Internat. J. Math. & Math. Sci.
(1985) 183-187
183
Vol. 8 No.
RESEARCH NOTES
A RELATIONSHIP BETWEEN THE MODIFIED EULER METHOD AND
RICHARD B. DARST
Department of Mathematics
Colorado State University
Fort Collins, Colorado
80521
THOMAS P. DENCE
Department of Mathematics
Ashland College
Ashland, Ohio
44805
(Received March 18, 198A and in revised form November
ABSTRACT.
f(x,y)
dy/dx
Approximating solutions to the differential equation
f(x,y) where
y by a generalization of the modified Euler method yields a sequence of
approximates that converge to e.
Bounds on the rapidity of convergence are determined,
with the fastest convergence occuring when the
parameter value
lized method reduces to the standard modified Euler method.
is
1/2,
so the genera-
The situation is similarly
examined when f is altered.
KEY WORDS AND PHRASES.
Euler method, modifi.d Euler method.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 65D20. 65L99
i.
INTRODUCTION.
The Euler method is known as a simple, but crude,method for approximating
solutions to differential equations.
The modified Euler method offers greater
Let us recall in this setting we wish to solve
the condition
YO" We let h denote a
x + kh. To approximate the exact solution y
positive increment in x and define xk
0
which
at Xk, Y(Xk)
a seaence of approximates yk
construct
we
yk
Yk’
converge to Yk" Proceeding inductively we get Yk+l by considering the sequence:
refinement, as shown in Ross [i].
the equation
dy/dx
f(x,y) subject to
Y(Xo)
(I), (2),
Yk+l
Yk+l
()
Yk + hf(xk,Yk)
(2)
Yk +
(h/2)[f(xk’Yk) + f(Xk+l’Yk+l (I))]
while in general
Yk+l
(n)
Yk
+
(h/2)[f(xk’Yk) + f(Xk+l’Yk+l (n-l))]
i. i)
(1.2)
184
R
B. DARST AND T. P. DENCE
hen successive terms in this sequence are close enough, we set their common value
:qual to
With this in mind, we can consider the equation
Yk+l"
Yk+l
Yk + hi 1/2f(xk,Yk) +
1/2f(xk+l,yk+l
(1.4)
defining the solution points by the modified Euler method
(MEM).
If we consider the specific differential equation with f(x,y)
y(O)
zondition
i/n
i, with an increment of h
Yn
2n + i
1
n
(i +
2n
y, and the side
we get the values
(n- .5) + .5
1
(1.5)
(as was to
This produces a sequence that converges to e
be expected since
y).
y’
The modified Euler method fits into a more general scheme given by
Yk+l
where 0
I.
p
Yk + h[pf(xk’Yk) + (l-p)f(xk+l’yk+l)
(i.6)
If we now apply this generalized method (call it M EM) to the same
P
get a general term of
differential equation as above, we
Yn
and clearly
Yn
n
n+p
(l-p)
approaches e.
1/2
the Euler method, and p
n
1
(17)
=[i+ n- (l-p)
n-
1 produces the same sequence as
We note here that p
produces the same sequence as the modified Euler method.
2.
MAIN RESULTS.
e.
One is now led to ask which value of p yields the sequence that best approximates
The expression for
suggests one could examine the family of functions
Yn
f
P
(x)
x
(i + i/x) +
(l-p).
(2.1)
These functions fall into one of three types, depending on the size of p.
function f
P
is decreasing for p _z .5, is increasing for p
> (-i + V5)/2,
The
and is
decreasing at first then eventually increasing for .5 < p
(-i + V5)/2. The reader
is referred to the articles by Darst, Dence and Polya [2-4] for further details on
this. It follows that p
.5 yields the best approximation to e because any value p’
greater than .5 can be improved upon by, say, (p’ + .5)/2. Perhaps Euler knew something that we haven’t given him credit for when he chose p
1/2 instead of an
alternate weighting system!
To
converges to e, we wish to find N such that x > N
quickly
determine,
e is bounded above by
implies
> O. To this end we have
how,
f.5
f.5(x)
i
f.5(x) f. si(x) (i +) x[ e" 501n(l
i
+
)
(. 50 i
+ i/x)
.49i)ini( i
.o()(.5o-a(
z
e .491n(i
+ i/x)
+ i/x)(i/i:
(2.2)
(2.3)
+ /x(/:)
+/- 0
i(.50i-l)(i/x)i(i/i!)
<.Ole
i=I
(2.5)
_
185
RELATIONSHIP BETWEEN THE MODIFIED EULER METHOD AND e
<
.Ole
21-ix
i=I
e/[ 50(2x
Since
f.5(x)
e
<
.5[f.5 (x)
large x, the difference between
I) ],
12xl >
for
f.51 (x)] e/[lOO(2x
f.5 and e can be made
greater than .511 + e/(iOOz)]. For example, with e
and get
2. 7182938 and the difference
f.5(137)
(2.)
.
(2.7)
i)] for all sufficiently
small enough by choosing x
.0OO1 we then choose x
f.5(137)
If we now consider the slightly more general initial value problem
f(x,y)
Ay with side condition y(O)
YO
Yl
+
I
then, using
(I/n)[pAYo + (I-p)AYl]
i
MpEM,we
+ (i/n)[pA +
> 136.A
.000012.
e
dy/dx
get
(2.8)
(l-p)AYl]
so
Yl
n
n+Ap
+ (p-l)A
(2.9)
and then
Yl + I/n)[pAYl + I-p)AY2]
n
given by Yn
Yl or
(2.10)
Y2
so
2
Y2
The n-th term is
Yl
Yn
[i + n
n
A
(2 II)
(l-p) A
A.
x + (I-p)A
Furthermore, since Yn is of the form (i + A/x)
insight into the behav-ior of
Yn can be gained by examining the related family of
sequences
Bn + C
+
so
Yn
converges to e
(i
A/n)
(2.12)
with A,B,C real.
We shall consider A as positive in what follows.
Case i.
(I + A/n) n + e and bn
Set a
n
a
(i + A/n) -n + and
define the number
IA) by
AI >
in(1 + A/2) ln(l +
ln(l + A) ln(l + A/2
The motivation for this is that
which a
that
n
[an
a
n+l"
if
y(A) is the limiting value of a as n tends to
for
By methods analagous to those used by the author in [4] we know
is increasing if e
then eventually increasing if
of
(2.1.3)
O.
Qy(A), decreasing if
< A/2.
7(A)
an it follows that the monotonicity of bn
A/2,
and initially decreasing
Because bn is basically a reciprocal
is increasing if e
-A/2,
decreasing
and initially increasing then eventually decreasing if -A/2
-y(A).
a
Set c n
(i A/n) n +e and d n (I A/n) -n +a with n > A, and define the
> -7(A),
Case 2.
umber
7(A) by
([A] + 2)in(l
y
(A)
in(l
[’]
A
A
[A] + 1
2
([A] + l)in(l
ln(l
A
[A] + 1
<, 0
(2.1A)
A
[A] + 2
where the brackets denote the greatest integer function. Similar to above we have
\
that
is increasing if
-A/2 decreasing if < y(A), and initially increasing
hen eventually decreasing if (A) <
is increasing if
< -A/2, and that
m
decreasing if
A/2 and initially decreasing then eventually increasing if
Cn
[dn
186
R.B. DARST AND T. P. DENCE
A/2
a
Because of cases I and 2 we can determine the monotonicity of (2.12’
<-(A).
from the identity
Furthermore, since
(2.11)
C/IBI
A (sgn B)n +
A Bn + C
(1+ ;)
[(1+ )
]
IBI
(2.15]
is of the form
x
(i + A/x) + (I-p)A
(2.16)
-
A
A/2, or p 1/2. This
it follows that the fastest convergence to e is when (1-p)A
n +
A
A/2, with the fastest converis decreasing to e for
is because (i + A/n)
gence at a
A/2. We remark here that some of the above monotonicity properties could
Bx +
be alternately derived by examining the logarithm of (I + A/x)
C.
The rapidity of this convergence can be discussed by considering the functions
f
(x)
p
given by
(2.16)
(same
and noting
before)
technique as
50Aln(l + A/x)
f.5(x) f.51 (x) < eA[e
< "OleA Ai(A/x)i21-i
A2eA/[50(2x A 2) ].
e
that
.A9Aln(l + A/x)
(2.17)
I
(2.8)
(2.19)
Table 1 lists some data for this situation.
i0
50
I00
/+00
f. 50 (x)
f. 51 (x)
f. 50 x)-f. 51 (x)
20.A3377
20.10259
20.08992
20. 08581
20.27357
20.067/+8
20.07211
20. 08131
16020
.03511
.01781
A2eA
(A
2)
32867
.03972
.01892
00/+ 57
00/+50
Table 1
A
50(2x
3)
For large enough x we have
f.50(x)- eA< 1/2[f.50(x)- f.51(x)]
.
<
A2eA
I00(2x-A2)
(2,20)
and for this difference to be less than
> 0 just choose x greater than
5[A 2 +
)]. For example, with z .001, we choose x 999 and get
(x) e 3 .0000/+.
A2eA/(IO0
f.50
3.
CONCLUDING REMARKS.
Noticing how critical the value p
1/2 is on the efficiency of convergence
orompts one to characterize those functions f(x,y) which fall under this classification.
Knowing this to be true for f(x,y)
the elementary functions
(we know y
2 3
h
i/n
we get
m,
f(x,y) =x
xm+l/( re+l)
and
Ay, we can now show it to be true for
with the side condition
y(1)
1/(m+1))
(0,0),
Using M EM of
P
and for m
(1 6)
and
0,i,
RELATIONSHIP BETWEEN THE MODIFIED EULER METHOD AND e
n
i=I
Yn
n-1
n
i=1
i=I
TM
i=I
a
m
1/2.
-
is expressable as a polynomial
i
Thus
Yn
I
a ni
i
+ (1/2
m
n
with a
m+l
i/(m+l)
and
i=1
m+l
n
+ am_i nm-i +
whenever
+
aln)
pnTM)
m+l
and this expression converges to
1/2
p(n)
can be written as
1
that p
(.)
m+l
n
m+1
m+l
n
n
But
187
f(x,y)
I/(m+l)
fastest when p
is a polynomial in x.
1/2.
Likewise it follows
Further classifications of f
appear to be more difficult to obtain.
REFEHENCES
2.
ROSS, S.L. Introduction to Ordinary_ Differential Equations (3rd ed. ,), John Wiley
and Sons, New York, 1980.
DARST, R.B. Comparison of estimates for e and e -I Amer. Math Monthly, 86 (1979)
3.
DENCE, T.P.
i.
772-773.
Monthly, 88
A.
On the monotonicity of a class of exponential sequences,
Amer. Math.
(1981), 3Ai-3AA.
POLYA, G., SZEGO, G.
Springer-Verlag, 1972,
Problems and Theorems in Analysis
38.
I (part i, No. 168)