ETNA
Electronic Transactions on Numerical Analysis.
Volume 23, pp. 329-338, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
NUMERICAL COMPUTATION OF THE EIGENVALUES
FOR THE SPHEROIDAL WAVE EQUATION WITH
ACCURATE ERROR ESTIMATION BY MATRIX METHOD
YOSHINORI MIYAZAKI , NOBUYOSHI ASAI , DONGSHENG CAI , AND YASUHIKO IKEBE Abstract. A method to compute the eigenvalues of the spheroidal wave equations is proposed, as an application
of a theorem on eigenvalues of certain classes of infinite matrices. The computation of its inverse problem (namely,
solving another parameter for given eigenvalue ) is likewise given. As a result, precise and explicit error
estimates are obtained for the approximated eigenvalues.
Key words. spheroidal wave equation, eigenvalue, numerical computation, error estimate, infinite symmetric
tridiagonal matrix
AMS subject classifications. 34L16
1. Introduction. The spheroidal wave equation
(1.1)
'
%$
$
!#"
&
'
(
*),+-
with an integer and real, is obtained when the Helmholtz equation is expressed in prolate
$
./$ by separation of variables. In the oblate (spheroidal coordispheroidal coordinates
followed
'10 +
nates)'3
case,
is replaced by
in (1.1). In this paper, we assume
and the prolate
2 +
case (
or the oblate case may be dealt with likewise).
The eigenvalue problem of the
5
7 8
spheroidal
wave equation is to find 4#" such
that
, the solution of (1.1), is regular 5in
6
9
. We call !#" an eigenvalue and
a spheroidal wave function denoted by :!; "
.
!#" is real. Eigenvalues sorted in an increasing order correspond to
As will be stated
later,
) '
) '
having <
,<
, ==&= .
The spheroidal wave functions occur in the solution of equations involving separation
of variables in spheroidal coordinates. The solution of the spheroidal wave equation plays
a significant role in the study of light scattering problem in optics, atomic and molecular
physics, and the like. Although there is abundant literature on this function, there are far
fewer publications on the eigenvalues of the differential equation. The paper [4] applies
an asymptotic iteration method to calculate the angular spheroidal eigenvalues, whereas [7]
$
discusses different methods
for the solution of the differential equation depending on the
value of the parameter .
In the present paper$ we propose a matrix method for determining the eigenvalues, which
applies for all values of , and we furthermore obtain an accurate estimate of the error of our
$
approximation. In addition,
our method also enables the solution of an inverse problem, i.e.,
the determination of corresponding
to a given eigenvalue.
9
The expansion of :!; "
by the associate Legendre functions gives
(1.2)
Q
:4; "
5>)@A ?
BDCFEHG
"9I BDJLKNMO
J BDJLK
5P-
Received January 4, 2006. Accepted for publication March 9, 2006. Recommended by F. Stenger.
Faculty of Informatics, Shizuoka University, Johoku 3-5-1, Hamamatsu, Shizuoka, 432-8011, Japan
(yoshi@inf.shizuoka.ac.jp).
Department of Computer Software, The University of Aizu, Tsuruga, Ikkimachi, Aizuwakamatsu, Fukushima,
965-8580,
Japan (nasai@u-aizu.ac.jp).
Institute of Information Sciences and Electronics, The University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki,
305-8573, Japan (cai@is.tsukuba.ac.jp).
Shimo-hirooka 702-16, Tsukuba, Ibaraki, 305-0042, Japan (ikebe@ibaraki.email.ne.jp).
329
ETNA
Kent State University
etna@mcs.kent.edu
330
O
J BDJLK
Z
where
is divided by .
5
is the
5
X
associate Legendre function and mod
is of the form
O "
/
5>)
O "
Y. MIYAZAKI, N. ASAI, D. CAI AND Y. IKEBE
' -WVDR )*SUT
<
YX
-[Z9
\ 9
O "
is the remainder when
9^]`_ba&cdae!fahg!Tikj9eTSUlnmdin
O "
oqp
with orthogonality over [-1,1],
p O s
r o p
9
pzy O
"
r
vuNw
)*xP-
9
t),+
O B
V
9|{}
t)
V
<
<
M
'
D~
'
<
D~ =
It is known that the substitution of (1.2) into (1.1) gives the following three-term recurrence relations [1, Chap 21, formula 21.7.3]. For convenience, let "9I B and !N" be simply
G
rewritten as B ' and , respectively, in the sequel.
G
When <
is even

E

(1.3)
‚ B
'
When <
B -
The symbols €
G
x)*+-
V '
'
Š*x B
x
Vdx
x '
p
BDJ
G
-WV-
)
G
p
p
E G
)
BDJ G
p
ˆ
Vd V '
V '
x
x
G‡† )
BDJ
G
†
Vdx
q
$
$
$
p x
=&== =
x)
V '
‡Š
-WV-
-WV-
=&== =
x„ŒŽP-
Vdx
-
BDJ
G
‹
x '
G
‡Š
‰
'
V '
V
$
Vdx
x„)
p
)
in the equations represent
B
G
p €
G
€ BDJ
xŒ3…D-me!
q
x x
V '
Vdx
V '
ˆ
)
‚ B
G
p
BDJ
V '
B )

, and ‚
B
G
B ƒ€ B

p
 B r
)
€ B

(1.4)
G
B
is odd
p
‚ BDJ
B r E € E
V '
Vdx
q
ƒˆ
$
x„ŒŽP-
‹
==&= . Since the two cases are treated in the same way, only the former case will
for
be discussed in this paper.
+-
2. Behavior of expansion coefficients. In this section, we discuss the behavior of y
-|V
=&== of (1.3).
First, let us begin with the next inequality:
(2.1)
A?
BDCFE G
B M&‘
'
-WVx 2
This is easily o
shown
as
p
*-“’”afWa
r
p}y
'
V
-•>)
V '
Vd•
V '
•D~
•4~
M
=
o…p
:!; "
‘
9|{z&
t)
p
)
r
A?
BDCFE!G
A ?
BPCFE!G
y
B O
J B
V
B`M V '
5D{z
‹ x
M
V '
VdxP~
2
VdxD~

=
G
{
B
x)
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NUMERICAL COMPUTATION OF EIGENVALUES FOR THE SPHEROIDAL WAVE EQUATION
331
Next, consider the recurrence relations with the same coefficients as (1.3), or

‚ B–!B
r
p
(2.2)
p
)
€ Ez– p
)
B–4B € B–4BDJ
E}–

p
–
x„),V-
–!B
- ‹ ˆ
==&= =
By [6, Theorem 2.3, case(a)], the existence of two independent solutions of (2.2) (say,
is guaranteed with the behaviors
—˜|™Hšœ› š
—˜|™Hšœ›
£ —9˜› q
 ž
¥¤ “¦¨§d©ª P«v¬Y­
4¤ “¦®§d©ª P«v¬¯±°/©Y¢²¯±³´«µ
¡&¢
D
{ x¶)·+- |
- V
obvious that y B
=
= = shows the same behavior
G
—˜› š1Ÿž |¡z
£ ¢
From (2.1), it is
the minimal solution of (2.2). Thus,
(2.3)
p
{ - y –4B {
y –4B 5
I
I
BDJ G
G
$
)
&¸
B
6
x .
ƒ¹
8F
x„Œ3
as
y – B {
I
,
=
3. (Ordinary) eigenvalue problem. In this section, a computation of eigenvalues is
shown, along with its error estimate.
In 1995, the following theorem was proved by Ikebe et al.:
pÁÀcomplex symmetric tridiagonal matrix
[9, Theorem 1] Given a non-compact
ÂÄÃ
¿
À
)¼»½
º
½
¿
†
+
+
À
À
½
½¾
¿
Ã
Ã
Ã
†
. .†
..
.
..
.
]Æ
º >ŒÈÇ -
Å
p.Ó
Ó
8ÕÔ
À
Œ

xŒ3…D-+ 2ËÉ B É92ÍÌ Tde!ÎÏ x),V- P-Æ
]
º >) y 6 ÐÒÑ - ÐLÑ ÓÉ 3
B
8ÕÔ×Ö
w
w
p
ˆ ==&=
where
===
6 ¿ ÐÒ
Ç {
)@+
+ )@Ø3) p
º
p.Ó Ñ -¿ Ó ÐÒÑ 8Õ-Ô
. Let
have a simple eigenvalue , and
==&=
6 Ø Ñ -.Ø Ñ º r
be an eigenvector corresponding to , and assume the existence of
.
===
YX
Then
x…)
-WV
x
º
º
Letting B
=&== denote the -th order principal submatrix of , there is a
{
º
p
Ó
Ú
Ó
y
Ô
YXªX
p
À
w B
sequence
of eigenvalues
of B Ñ B which converges
to .
Ñ BDJ
Ø
Ø )Ù+
Ø
ŒÛ+
x܌@
BDJ Ø
p.Ó +
2ÊÉ ¿
Letting
estimate:
.
and
, we have the following error
À
.is
)
BDJ
Ó
p.Ó
8Ý
Ø Ñ B Ø Ñ BDJ
6
xŒ3
Ô
¹
Ø
Ø
=
Ç …Þ
y 6R
p
- R ==&=
x…Œâ
8ÕÔ
p
] R -p R ===
º r
Ö
the complex Hilbert space
¹
,
is a
quantity
converging to zero as
, and
exists if
„)Ê+
)Ê+
º
is the only solution of
. Also, an eigenvalue is said to be simple if and only
if its corresponding eigenvector is unique (up to scalar multiplication) and also there are no
corresponding generalized eigenvectors of rank 2. These definitions are retained throughout
this paper.
This theorem is applicable to the computation of eigenvalues of the spheroidal wave
$ãw
w
equation.
'
)¥+
)ä+
T HEOREM 3.1. Given and
,
is an eigenvalue of (1.1) if and only if is
º
an eigenvalue of an infinite symmetric tridiagonal matrix acting as a linear transformation
Ç from å into defined below:
ß -|à
In this
p É theorem,
É 2
Ç B
p
á? C
R á Ÿ{
)
ETNA
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etna@mcs.kent.edu
332
Y. MIYAZAKI, N. ASAI, D. CAI AND Y. IKEBE

E
)
º
(3.1)
æ
»½
æ
‚ € E æ
½
æ
½
½¾
)
‚ € æ
‚9ç
Ö
ÂÄÃ
+
æ
Ã
Ã
‚5ç
€  æ
+


8
Ç ]dlnmdc 6 E =&==
Ö
y 
€ E æ
Ã
ç
..
.
..
..
.
.
Å
Ç {7è…Ç w
)Íé
where å
.
+
º
Moreover, if one lets an eigenvector of corresponding to be
å , or
(3.2)
holds,
é Ñá
º éê)
Ó YX
)
-WV-
==&=
é Ñá
(3.3)
p
ár
ë M&ì í C
)
p æ
p.Ó
6é Ñ
Þ
-é Ñ Ó
-
8ÕÔ¶Ö
==&=
é
Dw
)*+
are expressed with a scalar ë
Ó
-
as
€ í r á r =
æ ‚ í G
w
Ö
Proof. (1.3) in matrix form, with its symmetrization, directly
becomes
an eigenvalue
é )*+
é
º
What remains
problem
of
,
namely,
(3.2).
to
be
proved
is
and
å .
éî)ï+
)ï+ xã)ð+- -WV

p.Ó
Ó
Ö
means B
==&= , which is the trivial solution of (1.1) (which we
É
É Éé Ñ É
É
É Éé Ñ É
2 
é
é
)
G
E
MM&M
omit). In order to show
), one only
å (or ñ ñ

ò
has to prove
p.Ó
Þ
é Ñ BDJ
p.Ó Ú
(3.4)
Ó
ÑB
(3.5)
Ñ BDJ
ÑB
Ó
B
õ
õ
õ
?
p
B r
)ø÷ ì á CFE p € á
B p
ì á C r ‚ áù
Moreover, the
ratep.of
Ó convergence is

ÎôgUõ
õ
B r õ
õõ
é Ñ BDJ
õ
õ M õ
Ó
é ÑB
Ú 
õ 2
õ
=
õ
É Œ
B
(1.4).
r
from (2.3) and
from
x )
-WV
x
º
Ó
Ó
==&= . Then, one
be the -thº principal
submatrix
of
ÑB
ÑB
Œ
. And the following error
eigenvalue of
, such that é Ñ B Ó Œö+
õ
º ÑB õ
This holds since
õ
Ó
T HEOREM 3.2.Ñ B Let
can choose each , an
estimate is valid:
õ
Ó
ni lkS
BPó
)ú€ B
‚ B
BPJ 6
M G
B r G
õ
B
B rÔ B 6
é éG
M G
É
¹
8
)
õ
$
&¸
(
6
M x ç
8b
¹
xŒ3
¹
8b
=
x„ŒŽ
p
=
Proof. It only has to be shown that [9, Theorem 1] can be applied to the eigenvalue
º r
º
ûýü is easily
problem (3.2). To begin with, the existence of
need not be verified since
+
®
x
)ä+-|V º
‚
w þ
M&MM ,
û . Since € B M Ô BDJ proved to have an inverse
for
appropriately
taken
é
é
é )ä+
Ô
is real. So are and (up to scalar multiplication). Therefore
, é which
should lead
é )Ê+
é
is simple. To prove this, take the contraposition, or ‘ is not simple ÿ
’. First,
º
Ô multiplication. Therefore, it suffices to show that
w when
is uniquely determined up to scalar
é é„),+
é ),+
has an eigenvector of rank two,
holds. From the assumption, for given
, there
exists a vector such that w
”9ôÎ&-é
Ô
+
)*éU)
º
ü
éê) y º
Ô !ü
) º
ü
{
Ô º
º
j
ü
9é)
º
!ü
!ü
ϔa#Î.j9SUSUaPÏWfj7T º ^),+
),+
=
=
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NUMERICAL COMPUTATION OF EIGENVALUES FOR THE SPHEROIDAL WAVE EQUATION 333
pÓ Ú
Ó
À
p BDJ
Ñ
Ñ B Œö+ xŒ3
D
B
J
. This is, however, clear by (2.3).
What is left to show is
Hence, [9, Theorem 1] may apply the case of (3.2) and direct computation gives the error
estimate (3.4) and rate of convergence (3.5).
4. Inverse eigenvalue problem. In Section 3, the computation of (eigenvalue) was
$
presented. In this section, its inverse
case is discussed.
'
9
7 8 4.1. Given
, let any value of
D EFINITION
and
real
such
that
of (1.1) is
6
regular in
be called an inverse eigenvalue.
as Section
Let us begin,
3, with the prior result:
‹
p
À
complex
symmetric
tridiagonal matrix
[8, Theorems = and = ] Given an infinite
ÂÄÃ
¿
À
)
»½
½
G
½¾
+
Ã
†
..
¿
†
Ã
Ã
À
½
+
À
¿
†
..
,
Å
..
.
V- pÓ
Ó
B
w
8 Ô*Ö
ˆ w ==&= , representing a compact
where
, and B
)¥+
Ç + )¥Øq) 6 Ø Ñ -Ø Ñ Ç operator in . Let have a simple eigenvalue , and
=&==
X
G of corresponding to . Under the stated assumptions, we have
denote
an eigenvector
xƒ)
x
G -|VB
Letting
=&== be the -th order principal submatrix of , and B be an
{
Œ
pÓ Ú G
Ó
G
YXªX
YX
Ô we whave B
.
eigenvalue
of B . Then, taking y B properly,
{
Ø
Ø
,
)
+
Ø Ñ BPJ
Ø ÑB
y G B
Assuming that is taken
in the sense of ,
, and
is bounded
x
for all sufficiently large , we find the following estimate valid:
¿ B
Œ
+-
À
Œ
+
x,Œ
À
)
w
+
xð)
Ó
p.Ó
8Ý
Ø Ñ B Ø Ñ BDJ
6
xŒ3
Ô
Ø
Ø
¹
=
p
BDJ
)
B
À

.
.
‹
By applying [8,$ Theorems = and = ], the inverse problem, namely, • the-
computation
of
-
inverse eigenvalues is enabled. Beforehand, let us define new symbols B B B as
(4.1)
and also B

$
)Ÿ• B
€ B
-
as
B
(4.2)
Í
)
B )
'
'
x '
x '
x
x
>Š
$
x
$
- ‚ B ) B
B
xŒ3
-
=
Using (4.1) and (4.2), (1.3) is then rewritten as follows:
(4.3)
B
G
G
B r E
B
G
B • E
E •
w
)*+
B
Suppose B
$
)
G
)
BPJ G
E
$ B
E G
G
Ú
(4.4)
æ
G
)
»½
E Ú E
• E æ E ½
æ
æ
½
½¾
Ú
• E Ú æ
+
•
E Ú ç
-WVˆ
-
==&= =
$
w
'
-WV
)Ÿ+
Ú $ =&== for given
and real . Then,
is Ç an eigenvalue of an infinite symmetric
x),+-
T HEOREM 4.2.
is an inverse eigenvalue of (1.1) if and only if
tridiagonal matrix acting as a compact operator in
G
x)
B
æ
ç
, where
æ
ç
..
Ã
Ã
æ ç
Ú
ç
ÂÄÃ
+
Ú
•
.
ç
Ã
..
.
..
.
Å
=
ETNA
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334
Y. MIYAZAKI, N. ASAI, D. CAI AND Y. IKEBE
Ú $
Furthermore,
if one lets an eigenvector of corresponding to
Ç G
, i.e., if
(4.5)
holds, then
Ó YX
Ñá
)
-|V-
G
=&==
w
6
Þ
p.Ó
Ñ
-
Ñ
Ó
-
8 Ô¶Ö
=&==
) æ
ë M
p
ár p æ
á
r ì í C
•
í r æ
w
)Ÿ+
)
can be expressed in the form
Ó
Ñá
(4.6)
$
)
be
+
í
ár G
where ë
is a scalar.
Ö
Proof. (4.3) in matrix form, with its symmetrization, directly
becomes an eigenvalue
Ç problem of p.Ó , namely, (4.5). Also, from (2.3) it holds that
,
since
$
õ
(4.7)
õ
õ
Ñ BDJ G
Ó
ÑB
õ )
õ
õ
õ
õ
Ô T HEOREM
assuming
eigenvalues of
G
$
(4.8)
¶$
•
8
B r õ õ
B
z¸
6
Œö+ x„ŒŽ
õ )
M
õ
õ õ G
õ
x ¹
=
æ B
õ æ B r õ õ B r õ
Ó
õ
õ õ G
õ
ÑB
x
-WV
Ó
Ó xq)
w 4.3. Let
$
Ú submatrix
be the -th principal
of Ñ B
==&= .
Ó )
Ñ B
)
+
G
G
Ó , one can choose
ÑB
each
, where
represents one
$
$
Ó Œ
ÑB
, such that Ñ B
. And the following error estimate is valid:
ÑB
u
‡I B
æ
õ
B
/$u
Ó
)
)
V '
æ
M
8b
B 6
xŒ3…D-“’”afWa
I B Ô B r ¹
G
G
V '
V '
V
V '
V '
M ‹ x
‹ x
‹ x
ˆ
The rate of convergence
derived:$
$ is%thus
$
(4.9)
p.Ó Ú
$
p.Ó
¶Ñ $ BDJ
Ó
ÑB
)
ç
6
M x ç
‹
.
ƒ¹
8L
xŒ3
p.Ó Ú
Ó
J B Then,
of the
J =
Ó
=
‹
= and = ] to
É Œ×+ x¨Œ Ú 
$
Proof.
Let us prove a few conditions to apply
[8,É Theorems
(4.5).
ÑB
Ñ BDJ
ÑB
x
w large , since
Ô is bounded for all sufficiently
from
)ø+
(4.7). Also, from the assumption
, it is derived that the eigenvalue
of is
G
simple, in the same way as the proof of Theorem 3.2.
Direct computation leads to the error estimate
(4.8) and rate of convergence
(4.9).
)±+
‹
L EMMA 4.4. [8, Theorems = and = ] is applied also when B
for some nonx
negative integer in
(4.3).
'
'
x )ú+
)
)+
)
E
Proof. When
, or , the first line of (4.3) yields E
G
. Substituting this into the second line of (4.3) gives
Ñ BDJ
G
• E E (
G
•
G
ç )
$
G
=
With this and the subsequent equations, one can reformulate them into the matrix eigenvalue
p problem byx the same procedure.
)
• B
N)*+
B
q
B
B r BDJ When
is non-zero,
holds. Substituting this
x
x
G
G
into (
)-th and ( G )-th equations yields
•
•
B r B
B r B
)
B
r BDJ (
B
B
G
G
G
•
B BDJ B BDJ •
B J B J ç )
B r BPJ D
BDJ D
(
B
B
G
G
G
B r B r ç •
B r $B r B J $D
G
G
B r BDJ =
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NUMERICAL COMPUTATION OF EIGENVALUES FOR THE SPHEROIDAL WAVE EQUATION
‹
Likewise, [8, Theorems = and = ] applies to the new set of equations.
$
$
335
5. Geometrical properties ' of – graph. The - graph, created by the proposed
)*+
method, is shown in Fig. 5.1 (for
). The graph above -axis is the prolate output, while
the graph below is the oblate case.
F IG . 5.1. - graph (prolate and oblate cases combined for !#"%$ )
$
The following two geometrical properties are
proved
regarding graph.
0q'
'
R EMARK 5.1. In the prolate case, holds.
2…'
'
ÍSuppose
contrary, or , and let the contradition be derived.
Proof.
the
'
Vdx '
Vx
2 + x)Ÿ+- -|
V B ) =&== , the eigenvalue problem
&
Hp
)
$
Fp
-“’”afWa
(5.1)
&
)
r( '
)
) *
) r(' ) r+' »½
)
½
)
)
½
Hp
)/. æ
*
r+' )
)
*-,
r ' ,
+
r(' )
)
E
æ
E G
æ
• E ..
æ
G
•
- æ
P
æ
Ã
Ã
.
• E æ
ç
Ã
Ã
r(' ,
+
ÂÄÃ
+
) *-,
r(' ) r+' ,
,
)
r(' ½
½¾
)
r(' )
æ
æ
..
.
..
.
me
Å
Ô
ç Dç
G
=== 0
Since
ETNA
Kent State University
etna@mcs.kent.edu
336
Y. MIYAZAKI, N. ASAI, D. CAI AND Y. IKEBE
are both real, is obtained.
&
is found to be decomposed into
+
»½
;
½¾
æ
)ðx
O B
M76
5
V
SUT
ç†
q
x
V '
xH-|Vd
Vx
, where
Ã
-“’lÄÏW”
;43
..
+
)
; B
1
Ã
;
½
Ô
ÂÄÃ
+
; 1¶)
)21
&
V '
ˆ
..
.
V '
Å
.
x
Vdx
xU)ðVˆ
-
===
Lmde
=
Ú
&
7 A $ for a detailed proof), leading to a
This shows that is positive definite (refer to Appendix
&
2 +
Ú
$ are
w
contradiction (since the eigenvalues
of
assumed
).
'
¿ ¿
)*+
R EMARK 5.2. Given , we have
=
Proof. The next equation is obtained by applying [11, Corollary 1],
$
Ô (
M
)
$
M
é
é
Ô
with and respectively defined in (4.6) and (3.3). Since
the proposition obviously holds.
éDé
Ô
é
w
)ï+
é
( is non-zero and real),
6. Numerical experiments. Some experiments were conducted to show the validity of
the error estimates presented in (3.4) and (4.8). The computations were executed on Dell Dimension XPS (Pentium 4 CPU 3.00GHz, 1GB RAM), using double precision floating-point
arithmetic by Intel Visual Fortran Compiler (version 8). For computing eigenvalues of symmetric tridiagonal matrices, we used the FORTRAN subroutine COMQR
$ in EISPACK [12].
'
and . We first computed
E XPERIMENT 6.1. Computation of eigenvalues , given
Ó
an eigenvalue of a sufficiently large order principal submatrix of (3.1)
and regarded it as the
x
º ÑB
Ó
Ó true value .Ñ B Then, for each , we
computed
all
the
eigenvalues
of
and chose the closest
é ÑB
x%)
-WV
Ó
pÓ
w
=&== were obtained by backward-substitution
to to be . Theé Ñ9values
of
8
)3+
é Ñ98 r
)/: Ó )Ž+-:
pÓ
‘
by (1.3), initiating
for sufficiently large , and
(
shall be
é Ñ98
é Ñ98 r
Ó
doesn’t occur). These settings of
taken appropriately so that an
overflow
and
are
é ÑB Œ
+ xŒŽ
Ó
allowed from the behavior
.
ÑB
Ó
eigenvalues
Table 6.1 describes how fast the approximated
approach the exact eigenÑB
x)
-WV
Ó
Ó
8 estimates. In the table, ; #"
value with
error
the RHS of (3.4)
=&== represent
ÑB
6
ÑB )
$
#" .
without …¹
corresponding to the approximated eigenvalue
'
$
E XPERIMENT 6.2. Computation of inverse eigenvalues , given and . The proce
dure is almost the same as Experiment 6.1. The true value was computed from a sufficiently
x
Ó
Ó $
the eigenvalues
largeÑ Border principal submatrix of$ (4.4). Then,
all
Ó for each , we computed
ÑB
xƒ)
-|V
Ó
B
Ñ
of
and chose the closest to
to be
. The values of
=&== were obG
pÓ
Ñ98=<
)ä+
w
tained
by)A
backward-substitution
by (1.3), initiating
for sufficiently large ‘?> , and
Ñ@8=< r
:
)Ý
Ó +-:
pÓ
Ó
shall be taken appropriately so that an overflow doesn’t
(
occur). These
Ñ98=<
Ñ98=< r
ÑB Œ
+ xŒŽ
settings of
and
are allowed from the behavior $ Ó
.
Ó
B
Ñ
Table
6.2
describes
how
fast
the
approximated
eigenvalues
approach
the
exact eigen$
ÑB
x )
-WV
8 estimates. In the table, ;
$
value 6 with
error
the RHS of (4.8)
=&== represent
Ó
without …¹
corresponding to the approximated eigenvalue Ñ B .
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etna@mcs.kent.edu
NUMERICAL COMPUTATION OF EIGENVALUES FOR THE SPHEROIDAL WAVE EQUATION
337
TABLE 6.1
IÊJ-J"LKEM N$O$
P$$NQRSRTR ,
Actual
errors and estimates of (3.4). Given !B"C$EDª.F"HG$ , compute ?
š
UIãJ "WVXM KYOKO
PN$Ó QZRSRSR , U
IãJ Ó "?GGM V
[$N[
PPYÓ RSRR .
Ó
Ó
Ó
p ÑB
ÑB
Ñ Bp
Ñ Bp
ÑB
ÑB
x
E
E
E
E
EWE
E
E ; EE
; E
; E 2
3
4
5
6
-2.98E-02
-1.91E-04
-3.64E-07
-2.72E-10
Actual errors
GOEM PQOK[NKV`RSRR , $
x
E
2
3
4
5
6
7
-3.06E-02
-1.93E-04
-3.65E-07
-2.72E-10
-1.57E-02
-6.82E-05
-9.15E-08
-5.02E-11
-1.60E-02
-6.85E-05
-9.17E-08
-5.06E-11
-4.46E-01
-5.85E-03
-1.62E-05
-1.52E-08
-8.00E-12
-4.29E-01
-5.92E-03
-1.63E-05
-1.52E-08
-6.21E-12
TABLE 6.2
š
"
OEM Q
P[$EGKEGSPNRSRTR , and estimates of (4.8). Given !\"]$EDv^"_GO , compute J ]
"WNKEM K$NQ$NEGN`RRR .
Ó
Ó
Ó
¶$
$ p %$
$ ¶$
Ó
Ó
Ó
ÑB
ÑB
ÑB
ÑB
ÑB
ÑB
;
;
;
1.90E-01
5.21E-04
3.08E-07
6.62E-11
2.02E-01
5.23E-04
3.08E-07
6.62E-11
1.48E-00
5.64E-02
3.95E-04
9.00E-07
8.86E-10
1.65E-00
5.89E-02
3.98E-04
9.02E-07
8.87E-10
2.19E-00
1.04E-01
1.38E-03
7.56E-06
2.03E-08
2.99E-11
"
2.85E-00
1.10E-01
1.40E-03
7.60E-06
2.03E-08
3.00E-11
7. Concluding remarks. By the proposed method, one only has to compute the eigenvalues of the given matrices without further knowledge or skill. The eigensystem routines
$
such as EISPACK [12] with guaranteed
error estimates allow one to obtain accurate eigen
values and inverse eigenvalues . The theorems in [8] and [9] are powerful tools for
computing eigenvalues of certain classes of infinite matrices, which arise in solving some
types of linear differential equations. Further applications of the theorems will be sought by
us.
Appendix A.
&
8 Ô¶Ö
p
in (5.1) is positive 6 definite.
P ROPOSITION . A.1 The matrix Ô defined
0 +
ð)
-
& Ô
Proof. One needs to show that
holds for all
=&==
ð)*+
)a1
1
&
equality is valid only when
. Since
,
Ô
Ô
&
)È+^b
The proof for
Conversely, if one assumes
Ô
ö)È+
ð)
&
+
»½
½
Ô
1,)
p
ÂÄÃ
Ã
;
ç
..
; 3
..
+
0
½
½¾
Å
ÂÄÃ
)
;
; †
p
-.
†
)
ç
;
;E3 ;
; »½
)
Å
..†
.
.
†
p
=
3)ø+
½
½
½
½¾
;
-
===
-.
"
)
.7
Ñ" r
†
&
ÂÃ
;Ec
p.Ó
" r
ì áC
Ã
Ã
† ;E3
..
.
Ã
;
Ô
leads
+
p
; ç ; Therefore,
+
Ã
»½
.
..
.
ñ
Ã
Ã
†
1
ñ
follows.
It is obvious
1^
),+
ñ
, one finds
Ã
;
½
½¾
ñ
1
+
; 1,)
Ô
,)Ÿ
&
Ç ç†
p
p
Ã
)*+
=
Å
; á p
(
; áJ
p
=
and the
)È+
.
ETNA
Kent State University
etna@mcs.kent.edu
338
Y. MIYAZAKI, N. ASAI, D. CAI AND Y. IKEBE
This shows
ñ
ñ
p
)Ÿ p
has to vanish, since, otherwise,
ú
;
; (
†
contradicts the premise
p
Ö
ú
Ç ;
; (
†
ç
;
;43
(
p
ŒŽ
M&MM
. This argument leads to
-
)*
)
†
j
==&=
iklnS
áó
),+
?
; á p
; áJ
or
)
,)*+
(
.
REFERENCES
[1] M. A BRAMOWITZ AND I. A. S TEGUN , Handbook of Mathematical Functions, Dover, N.Y., 1972.
[2] A KHIEZER , N. I. AND G LAZMAN , I. M., Theory of Linear Operators in Hilbert Space, Volume I, Pitman,
Boston, MA, 1981 (English Translation).
[3] T. M. A POSTOL , Mathematical Analysis (second edition), Addison-Wesley, Boston, MA, 1974.
[4] T. B ARAKAT ET AL ., The asymptotic iteration method for the angular spheroidal eigenvalues, J. Phys. A, 38
(2005), pp. 1299–1304.
[5] F LAMMER , C., Spheroidal Wave Functions, Stanford University Press, Palo Alto, CA, 1957.
[6] W. G AUTSCHI , Computational aspects of three-term recurrence relations, SIAM Rev., 9 (1967), pp. 24–82.
[7] C. H UNTER AND B. G UERRIERI , The eigenvalues of the angular spheroidal wave equation, Stud. Appl.
Math., 66 (1982), pp. 217–240.
[8] Y. I KEBE , Y. K IKUCHI , I. F UJISHIRO , N. A SAI , K. TAKANASHI , AND M. H ARADA , The eigenvalue problem for infinite compact complex
symmetric matrices with application to the numerical computation of
š
complex zeros of dEJEe@fhg(ikjd elfhg and of Bessel functions dEmUe@fhg of any real order ! , Linear Algebra
Appl., 194 (1993), pp. 35–70.
[9] Y. I KEBE , N. A SAI , Y. M IYAZAKI , AND D. C AI , The eigenvalue problem for infinite complex symmetric
tridiagonal matrices with application, Linear Algebra Appl., 241-243 (1996), pp. 599–618.
[10] Y. M IYAZAKI , N. A SAI , D. C AI , AND Y. I KEBE , The computation of eigenvalues of spheroidal differential
equations by matrix method, JSIAM Annual Meeting, (1997), pp. 224–225 (in Japanese).
[11] Y. M IYAZAKI , N. A SAI , Y. K IKUCHI , D.S. C AI , AND Y. I KEBE , Computation of multiple eigenvalues of
infinite tri-diagonal matrices, Math. Comp., 73 (2004), pp. 719–730.
[12] B. T. S MITH , J. M. B OYLE , J. J. D ONGARRA , B. S. G ARBOW, Y. I KEBE , V. C. K LEMA AND C. B.
M OLER , Matrix Eigensystem Routines - EISPACK Guide, Second Edition, Springer-Verlag, Berlin, 1976.
[13] J. A. S TRATTON , P. M. M ORSE , L. J. C HU , J. D. C. L ITTLE , F. J. C ORBATO , Spheroidal Wave Functions:
Including Tables of Separation Constants and Coefficients, The MIT Press, Cambridge, MA, 1956.
[14] S. YAMASHITA , On eigenvalues of spheroidal wave function, Proceedings of the 23rd Numerical Analysis
Symposium, (1993), pp. 79–82 (in Japanese).
[15] J. W IMP , Computation with Recurrence Relation, Pitman Publishing, Boston, MA, 1984.