, Eigenvalues and pseudo-eigenvalues of Toeplitz matries,
Linear Algebra Appl., 162{164 (1992), pp. 153{185.
[10℄ P. Shmidt and F. Spitzer, The Toeplitz matries of an arbitrary Laurent polynomial, Math.
Sand., 8 (1960), pp. 15{38.
[11℄ P. Tilli, Some results on omplex Toeplitz eigenvalues, J. Math. Anal. Appl., 239 (1999),
pp. 390{401.
[12℄ L. N. Trefethen, Pseudospetra of matries, in Numerial Analysis 1991, D. F. GriÆths and
G. A. Watson, eds., Harlow, Essex, UK, 1992, Longman Sienti and Tehnial, pp. 234{
266.
[13℄ , Pseudospetra of linear operators, SIAM Review, 39 (1997), pp. 383{406.
[14℄ E. E. Tyrtyshnikov, Singular values of Cauhy{Toeplitz matries, Linear Algebra Appl., 161
(1992), pp. 99{116.
[15℄ H. Widom, Eigenvalue distribution for nonselfadjoint Toeplitz matries, Oper. Theory Adv.
Appl., 71 (1994), pp. 1{8.


Piecewise continuous Toeplitz matrices and operators: slow approach to infinity

The pseudospectra of banded finite dimensional Toeplitz matrices rapidly converge to the pseudospectra of the corresponding infinite dimensional operator. This exponential convergence makes a compelling case for analyzing pseudospectra of such Toeplitz matrices, not just eigenvalues. But what if the matrix is dense and its symbol has a jump discontinuity? The pseudospectra of the finite matrices still converge, but it is shown here that the rate is no longer exponential in the matrix dimension, only algebraic.  The second and third authors were supported by UK Engineering and Physical Sciences Research Council Grant GR/M12414

Oxford University Research Archive

PIECEWISE CONTINUOUS TOEPLITZ MATRICES ANDOPERATORS: SLOW APPROACH TO INFINITYALBRECHT BOTTCHER, MARK EMBREEy, AND LLOYD N. TREFETHENyAbstrat. The pseudospetra of banded nite dimensional Toeplitz matries rapidly onverge tothe pseudospetra of the orresponding innite dimensional operator. This exponential onvergenemakes a ompelling ase for analyzing pseudospetra of suh Toeplitz matries, not just eigenvalues.But what if the matrix is dense and its symbol has a jump disontinuity? The pseudospetra of thenite matries still onverge, but it is shown here that the rate is no longer exponential in the matrixdimension, only algebrai.Key words. Toeplitz matrix, pieewise ontinuous symbol, pseudospetraAMS subjet lassiations. 47B35, 15A60Let T be a Toeplitz operator (singly innite matrix) on `2(N) with symbol a 2L1(T), where T is the unit irle. If a is ontinuous, then the spetrum spT isthe urve a(T) together with all the points this urve enloses with nonzero windingnumber [6℄. This result generalizes to pieewise ontinuous a: If a#(T) is the urveonsisting of the omponents of a(T) onneted by straight segments at points ofdisontinuity, then spT is a#(T) together with all the points it enloses with nonzerowinding number; see [5, x1.8℄.A long-reognized anomaly is that the spetra of Toeplitz matries TN of nitedimension N look very dierent, typially onsisting of points distributed along urvesrather than aross regions, even asN !1 [1, 5, 10, 11, 15℄. Some kind of resolution ofthis anomaly was obtained with the disovery that although the spetra of the matrixand the operator do not agree, the "-pseudospetra may agree very losely [8, 9℄.(The "-pseudospetrum sp" A of a matrix or operator A is the set of points z 2 Csatisfying k(zI A) 1k  " 1, where we write k(zI A) 1k =1 when z 2 spA; see,e.g., [12, 13℄.) In partiular, if TN is banded, then for eah point z enlosed by a(T)with nonzero winding number, k(zI   TN ) 1k grows exponentially as N !1 [3, 9℄;the ondition number kVNkkV  1N k of any matrix VN of eigenvetors of TN is likewiseexponentially large. As illustrated by numerial examples in [9℄, the result is that forsmall ", the "-pseudospetra of TN typially look muh like the spetrum of T forvalues of N on the order of hundreds.A more general onvergene result for sp" TN has been proved in [2℄. If a 2 L1(T)is pieewise ontinuous, then for eah " > 0, sp" TN onverges to sp" T as N ! 1.The question arises: If a is disontinuous, is the onvergene still fast enough to beompelling for modest values of N?We have found that the answer is no. If the symbol is disontinuous, the rate atwhih k(zI TN) 1k and kVNkkV  1N k inrease as N !1 may drop from exponentialto algebrai, hanging the qualitative nature of the pseudospetra strikingly.We onsider the following simple example. Take a suh that a(T) is the right halfof the unit irle, speially, a(ei) = ie i=2 for  2 [0; 2). Then spT is the losedright half of the unit disk, and TN is a dense Toeplitz matrix whose entries are givenFakultat fur Mathematik, TU Chemnitz, 09107 Chemnitz, Germany (albreht.boetthermathematik.tu-hemnitz.de).yOxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD,UK (embreeomlab.ox.a.uk, LNTomlab.ox.a.uk). Supported by UK Engineering and PhysialSienes Researh Counil Grant GR/M12414. 12 A. BOTTCHER, M. EMBREE, AND L. N. TREFETHENN = 102 N = 103 N = 104Fig. 1. Eigenvalues and "-pseudospetra for the Toeplitz matries TN given by (1) for threevalues of N with " = 10 1, 10 2, and 10 3 (from the outside in). The ross (+) marks the origin.Exept in the rst image, the eigenvalues are so numerous that they appear fused into a urve. Thethikness of this urve is atually due to the boundaries of the 10 2- and 10 3-pseudospetra; theboundary of the 10 3-pseudospetrum also aets the thikness of the middle eigenvalues in the rstplot. We believe these images are orret to plotting auray.by the Fourier oeÆients of the symbol,(TN)jk := 1(j   k + 12 ) ; j; k = 1; : : : ; N:(1)Figure 1 shows numerially omputed "-pseudospetra of TN for N = 100, 1000, and10 000 with " = 10 1, 10 2 and 10 3. Note how far they are from spT for thesmaller values of ", and how the interior ars approximate irles passing throughi. Figure 2 shows resolvent norms as a funtion of N for points on the real axis.For z = 12 , the bound k(zI   TN) 1k grows roughly like 3:8N0:30. At this rate, theresolvent norm will not exeed 105 until N  1015. For z = 0, k(zI   TN ) 1k growsroughly like 0:4 logN+1:5; it will not exeed 105 until N  10108572. This behavior isrelated to the \Moler phenomenon," the observation that the norm of the matrix (1)approahes 1 spetaularly fast as N ! 1 while the smallest singular value deaysto 0 very slowly [5, x4.5℄, [14℄.Here is a mathematial foundation for these observations. Let a be a pieewiseC2 funtion with at most one jump disontinuity, say at ei0 2 T. For z outside a(T),let arg(a   z) be any ontinuous argument of a   z on T n fei0g. Dene z, theCauhy index of a with respet to z, byz = 12 arga(ei(0+2 0))  z  arga(ei(0+0))  z ;and put z = jz j. If z < 12 , then zI   TN is invertible for all suÆiently large N ,and it is well known that k(zI   TN ) 1k = O(1) in this ase [7℄. If z  1, thenk(zI   TN ) 1k may grow exponentially, as trigonometri polynomials (i.e., bandedmatries) with nonzero winding number about z show. The following result tells usthat for 12  z < 1, we have just algebrai growth at a known rate.PIECEWISE CONTINUOUS TOEPLITZ MATRICES 3PSfrag replaementsz =   14z = 0z = 14z = 12z = 34k(zI   TN) 1kN100101102103101 102 103 104Fig. 2. The resolvent norm as a funtion of N for the lass of matries (1). The growth isalgebrai for z = 14 ; 12 and 34 and logarithmi for z = 0. For z =   14 , k(zI   TN ) 1k is boundedby 4 (see Theorem 3:19 of [5℄).Theorem. If 12  z < 1, then for every Æ > 0 there exist positive onstants Czand Dz;Æ suh that CzN2z 1  k(zI   TN ) 1k  Dz;ÆN2z 1+Æ(2)for all suÆiently large N .In the example (1), we have z < 12 for all z outside spT and z = 12 for z 2 ( i; i).For z in the interior of spT , we havez = 1  1 artan 1x;(3)where x 2 (0; 1) is the point at whih the irular ar through  i; z; i intersetsthe real line. In partiular, 12 < z < 34 , and hene, by our theorem, the resolventnorm inreases at most like O(N1=2) for z in the interior of spT , explaining the slowonvergene seen in Figure 1. Moreover, formula (3) also reveals why the interior arsof Figure 1 are lose to irles passing through  i and i. Finally, our theorem explainsFigure 2. For z = 12 , for example, we have 2z   1 = 0:295 : : :, in good agreementwith the growth 3:8N0:30 estimated numerially.Sketh of the proof of the theorem. The proof of the upper bound in (2) an bebased on the argument used to prove Theorem 6.1() of [4℄: A theorem by Verbitskyand Krupnik (see, e.g., Theorem 7.20 of [5℄) states that the resolvent norm is uniformlybounded on ertain weighted `p spaes, and appropriate hoie of these spaes togetherwith Holder's inequality gives the `2 estimate O(N2z 1+Æ). To prove the lower boundin (2), assume that 12  z < 1. (The ase  1 < z    12 an be redued to this aseby passing to adjoints.) We an write a   z = z'z , where z is a ontinuous andpieewise C2 funtion with no zeros on T and with zero winding number and where4 A. BOTTCHER, M. EMBREE, AND L. N. TREFETHENFig. 3. Slow growth for the symbol a(ei) = ei. On the left, omputed eigenvalues and "-pseudospetra for the Toeplitz matrix of dimension N = 1000 for " = 10 1, 10 2, 10 3, 10 4. (Theeigenvalues appear fused into a urve near the essential range of a.) The shaded region shows thespetrum of the orresponding innite dimensional operator. On the right, ontour lines of onstantz for z = 0:5; 0:55; : : : ; 1:45 (lokwise from right).'z is a ertain anonial pieewise ontinuous funtion with a single jump (see, e.g.,pp. 170{171 and 182 of [5℄). Here z is a omplex number whose real part equals z.By Cramer's rule, the (N; 1) entry of (zI   TN) 1 is ( 1)N+1 times the quotient oftwo Toeplitz determinants,(zI   TN) 1N;1 = ( 1)N+1DN 1(z'z 1)DN(z'z ) ;and sine jRe z j < 1 and jRe z 1j < 1, we an invoke Renement 5.46 of [5℄ (whihproves an important speial ase of the Fisher{Hartwig onjeture) to onlude thatthe absolute value of [(zI   TN) 1℄N;1 is asymptotially equal to a nonzero onstanttimes N (z 1)2N 2z  = N2z 1 = N2Re z 1 = N2z 1:As the norm of (zI   TN) 1 is greater than the modulus of its (N; 1) entry, we arriveat the lower bound of (2).For z = 0, the estimate (2) asserts that C  kT 1N k  DÆNÆ for arbitrary Æ > 0.Tyrtyshnikov [14℄ showed that in this ase we atually haveC logN  kT 1N k  D logN:We may summarize our observations as follows. Sine the pseudospetra, orresolvent norms, onverge, TN must \behave" as if spTN = spT for suÆiently largeN . However, it is worth bearing in mind that a typial marosopi system has onthe order of 108 or 1010 atoms or moleules in eah diretion (on the order of the uberoot of Avogadro's number or somewhat more). Thus for TN to behave like T , thedimension N will have to be larger than the numbers that usually pass for innity inthe physis of gases, liquids, and solids.Figure 3 presents a further example, the Toeplitz matries assoiated with thesymbol a(ei) = ei. The eigenvalues of these nite Toeplitz matries have beenPIECEWISE CONTINUOUS TOEPLITZ MATRICES 5studied by Basor and Morrison [1℄. Our theorem provides us with the growth rate ofthe resolvent norm as N !1 in the regions where z < 1. Computational evidenesuggests that the same rate is valid throughout the interior of the spetrum, althoughthe values of z range up to 32 .One ould attempt to generalize our theorem and to raise onjetures suggestedby our omputations, but we will not pursue this here as our purpose is to point outthe slow onvergene phenomenon as briey as possible.Aknowledgements. Some of our alulations were performed on the SGI CrayOrigin2000 at the Oxford Superomputing Center; the third plot of Figure 1 involvedomputation of the minimum singular value of 104 dense square matries eah of di-mension 104. We thank Rihard Brent and Walter Gander for suggestions onerningfast Toeplitz algorithms, and Nik Birkett, Jeremy Martin, and TomWright for advieonerning implementation and exeution.REFERENCES[1℄ E. L. Basor and K. E. Morrison, The Fisher-Hartwig onjeture and Toeplitz eigenvalues,Linear Algebra Appl., 202 (1994), pp. 129{142.[2℄ A. Botther, Pseudospetra and singular values of large onvolution operators, J. IntegralEquations Appl., 6 (1994), pp. 267{301.[3℄ A. Botther and S. Grudsky, Toeplitz band matries with exponentially growing onditionnumbers, Eletron. J. Linear Algebra, 5 (1999), pp. 104{125.[4℄ A. Botther and B. Silbermann, Toeplitz operators and determinants generated by symbolswith one Fisher{Hartwig singularity, Math. Nahr., 127 (1986), pp. 95{124.[5℄ , Introdution to Large Trunated Toeplitz Matries, Springer-Verlag, New York, 1999.[6℄ I. Gohberg, On the appliation of the theory of normed rings to singular integral equations,Uspekhi Mat. Nauk, 7 (1952), pp. 149{156. In Russian.[7℄ , Toeplitz matries omposed of the Fourier oeÆients of pieewise ontinuous funtions,Funkts. Anal. Prilozh., 1 (1967), pp. 91{92. In Russian.[8℄ H. J. Landau, On Szeg}o's eigenvalue distribution theory and non-Hermitian kernels,J. d'Analyse Math., 28 (1975), pp. 335{357.[9℄ L. Reihel and L. N. Trefethen, Eigenvalues and pseudo-eigenvalues of Toeplitz matries,Linear Algebra Appl., 162{164 (1992), pp. 153{185.[10℄ P. Shmidt and F. Spitzer, The Toeplitz matries of an arbitrary Laurent polynomial, Math.Sand., 8 (1960), pp. 15{38.[11℄ P. Tilli, Some results on omplex Toeplitz eigenvalues, J. Math. Anal. Appl., 239 (1999),pp. 390{401.[12℄ L. N. Trefethen, Pseudospetra of matries, in Numerial Analysis 1991, D. F. GriÆths andG. A. Watson, eds., Harlow, Essex, UK, 1992, Longman Sienti and Tehnial, pp. 234{266.[13℄ , Pseudospetra of linear operators, SIAM Review, 39 (1997), pp. 383{406.[14℄ E. E. Tyrtyshnikov, Singular values of Cauhy{Toeplitz matries, Linear Algebra Appl., 161(1992), pp. 99{116.[15℄ H. Widom, Eigenvalue distribution for nonselfadjoint Toeplitz matries, Oper. Theory Adv.Appl., 71 (1994), pp. 1{8.

English

The pseudospectra of banded finite dimensional Toeplitz matrices rapidly converge to the pseudospectra of the corresponding infinite dimensional operator. This exponential convergence makes a compelling case for analyzing pseudospectra of such Toeplitz matrices, not just eigenvalues. But what if the matrix is dense and its symbol has a jump discontinuity? The pseudospectra of the finite matrices still converge, but it is shown here that the rate is no longer exponential in the matrix dimension, only algebraic.

The second and third authors were supported by UK Engineering and Physical Sciences Research Council Grant GR/M12414

Piecewise continuous Toeplitz matrices and operators: slow approach to infinity

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