Coefficients calculated for Taylor series expansion about two points - application of Taylor expansion to two-body proble

Estes, R. H.

Lancaster, E. R.

English

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TWO-POINT TAYLOR SERIES 
EXPANSIONS 
BY 
R. H. E S T E S  
E. R. L A N C A S T E R  
DECEMBER 1966 
I 
’ GODDARD SPACE FLIGHT CENTER 
GREENBELT, MARYLAND 
https://ntrs.nasa.gov/search.jsp?R=19670014636 2020-03-24T00:54:06+00:00Z
TWO-POINT TYALOR SERIES EXPANSIONS 
R .  H .  Estes  
E.  R .  Lancaster 
Laboratory for Theoretical  Studies  
December 1966 
GODDARD SPACE FLIGHT CE3TTER 
Greenbelt, Maryland 
INTRODUCTION 
The Taylor s e r i e s  has been general ized i n  mariy d i r ec t ions .  
i s  o u r  purpose here t o  extend the  se r i e s  t o  in- lude two poin ts  about 
which a funct ion may be exparid-$.:. The v i r t u e  of t h i s  procedure l i e s  
i n  t h e  f a c t  t h a t  t he  accuracy of using 
Taylor s e r i e s  i s  a t t a ined  using only n 
expansion. 
It 
2n der iva t ives  i n  a one-point 
der iva t ives  with the two-point 
The c l a s s i c a l  in te rpola t ion  problem i s  concerned with approximating 
a function, f ,  with a polynomial of degree n such t h a t  the  polp-omial 
i s  i n  agreement with the  values of the funct ion a t  each of t he  points  
x i=o  , 1, 2, . . . , n. If i n  addi t ion t o  the funct ion values a-t each 
poin t  the-e a r e  ava i lab le  a given number o f  der iva t ives  of t he  funct ion 
a t  each goint ,  we have the  well-known general  Hermite in t e rpo l s t ion  
formula. 
the  Hermite formula with only one point XI and n der iva t ives  of f 
given a t  xl. The spec ia l  case o f  the Hermite formula for two po in t s  
i' 
Thus the  n+ l  term Taylor s e r i e s  i s  simply a spec ia l  case of  
1 
XI and x2 c3.n be wr i t t en  i n  the  form 
n - l  i n-1  i B. ( x - x ~ )  A. ( x - x ~ )  
y h )  = (x-x1) 1 + (x-x2)n 1 
A. = - n 
x=x1 
where 
1 
It i s  the purpose of t h i s  repor t  t o  develop a s e r i e s  expansign of 
a f w c t i o n  f about two pcsjnts, XI m d  x2, such t h a t  the coe f f i c i en t s  
a r e  f i n a l .  Hence i m r e a s i n g  the number of der iva t ives  used involves 
only the addi t ion of  rn(x-3 te-"ms t o  the o r ig ina i  s e r i e s ,  
C A L C U T T Q N  OF SE3IES CIOEr;i'T@TZWS: 
dif 
3-x 
Assume that, - , i = 0, 1, . k, a r e  known a t  the  two poin ts  
XI and x2. %eL 
k-l 
is a (2k-l.)th degree polynomial. approximation of f ( x )  i f  a and b .  a r e  i 1 
agree a t  x1 and x2 f o r  i = 0, 1, dL f determined so t h a t  and -
dxi dxi 
k-1, 
For ease i n  the  calcialatplo7s which follow, we introduce the  
transformation 
t =  
Then x-x1 
x.-x2 
and 
Y W  
x - X I  
X2'Xl  
- -  -- y, l e t t i n g  x2-x1 = h. 
= h t  
= h(  t-l), 
k- 1 
= [Ai(t- . l . )  + B . t ]  1 [t(t-1)li 
. I  
. I  
i-0 
2 
I 
The coe f f i c i en t s  of the  transformed s e r i e s  (2 )  a r e  r e l a t e d  t o  those of 
t he  o r i g i n a l  s e r i e s  (1) by 
2 i+la  A. = h 
1 i 
Bi = h 2 i+lb  
i '  
This  transformation e f f e c t i v e l y  transforms the  i n t e r v a l  rxl, xz] onto 
d d Y - h J d J y  j 
[0, 13. Now dy - - dy 2 = h 2 and i n  general  - - 
at j d x j  a dx d t  dx 
Rearranging (2)  i n  the form 
k- 1 k-1 
y ( t )  = 1 A. 1 ti(t-l)i+l + 1 Biti+l(t-l) i  
and d i f f e r e n t i a t i n g  we obtain 
k-1 k-1 
It can be shown from the  binomial theorem t h a t  
t = O  
when 2 i + l  2 j 2 i+l 
. 
t = O  
3 
wher: 291 2 ,j 2 5 
whel; 21+1 2 j 2 i 
4 
I 
f o r  j = O ,  1, 2, ..., k 
where 1 i s  de le ted  f o r  j = O  and [A] denotes the  in t ege r  p a r t  of - j 
2 2 -  
The remainder for t h e  c l a s s i c a l  i n t e rpo la t ion  problem involving m 
d i s t i n c t  da t a  poin ts  with cu . -1 ,  j=1, 2, . . ., m, der iva t ives  ava i l ab le  
J 
a t  each poin t  i s  well-known 2 . For ou r  case where m = 2 and c y 1  = cu2 = k 
t he  remainder i s  
is (2k) l i e s  i n  the  i n t e r v a l  defined by x, XI, and x2, and f where 
continuous on t h a t  i n t e r v a l .  
Taylor s e r i e s  i s  
5 
Thus the  f i n i t e  form of the  two-point 
EKAlWU OF APPLICATION: 
The so lu t ion  o f  t he  two-body problem, given i n i t i a l  coordinates and 
v e l o c i t y  com,nonents a t  a time t=to i s  
x = Xlt ) ,  y = y h ) ,  2 = z ( t >  
5 
and f o r  shcrf i n t e rva l s  cf time these  s - l u t i m s  may be expmded i n  a 
Taylor series about to: 
From the  d i f f e r e n t i a l  equat iow reppesenticg the  mi?tion cf the t w e  
body problem we have 
or 
where v = - and p i s  the  g rav i t a t iona l  constant  times the  mass 
of  t he  a t t r a c t i n g  body. 
3.' r 
C,h To obtain an algorithm f o r  +,he E- time der iva t ive  of x, y, and z 
we consider a power s e r i e s  so lu t ion  of the  equatAorrs of motion. 
Let 
03 
i 
03 
y z  r; Yn ( t - t o ) "  
, 
Then 
, 
= R (t-to)n n 
n=O 
OD 
* = L (k+ l )  
k=O 
= (k+ l )  
k=O 
00 
7 .. 
X = 1 (k+2) 
k=O 
m .. 
Y = 1 (k+2) 
k=O 
43 
c .. 
Z = ) (k+2) 
L 
( k+l)  
( k+l) 
(k+ l )  
k=O 
Substituting into the equations of motion and equating coefficients for 
the powers of (t-to) we obtain 
RO 
VO 
XO 
2, 
X1 
= ro 
1 vo = - - 
ro 
= Yo 
= ; r o  
= $0 
7 
ana 
I 
and hence 
= a! 
= n,' 
t=t, 
= E t  
t=to 
, 
A s  a spec ia l  case of t h e  preceding two-body problem consider t h e  
motion t o  be confined t o  the  x-y plane and suppose t h a t  we a r e  given 
and 
X1,  yl, il, and jrl a t  t i m e  tl 
x2, y2, k2, and i 2  a t  time t 2 .  
For convenience we s e t  t he  period of t h e  o r b i t  
major a x i s  a = 1 so t h a t  the  constant ~1 = 1. 
T = 211 and the  semi- 
Obtaining the quan t i t i e s  
by the  general  procedure out l ined above, we may i n  t u r n  ca lcu la te  the  
c o e f f i c i e n t s  a and b.  f o r  ;he two-point Taylor expansionfrom equa- 
i l 
A comparison of  the  r e su l t i ng  solut ions f o r  t h e  two-body problem 
from the two-point Taylor s e r i e s  and one-point Taylor Eeries expbr.dzd 
about tl i s  presented i n  Figure I. 
e = - and tl, t 2  were chosen t o  be time a t  p?rigee and time a t  apoge.: 
( 0  and ll) respec t ive ly .  The f igu re  i s  a p l o t  of the  e r r o r  i n  the  ca l -  
Here the  o r b i t  i s  zf eccen t r i c i ty  
1 
10 
cu la t ion  of the magnitude of the radius vector  vs .  the  time of the  
o r b i t .  A l l  ca lcu la t ions  were performed i n  double precis ion on an 
IBM 7094. 
9 
0 
0 
c 
I 
0 
c 
3 
t 
0 
I 
0 
w 
REFERENCES 
1. Davis, Philip J., "Interpolation and Approximation", Blaisdell 
Publishing Co., New York, 1963, page 37. 
2. For t ,  Tomlinson, "Finite Differences and Difference Equations in 
the Real Domain", Oxford Press, 1948, page 86. 
11 


Two-point Taylor series expansions

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