ETNA
Electronic Transactions on Numerical Analysis.
Volume 17, pp. 1-10, 2004.
Copyright  2004, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
ON THE ESTIMATION OF THE -NUMERICAL RANGE OF MONIC MATRIX
POLYNOMIALS For a given the -numerical range of an matrix polynomial "!
# Abstract.
%$(& !*)+))+! # & ! #-, is defined by ./
0132457698;:=<>?@ABC
=ADE:FG6IHJ=A<4AFK:=<4:L
%$
'
&
;
:M<>AN9O . In this paper, an inclusion-exclusion methodology for the estimation of .P/
0 is proposed. Our
approach is based on i) the discretization of a region Q that contains . / ?0 , and ii) the construction of an open
circular disk, which does not intersect . / 0 , centered at every grid point RPQPST. / 0 . For the cases UV
and XWYWZ; an important difference arises in one of the steps of the algorithm. Thus, these two cases are
PANAYIOTIS J. PSARRAKOS
discussed separately.
Key words. matrix polynomial, eigenvalue,
Wielandt shell.
-numerical range, boundary, inner -numerical radius, Davis-
AMS subject classifications. 15A22, 15A60, 65D18, 65F30, 65F35.
1. Introduction and definitions. Let
and suppose that
(1.1)
[]\
be the algebra of all
^B_%^
complex matrices,
`Xacb'dfe]ghb(i9jlk i7mon b(iBmon0jZpqpqp=jk n bjkBr
^Y_3^ monic matrix polynomial, where kFsKt [ \ avuweyx{zq|zq}~}q}z+€‚|
d
b
is an
and
is
a complex variable. One encounters matrix polynomials for instance when studying systems
of ordinary differential equations or difference equations, with constant coefficients. The
suggested references are [5, 11, 19].
t…„ x{z~|†‡z the ƒ -numerical range of `Xaˆb(d in (1.1) is defined by [14, 15]
For a given ƒ
‰VŠ@aE`dXeŒ‹MbCt 6Ž `Xacb'd‘KeŽx{zJ‘-z  t 6 \ zJ‘ ‘Ke   e’|z  ‘Ke ƒh“ }
‰ Š ac`d is always closed and contains the spectrum of `Xacb'd , that is, the set of all
Evidently,
`Xaˆb(d , ” aE`d5e•‹=blt 6– det`Xacb'd5e—x “ . For ƒ e˜|z we have the classical
eigenvalues of
`Xacb'd
(1.2)
numerical range of
[10, 12], namely,
‰–ac`dN™š‰ n aE`dXeŒ‹MbCt 6Ž ‘ `Xacb'd‘KeŽx{zJ‘t 6 \ z{‘ ‘Ce’| “ }
t’acx{z~|† and `Xacb'd›eœgDb€ ƒ k for some kšt [ \ z then ‰"ŠaE`d coincides with the
If ƒ
k Šack7dež‹ { kF‘  ‘Tz  t 6 \ z{‘ ‘9e 1 e]|@z { ‘Ÿe ƒh“ . For
range of , 
ƒ -numerical
š
e
|
z
k
aEk7d™  n aEkd›e]‹~‘ kF‘  ‘Zt 6 \ zJ‘ ‘Ÿe]| “ [4].
ƒ
the numerical range of is 
k
Moreover, the (outer) ƒ -numerical radius and the inner ƒ -numerical radius of are defined
by
¥ t  Š@aEk7d “
Šack7dNež¡X¢£(‹¤ ¥¦¤  "
Šack7dL«­¬k¬®
respectively. Note also that § ŠaEkdXež¡f¨v©o‹¤ ¥¦¤  ¥VtCª  Š@aEkd “ z
t…„ x{zq|q† [8], where ¬-pq¬q® denotes the
for every ƒ
and
norm induced by the standard inner product.
‰–aE`d has attracted attention, and several
During the last decade, the numerical range
results have been obtained (see, e.g., [2, 6, 7, 10, 12, 13, 16, 18]). These results are helpful
< Received February 21, 2003. Accepted for publication June 30, 2003. Recommended by D. Szyld.
Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece. E-mail:
ppsarr@math.ntua.gr
1
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On the estimation of the -numerical range of monic matrix polynomials
in investigating and understanding matrix polynomials, and some of them have been generalized to the case of the ƒ -numerical range [3, 14, 15]. Furthermore, applications of the
numerical range of matrix polynomials in spectral analysis, factorization and stability of mae x{z then
trix polynomials can be found in [6, 13, 18], respectively. It is easy to see that if ƒ
‰"ŠaE`dI™ 6 . Hence, in the remainder of this note, we assume that x ƒ « | . As a conse`Xaˆb(d in (1.1) is the identity matrix, the ƒ -numerical
quence, since the leading coefficient of
"
‰

Š
E
a
`
d
 connected components [10, 14].
range
in (1.2) is compact and has no more than
of the matrix polynomials
k7b ® iAlgorithms
b i boundary
k7b i
ofothe
jCa3j i 5drange
kz z
oi jPb for
i
oplotting
i j5the
i numerical
i z where
and
the matrices
are Hermitian, can be found in [7, 16, 17]. Moreover, the point equation of the boundary of
the numerical range of a general matrix polynomial is studied in [2]. The numerical approx`Xacb'd in (1.1) is still an
imation of the ƒ -numerical range of the monic matrix polynomial
open and challenging problem. The “brute force” approach would be to plot the roots of the
`Xaˆb(d‘ for a large number of randomly chosen unit vectors ‘Tz  t 6 \ satpolynomial‘ (e ƒ .‰ But
isfying (
be too costly, and it would probably not accurately depict
d . Inwould
Š aE`that
this paper, a methodology for the estimation of the ƒ -numerical
the boundary of
‰ Š aE`d is proposed. This method is the first method for drawing ‰ Š aE`d besides the
range
application of the definition, and can also be used for the approximation of the boundary
ª ‰ Š aE`d . In the next section, we describe the general inclusion-exclusion algorithm, which
‰ Šac`d and a result on open disks that do not intersect ‰…Šac`d
is based on the boundedness of
[12, 14], and requires the computation of the inner ƒ -numerical radius of the (fixed) matrix
`Xa ¥Td for several ¥lt 6 ‰"Š@aE`d . In Sections 4 and 5, algorithms for the calculation of the
e­| and x ƒ |@z reinner ƒ -numerical radius of a square complex matrix for the cases ƒ
spectively, are given. Furthermore, numerical examples are presented to illustrate our results.
For all the experiments, the computations were performed in MATLAB 4.2 on a PC Celeron
600.
`Xaˆb(dKeŒghb i j
polynomial
an ^ _ ^ matrix
k i7mo2.n b The
j—pqp~p{j­algorithm.
k n b3j’k r Consider
š
t
E
a
1
x
q
z
|
†
¥ be a complex
Bi mon general
as in (1.1) and a real ƒ
. Let
X
`
ˆ
a
(
b
d
¥
number,‰ which
not belong to the ƒ -numerical range of aE¥Uz,Ji.e.,
in the¥ open
d with lies
Š ac`d .does
set 6
Then, we can construct an open circular disk center at and
| such that a ¥Uz Jd ‰ Š ac`dP
e . The closure of a ¥Uz{d is denoted by
radius
a E¥Uz Dd .
`Xaˆb(d%eZgDb i jKk iBmon b iBmon jVp~pqp‡jKk n bLjKk r and ƒ t acx{z~|†‡z
T HEOREM
2.1.
Suppose
"
¥
t
‰
E
a
`
d
a ¥Uz{d with radius
Š
6 and let
. Then the open disk § Š aE`XaE¥Td d
e
§ Š@aE`Xa ¥Td+dTj¡f¢£hs n "!"!"! i ¬ s n # `%$ s& aE¥Tdq¬®
‰ Š aE`d .
does not intersect the ƒ -numerical range
Proof. Consider the matrix polynomial
`' acb'dNež`XaˆbPj¥TdNežghb i j Bi mon b i7m n jZpqp~p
j n bPj r
e g . It is well-known that ‰ Šac` dXe]‰"Šac`d€›¥ [10, 14]. Thus, the origin
and denote i
‰"Š@aE` d , or equivalently, x t (  Š@a Br=dGaˆ™  Š@aE`Xa ¥Td+d d . By [14, Theorem
does not belong to
t‰ Š ac`*{dz
1.4 ] (see also [12, Theorem 3.1]), for every
)
we have that
ŠaBrMd
Š § a r dTj…¡X§ ¢£ s n "!"!"! i Š a s d « ¤ ) ¤ }
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Panayiotis J. Psarrakos
Since
Š a s dI« ¬ s ¬ ® (ue­|z zq}q}~};z 
), it follows that
§ Š a r d
« ¤) ¤}
§ Š@a 7r
dTj¡X¢M£hs n "!"!"! i ¬ Fs¬®
acx{zDd ‰ Š aE`'{de Jz or equivalently, aE¥Uz Dd*C‰ Š aE`dLe Hence, '
`
acb'd¦eŽ`Xacbj…¥Td are given by
cients of
Fs3e u| ` $ s & a ¥Td˜ueZx1zq|zq}q}~};z "z
. Since the coeffi-
the proof is complete.
‰ Šac`d is bounded, and thus, we can always find a bounded
The ƒ -numerical range
‰ Š@aE`d . For example, it is known that [14,
region in the complex plane that contains
Theorem 1.4]
‰VŠ@aE`d xPzT|%j s r ¡f¢£ Š ack s d xfzo|%j sor ¡f¢£ ¬k s ¬ ® }
" !"!"!" i7m n ƒ "!"!"! i7m n ƒ ‰ ŠaE`d by plotting its complement with respect to .
Then we can approximate
(2.1)
Algorithm 1 (The general inclusion-exclusion procedure)
‰…ŠaE`d .
Step I Obtain an open bounded region 96 such that
Step II Construct a grid of .
¥"t z repeat the following:
Step III For every grid point
¥
t
(
"
‰
@
Š
E
a
`d;z or equivalently, if x t (  Šac`Xa ¥Td d ,
(a) check
if ac`Xa ¥Td dz
x
t
(
Š

(b) if
then compute the inner ƒ -numerical radius
the matrices
(c)
§ Š aE`XaE¥Td d
and
| $ s &
s e u ` Ea ¥TduPeŽx{zq|zq}~}q}z+Vz
aE¥Uz d Ÿ6 ‰VŠ@aE`d with radius
construct the open circular disk rMd
e Š a r dTj¡X § ¢MŠ£ a 7
s n "!"!"!" i ¬ s ¬ ® }
§
Step IV The set
is an approximation of
‰ Š aE`d
aE¥Uz{d
r $ $ & &
and always contains
‰ Š ac`d .
An important feature of the above methodology is that it does not depend strongly on
 of `Xacb'd , which appears only in the computation of the matrices s (uŽe
the
degree
x{z~|@zq}~}q};z+ ). Note
also that the most expensive part of Algorithm 1 is the calculation of the
ŠaE`XaE¥Td d in Step III (b). This step will be further discussed in
inner ƒ -numerical radius §
Sections 4 and 5.
The two inclusion regions given by (2.1) are not always satisfactory, since they are
centered at the origin. By a simple MATLAB code, one can plot the roots of a few poly`Xaˆb(d‘Tz where ‘Tz  t 6 \ are unit vectors satisfying ( ‘Œe ƒ .
nomials of the form (
e a"!$#&% 'Jz(!#&)(*d _ a i + #&% 'Jz i + #&),*@d
In this way, a first approach of an open rectangle ETNA
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On the estimation of the -numerical range of monic matrix polynomials
! #&% ' z(! #&)(* z #&% ' z #&)(* t
‰ Š aE`d
z
+
+
), which contains
, is obtained. After choosing we
(
can construct either a constant or a variable grid. In our experiments, we use two grids, where
we move rightwards on each grid line and upwards on each grid column. The first grid, dea @d;z is formed by a partition of the intervals a !#&% 'Jz(!#&)(*@d and a + #&% '{z + #&),*@d
noted by n
Jz where all grid points within the disks generated by Step III (c) are
with constant length
®a @d;z
excluded. The second one, denoted by $ by a partition of the interval
a"!#&% 'Jz(!#&)(*d with constant length and a partitionisofformed
a + #&% '{z + #&),*@d with varithe
interval
f
¡

¢
£
‹
J
z
z
able length
“ where is the radius of each disk generated by Step III (c).
`Xacb'd
3. Approximating the boundary. Let
be an ^C_P^ matrix polynomial as in (1.1)
taEx{z~|† . For a ¥ t 6 ‰"Šac`d;z recall the
aE¥Uz d in Theorem 2.1 and
and let ƒ
disk s& aEopen
$
X
¡
M
¢
£
s
¬
`
T
¥
q
d
q
¬
®
n
s
#
n "!"!"!" i
consider a positive real
. Then from the relation
Šac`Xa ¥Td+d
Š@aE`Xa ¥Td+d
Š § @aE`X§ aE¥Td d-j « « § Šac§ `Xa ¥Td d-j | z
it follows that
§ Šac`Xa ¥Td d« |I € z
(3.1)
`Xa ¥Td
and it is clear that the radius
is small if and only if the inner ƒ -numerical radius of
is sufficiently small. ac`Lz ¥Td
¥
‰ŠaE`d . By Theothe distance between and the compact set
Denote now by
rem 2.1, this distance is greater than or equal to . Moreover, we have the following result.
`Xacb'd¦e‚gDb i jYk iBmon b i7m n jŽpqp~p
j
For an ^"_K^ matrix polynomial
k n bTj HEOREM
kFr and3.1.
˜
¥
t
"
‰
@
Š
E
a
`
˜
t
c
a
{
x
q
z
q
|
ˆ
†
z
d defined in
consider
6 ‘'rz r d t and \ the disk aE¥Uzr ‘1 rŸ
ƒ
Theorem 2.1. Suppose that for two unit

6 such that 1 e ƒ and
¤ {r `Xa ¥Td‘1rJ¤@e § Šac`Xa ¥Td+d , we have 1r ` $ n & a vectors
¥Td‘1r eŽx . Then
for sufficiently small ,
a E`Lz ¥TdK« a|I€ { od ¤ ` $ & aE¥Td‘ r ¤ }
 r n
‘'rz  r5t 6 \ satisfying
Proof. Consider the two unit vectors
$ &
x{}
 r ‘1rVe ƒ z ¤  r `XaE¥Td‘(rD¤@e § Š@aE`XaE¥Td d;z and  r ` n a ¥Td‘1reZ
x such that for any bCt a ¥Uz d;z
Then there is a real $ &
 r `Xacb'd‘(re  r ‹
`XaE¥TdojZaˆbX€"¥Td` n a ¥Td-jZ& aˆbX€"¥Td facbTz ¥Td “ ‘1r
e  r `Xa ¥Td‘1rjŽacb €V¥Td  r ` $ n a ¥Td‘(rLj  r facbTz ¥Td‘1r ›z
(3.2)
¬facbTz ¥Tdq¬IeJa|=d
¤ b0€5¥¦¤ x
x , can be chosen so small
r ` $ n & a ¥Td‘1reŽ
a ¥Uz d . Furthermore,
for sufficiently
where
. Since 1
¤ r ` $ n & a ¥Td‘1rJ¤ ¤  asr facbTz ¥Td‘1r{¤ for
b"t
that  every
small , by (3.1), we can assume that
¤  r ` $ n & a ¥Td‘1rj  r facbTz ¥Td‘1r{¤ }
bšt a ¥Uz d;z
b(r t a ¥Uz d such that
Then, since (3.2) holds for every
r1 `Xaˆb{rMd‘(re x1z i.e., b1rt‰"Š@ac`d . Thus, there exists a
Šac`Xa ¥Td d
Š@aE`XaE¥Td d
ac`Lz ¥TdC« ¤ b1rF€"¥¦¤0e ¤  r ` $ n & a ¥Td‘§ r j  r facb r z+¥Td‘ r ¤ « ¤  r ` § $ n & aE¥Td‘ r ¤ z
§ Š@aE`Xa ¥Td+dK«
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5
Panayiotis J. Psarrakos
and the proof is completed by (3.1).
¥ tV
( ‰ Š aE`dz the origin does not belong to the ƒ -numerical range of
Notice `X
that
for any
a
T
¥
d
‘ rz  r3t 6 & \ such that
the matrix
and there are infinitely many pairs of unit vectors
¤
X
`
E
a
T
¥
d
1
‘
J
r
@
¤
e
E
a
X
`
a
T
¥
+
d
d
(
‘
7
r
e
r ` $ n a ¥Td‘1r e x{z
§
1r ƒ and 1r . If one of these pairs also satisfies (
is small if
E
a
L
`
z
T
¥
d
then Theorem 3.1 and the inequality
imply that the radius
¥
‰
@
Š
E
a
`
d
and only if the point is sufficiently
to the boundary of
. Recall also that the
¡X¢M£Ds n "!"!"! i ¬ s n # ` $ s& aˆb(d~¬® isclose
function
bounded. Hence, if we choose a sufficiently small
x{z then we can approximate the boundary ª ‰…ŠMaE`d by using Algorithm 1 and plotting
a ¥Uz d ‰"ŠaE`d with radius (see Examples 4.2 and 5.1 below).
the disks e |
`Xaˆb(dLe‚ghb i jYk
b i7mon jŽpqp~p=jlk bPjYk r
i7m n
n
4. The case ƒ
. Let
be an ^ _K^

e
|
ƒ
monic ‰–
matrix
polynomial
and
let
.
It
is
clear
that
for
the
estimation
of
the
numerical
ac`dPa‡™œ‰ n aE`d+d via the algorithm in Section 2, a method for the computation
of
range
the inner numerical radius of an ^ _K^ complex matrix is needed. By [1, Theorem 2.1] (see
k’t [ \ is given by
also [4, Chapter 1.5]), the inner numerical radius of a matrix
§ aEkdN™ § n aEkdNe
¡f r ¨v© ®
b #&),* ap d
b #&),*
kYj m i k i
z
where
denotes the maximum eigenvalue of a Hermitian matrix. As a consequence,
aEk7d and classifies the origin as belonging to  aEkd*™
the following procedure produces §
E
a
k
d
 n
or not.
Algorithm 2
Step I Construct a grid
„ x1z †
of the interval
Step II For every choice of
tian matrix
e u  uPeŽx{zq|zq}~}q};z 
€Y|
some positive integer .
z forcompute
b#&),*{aa d d
the largest eigenvalue
of the Hermi-
dNe | i k9j m i k }
b #&),*{aa d d and the corresponding angle r . The absolute
Step III Find the minimum of
aEkd .
value of this minimum is an approximation of the inner numerical radius §
E
a
k
d
Moreover, the origin does not belong to the numerical range 
if and only if the
b #&),*{a a r=d d is negative.
eigenvalue
a
The above algorithm can be used for Step III (a),(b) of Algorithm 1. As a consequence,
‰–ac`d , which
we have a complete methodology for the approximation of the numerical range
is illustrated in the following examples.
E XAMPLE 4.1. For the matrix polynomial
€ | 5
€ | ® j x i
` n ˆa b(dNe]ghb%j 5
x € b | 5
€ |
bj |
x
i!
"
z
‘ ` acb'd‘ aE‘œt
the® roots of a few thousand (randomly chosen) polynomials of the form n
Observe that we do not have a clear
6 z{‘ ‘Ce ‰–|=daE` ared;z sketched in the left part of Figure
` acb'4.1.
d (marked
n
picture of
and that three eigenvalues of n
‘+’) appear to lie out of
‰–aE` n d . Moreover, it seems that ‰–ac` n d lies in the open rectanglewith
e a €$#{z#hd _ a € i }%Jz i #d .
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4
4
3
3
2
2
Imaginary Axis
Imaginary Axis
On the estimation of the -numerical range of monic matrix polynomials
1
0
1
0
−1
−1
−2
−2
−3
−4
−3
−2
−1
0
Real Axis
1
2
3
−3
−4
4
−3
−2
−1
0
Real Axis
1
2
3
4
F IG . 4.1. A numerical range with three connected components.
acx{} x d;z
3
3
2
2
1
1
Imaginary Axis
Imaginary Axis
For the grid n
by Algorithms 1 and 2, an approximation of the numerical range
‰–aE` n d is drawn in the right
of the figure. The number of disks is 1076, and now we
‰–ac` n d has three part
aE` n d in its
can see that
connected components and contains the spectrum ”
interior.
0
0
−1
−1
−2
−2
−3
−4
−3
−2
−1
0
Real Axis
1
2
3
4
−3
−4
−3
−2
−1
0
Real Axis
1
2
3
4
F IG . 4.2. A connected numerical range.
E XAMPLE 4.2. For the matrix polynomial
`-®acb'dNešgDb j
|
€5|
b® j
x
| 5
€ i|
bPj
|
€$#
x
z
‘ ` ® aˆb(d‘Za ‘yt
the® roots of a few thousand (randomly chosen) polynomials of the form 6 z1‘ ‘9e |
d are
the left part of Figure 4.2. We have a clear picture only for
‰–acsketched
` ® d and anin eigenvalue
` ® acb'd (marked with ‘+’) appears to lie out of
the
left
part
of
of
‰–aE`®=d . Using our methodology and choosing
acx{} x d of the inclusion domain
the grid $ n
e
a
$#{z #d
€
a
€
z
%@d;z
}
‰–ac` ®=d is drawn in the
_ i i
an estimation of the numerical range
–
‰
E
a
U
`
=
®
d
right part of the figure and gives a better visualization of
. The total number of disks
aE¥Uz d is
3812. With respect to the same grid, in Figure 4.3, we plot exactly the open disks e x1} x ! (left part) and e x1} x (right part). In this
‰–aE`®
d with radius for
–
‰
E
a
`U®
d , confirming Theorem 3.1 and the discussion in the
way, we approach the boundary of
previous section.
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3
3
2
2
1
1
Imaginary Axis
Imaginary Axis
Panayiotis J. Psarrakos
0
0
−1
−1
−2
−2
−3
−4
−3
−2
−1
0
Real Axis
1
2
3
−3
−4
4
−3
−2
F IG . 4.3. Approximating the boundary of
−1
0
Real Axis
1
2
3
4
.…> .
`Xacb'd
4
4
3
3
2
2
Imaginary Axis
Imaginary Axis
The above method for the estimation of the numerical range of
be considered
‰–aE`d is a regular closed set, i.e., when it coincides with can
reliable when
the
closure of its
ª ‰–aE`d is smooth. If ‰–ac`d is not a regular closed
interior, and when the boundary
set and/or
ª ‰–aE`d contains non-differentiable points, then our algorithm may become less satisfactory
(locally), as one can see in the next example.
1
0
1
0
−1
−1
−2
−2
−3
−5
−4
−3
−2
−1
Real Axis
0
1
2
3
−3
−5
−4
−3
−2
−1
Real Axis
0
1
2
3
F IG . 4.4. A numerical range with line segments.
E XAMPLE 4.3. The boundary of the numerical range of the quadratic matrix polynomial
` aˆb(dNešghb ® j
€
x
|
x
i
x
i x
|
bj
| |
x x
| x (with Hermitian coefficients) is accurately sketched in the left part of Figure 4.4 by an alea€$#1} {z }%@d
gorithm described in [7]. In the right part of the same figure, choosing a€ i }%Jz i #d and its grid ®haEx1} x #dzB‰–aE` d is approximated by Algorithms 1 and 2 (the_
` acb'd are marked
number of disks is 8356). In both parts of the figure, the eigenvalues of ‰–aE` d intersects
with asterisks, and we remark that the non-real part of the boundary of
the real axis orthogonally [7, Theorem 5.2]. Clearly, the existence of real intervals and nonª ‰–aE` d affects the accuracy of the methodology.
differentiable points on the boundary
ETNA
Kent State University
etna@mcs.kent.edu
8
On the estimation of the -numerical range of monic matrix polynomials
x |
x |
ƒ
ƒ
. In this section, we assume that
. In [9], Li and
5. The case
ƒ
9
^
_ ^ complex
Nakazato
describe
an
algorithm
for
drawing
the
-numerical
range
of
an
k
matrix . Their method is based on the construction of the (convex) surface of the Davisk
Wielandt shell of , that is,
‰3aEkdNeŒ‹haE‘ Fk ‘-z ‘ k kB‘'dLt 6‚_
and the relation
 ‘3t 6 \ {z ‘ ‘*e | “ z
z
ƒ')
a|I€ ƒ ® qd af€ ¤ ) ¤ ® d  a ) z(dtCª ‰ 3aEkdPz
ŠaEkd and for verifying
and can be used for the calculation of the inner ƒ -numerical radius §
@
Š
E
a
k
d
whether the origin belongs to 
or not.
(5.1)
 Š ack7dNe Algorithm 3
Step I Construct a grid on the unit sphere in
a
+¨v©
with e
and
Že
z dNe
d
z qz }~}q}qz a €9|
d z d;z repeat the following:
d d of the Hermitian matrix
a +¨© ' d kYjl k jKa
+ ¨v© +¨©'d k € k jKa dqaEk
for some positive integers
and .
a ¨©
-z +¨v© +¨© -z
Step II For every choice of
b #&),*Ja a z
(a) compute
the largest
eigenvalue
a
using spherical coordinates
Tz ¨© +¨v©-z z ~z }q}~}z a ž€Y|=d z a corresponding unit eigenvector  ) a z'dXe  k  t 6 \
and
kd;z
and the scalars
Ta z dNe
(b) consider the closed circular disk
3a z dNe ƒ') a
¤ a z dq¤ ®
and notice that if )
Š ack7d .
belongs
to 
3a
Step III By (5.1), the union 3a
z
;
d
z
x
then we say that
i
 k k  z
z dz a |I€ ƒ ® d a-a z'dU €Ÿ¤ ) a z d~¤ ® d z
Š Ta z d0€Ÿ¤ a z d~¤ ® z
« n>m Š )
then the origin
z d is an approximation of  Š ack7d . If x t (
t (  Š ack7d and € a|I€ ƒ ® d a Ta z d € ¤ ) a z dq¤ ® d }
§ Šack7de ¡f ¨© ƒ ¤ ) a z dq¤M
`Xacb'd is a monic matrix polynomial as in (1.1) and x ƒ ® ƒ n « |@z then it is known
If
V
‰
Š aE`d –‰VŠ aE`d [15, Theorem 10]. This result is confirmed by comparing Figure
that
4.1 with the figure in the following example.
` acb'd in Example 4.1. In the left part
E XAMPLE 5.1. Recall the matrix polynomial n
® of a few thousand (randomly chosen) polynomials of the form
of Figure 5.1, the roots
Observe the gaps around
1 `Xaˆb(d‘…aM‘Tz  t 6 ` z{ac‘ b' d ‘Ye 1 e|z 1 ‘Yeyx1} " d are
‰Vr plotted.
! ac` n d appears to lie in the open
two eigenvalues of n
(marked with ‘+’) and that
!
ETNA
Kent State University
etna@mcs.kent.edu
9
3
3
2
2
1
1
Imaginary Axis
Imaginary Axis
Panayiotis J. Psarrakos
0
0
−1
−1
−2
−2
−3
−3
−2
−1
0
Real Axis
1
2
−3
3
−3
−2
F IG . 5.1. The -numerical range of
e a €$#{z #d
a€ 1# z # d
−1
0
Real Axis
1
2
3
& ? @ .
square _ i i . In the right part of
using Algorithms 1 and
‰"r Figure
aE` n d 5.1,
!
3, we sketch an approximation of the boundary of
by drawing
638 open disks
a ¥Uz d ‰w r ac‰w` r d ! aE` n d with centers in ®haEx1} xd and radius x1} x . It is easy to
! n is connected with smooth boundary and contains the numerical range
verify that
‰–aE` n d in Figure
4.1.
Algorithm 1 is simple and robust, but it is also expensive (even for medium sized matrix
polynomials) because of the required computations in Step III (b). Furthermore, Algorithm
2 is based on polar rotations (two dimensional) rather than spherical rotations (three dimensional), which are required by Algorithm 3. Consequently, the cost of Algorithm 3 is much
higher than the cost of Algorithm 2. So, the problem of the design of less expensive algorithms for the computation of the inner ƒ -numerical radius of a general square matrix is still
challenging.
!
!
!
REFERENCES
[1] S.-H. C HENG AND N. H IGHAM , The nearest definite pair for the Hermitian generalized eigenvalue problem,
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ETNA
Kent State University
etna@mcs.kent.edu
10
On the estimation of the -numerical range of monic matrix polynomials
[14] P. P SARRAKOS AND P. V LAMOS , The -numerical range of matrix polynomials, Linear and Multilinear Algebra, 47 (2000), pp. 1-9.
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