TECHNICAL NOTES AND SHORT PAPERS
Characteristic
Exponents
of Mathieu Functions
By T. Tamir
The study of the properties and solutions of the Mathieu second-order differential equation has been very extensive in the last decades, due to the special
interest presented by physical problems involving periodic media and problems
separable in elliptic coordinate systems, [1], [2]. The purely periodic solutions of
the Mathieu equation have been extensively tabulated with high accuracies over
large ranges of the parameters involved [3], [4].
On the other hand, numerical tables for the non-periodic Floquet-type solutions
are scarce [5], [6], [7]; in addition, the increments between two successive tabulated
values are relatively large for some of these tables and the accuracy is relatively
low in the other tables. Consequently, the applicability of the available data is
rather limited.
The regions of "stable" solutions of the Mathieu equation are of particular
interest in applications involving the wave equation with periodic boundary conditions, since these regions correspond to the propagating waves in the corresponding media. The first three regions of stable solutions were tabulated with this type
of problem in mind, the numerical results being given in Tables I, II, and III.
The canonical form of the Mathieu equation considered is given by
(1)
y" + (p - 2q cos 2z)y = 0,
with p and q real. Every non-periodic solution of (1) is a linear combination of two
linearly independent Floquet-type solutions of the form
(2)
y = eir°P(z),
where P(z) is a periodic function of z, and v is the so-called characteristic exponent.
This exponent is a function of the parameters p and q only. When v is real, the
solutions (2) are called "stable" since they are uniformly bounded for all real z;
then v = f(p, q) may be chosen to be single-valued. (Note: In the tables and curves
shown here, v is not reduced to 0 ^ v ^ 1 as is usually done, but its actual value
is taken, thus defining the particular stable region it represents.)
It is shown in the references that all values of v are solutions of the continued
fraction equation
(3)
¿.¿L
11-^-...+
A
\Dt
IDs
l!
l!
Z)_i
|D_2
l!
IZL
where
(4)
P.-<2"
+ ->'-P,
(m = 0,±l,±2...).
Received June 12, 1961. The work described in this paper was supported
Cambridge Research Laboratories.
100
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by the Air Force
CHARACTERISTIC EXPONENTS OF MATHIEU FUNCTIONS
For computation
101
purposes, it is convenient to define
(5)
x —v —p
so that equation (3) will take the form
2
X
(6)
m
■
i
q
4(1 + v) + x
g2
2
1
|_q_I
I.
4(1 - v) + x
18(2 + v) + x
g2
2
_
'"
\in{n + v) + x
1
18(2 - v) + x
1
_ _q_|
g2
_
1
\in(n - v) + x
(n=
1,2,3, •••)
The continued fractions are convergent since they satisfy the convergence test
[2] which requires that \D„\ è 2 for n > N, where N is a finite integer.
To find the roots of equation (6), it is noted that this equation is already expressed in a suitable form for solution by means of an iteration process, with x
regarded as the unknown. Taking q and v as variable parameters, such an iteration
procedure was programmed on a computer which obtained p with an accuracy of
10-5. As is proved in the Appendix, the iteration process converges in an alternating
fashion, i.e., the exact result always lies between the values obtained by two successive iterations; hence, the iteration process must be carried out only until the
difference between two successive results is less than the maximum desired deviation.
It was also noted that the number of iterations required to solve equation (6)
was extremely large for a large range of parameters if a simple iteration procedure
was used. This was due to a very slow convergence of the iteration process in some
regions of q and v. To improve this situation, the computer was programmed to
differentiate between two cases:
a) "Fast" convergence: x'n+\ contained within the interval (xn , xn+i),
b) "Slow" convergence: x'n+i outside the interval (xn , xn+2), where xn+i is
the result of the ?ith iteration, using xn as the trial value; x'n+l is the arithmetic
mean of xn+i and xn .
After using the test to determine whether the convergence is "fast" or "slow,"
as defined above, the computer used either xn+%or a;'„+2, respectively, as the next
trial value for the n + 2 iteration. This procedure reduced the number of iterations required from more than a thousand to less than twenty for the most slowly
converging cases.
The results thus obtained are given in the appended tables, while Figure 1
shows these values plotted in the p — q plane. Note that, for greater clarity,
P = ± VV
rather than p is plotted in the figure. Thus p = P2 if P è 0; p = - P2
if P < 0.
Appendix. It is shown below that an iteration process based on equation (6),
if converging, will do so in an alternating manner; i.e., the value of the exact solution always lies between the values of the results of any two successive iterations.
The iteration process may be written in the form
Xn+l = Ro (Xn) +
Ro (Xn)
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102
T. TAMIR
SECONDREGION
THIRDREGION
Fig. 1—Curves of g vs P for constant values of v. The values of v pertaining to each
curve are given directly by the interceptions with the P axis, p = P2 if P & 0; p = — P2 if P < 0.
where
Ro \Xn)
1
atpxn + b0±
— —±
1
lafyn + b^
IO^Xn + bm±
Then
Km
\ %n )
'
Om^Xn +
bjc
is a "remainder" function or the "tail"
terms have been omitted.
Disregarding the ± sign, we have
Ro(xn)
Differentiating
=
| a*+i xn +
6*+i
of a continued
fraction after the first m
1
«o xn + b0 — Ri(xn) '
with respect to xn , we obtain
Ko'(zn)
= -
a°~1Rï'{x;\
„, = -Ro2[a0
[ai xn + h - Äi(a;n)]2
Noting that Rm(xn) has the same functional
Ro'(xn)
= -[aoßo2
In the case considered,
+ a^RoRi)2
am = — or 1; hence
-
ñ/(^)].
form as Ra(xn), one finds
+ a2(Aoñ!A2)2
+
am > 0. Rm(x„)
• • •].
—> 0 as m —> as
so that Ro'(xn) has a finite negative value for any finite xn ■ Consequently, the
derivative of xn+i is finite and negative so that the iteration process based on
equation (6), if convergent, will converge in an alternating fashion.
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103
CHARACTERISTIC EXPONENTS OF MATHIEU FUNCTIONS
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CHARACTERISTIC EXPONENTS
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106
ROBERT D. STALLEY
No absolute convergence proof was found for the iteration process itself; on the
other hand, no convergence instabilities were experienced in the actual computation
for the range of parameters that was tabulated.
Acknowledgement. The author wishes to express his indebtedness to Professor
A. A. Oliner for his interest and help in connection with this paper, and to Dr.
Gertrude Blanch, Aeronautical Research Laboratory,
for providing valuable
references and information. Special thanks are due to John Segal for his patience
and ingenuity in carrying out the programming and obtaining the numerical
results.
Microwave Research
Polytechnic Institute
Institute
of Brooklyn
Brooklyn, New York
1. N. W. Mclachlan,
Theory and Application
of Mathieu
Functions,
University
Press,
Oxford, 1951.
2. J. Meixnek
& F. W. Schäfke,
Springer-Verlag,
Berlin, 1954.
3. Nat.
Standards,
Bur.
Mathieusche
Tables Relating
Funktionen
to Mathieu
und Sphäroidfunktionen,
Functions,
Columbia
University
Press, New York, 1951.
4. J. C. Wiltse
& M. J. King,
Values of Mathieu Functions,
Johns Hopkins
Radiation
equations,"
Phil. Mag.,
Laboratory, Technical Report No. AF-53, 1958.
5. J. G. Brainerd
& C. N. Wetgandt,
"Solutions
of Mathieu's
v. 30, 1940, p. 458-477.
6. H. J. Gray,
R. Merwin
& J. G. Brainerd,
Inst. Electr. Engrs., v. 67, 1948, p. 429^41.
"Solution
of the Mathieu
7. S. J. Zaroodny,
An Elementary Review of the Mathieu-Hill
Based on Numerical Solutions, Ballistic Research Laboratories,
Equation
Aberdeen
equation,"
Amer.
of Real Variables
Proving Ground
Maryland, Memo. Rep. No. 878, April 1955.
An Algorithm
Equations
for Solving Certain Polynomial
with Coefficient Parameters
By Robert D. Stalley
1. Introduction and General Method. We describe a storage-saving procedure
that can be used in real time simulation or other situations in which the roots of
an equation of the form
n
(1)
J2 Fj{xi, z2, • ■• , xm)yj =0,
y-o
m ^ 2,
0 ^ e0 Û ei ^ • • • ^ en ,
ey integral (j = 0, 1, • • • , n),
must be obtained within severe time limits, i.e., from storage, for values of the
coefficient parameter m-tuple (xi, x2, • ■■ , xm) from some given set. The exponents ey are defined so as to be monotonically increasing rather than strictly increasing for later notational convenience.
Suppose there exist relations
(2)
Ui = <Pi(xi, x-2, ■■■ , xm),
(i = p., ¡i. + 1, • • • , m),
Received February 21, 1961.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
2 ¿ fi ¿ m,