Chebyshev approximations for Stumpff series of orders four and fiv

Salzer, H. E.

English

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Interim Report 6 
3 CHEBYSHEV APPROXZMATIONS FOR THEBTUMPFF SERIES - 
OF ORDERS FOUR AND FIVE 9 
6 Herbert E.  Salzer 9 
GPO PRICE S 
q January 1967 10 
1 Analytical Mechanics Associates, Inc. 
Westbury, Long Island, N. Y. 
https://ntrs.nasa.gov/search.jsp?R=19670018315 2020-03-12T11:37:29+00:00Z
Summary 
This report gives the coefficients in the Chebyshev series for the four 
functions 
10 
X 
2 
X X + 1 
f (x )  = T(*x) = - T - + - T - - -  i !  (1+2)! (1+4)! (i+20) ! 
i = 4 , 5  O < - X S l  
and the two functions 
f (x)  = I?(.) i = 4 , 5  - 1 S x S l  
With these coefficients, f (x)  can be found by a simple recurrence formula, 
without the need to calculate any Chebyshev polynomials. This form for f (x)  
provides a single series which can be truncated at  any term to meet varying 
needs in accuracy, and also avoids the considerably larger coefficients occur- 
ring in the explicit polynomial expressions for the approximations to E( x )  . 
1 
1 
We wish to approximate the Stumpff series given in [ l ] ,  p. 6 and 
[2] ,  p. 4 ,  in the notation 
2 3 
+ - - -  1 a! + - - -  a a- 
= - (i+2)! (i+4)! ( i+6)!  
k=O 
2 
(We here replace a! by x for  notational convenience.) In (l), cy = x = 8 
where 0 may be real or  purely imaginary. 
2 
The use of reduction formulas for 1 8 1 enables us to concentrate 
2 
on \ 01 \ = \ x  1 = I 8 1 S 1. These reduction formulas which a re  due to S. Pines 
are given for F (x) and F5(x) in[:1], p. 6, eq. (16) and for F6(x) and 
F (x)  in [ Z ] ,  p. 4, eq. (13). As an illustration, for F and F5, with which 
this present report is concerned, we have 
4 
7 4 
and 
The simple recursion formula 
1 T(x) = 7 l! - x F ~ + ~ ( x )  
enables us to concentrate further on (1) for two conveniently located values of 
i ,  say i = 4 and 5. After discussion with some programmers, it appears 
2 
that it might be helpful to have some way of approximating the series for P(x)  
1' ' 10 as far as the term x /( i+20) ! inclusive. Thus this present note will be 
concerned with 
2 10 1 X F4(x) = - - - 
4! 6 !  8! 24 ! 
X 
f -  + x - - - -  
and 
2 10 
5 5 !  7! 9 !  25 ! 
x x  + - - - - -  + z- 1 F ( x )  = -  - - 
for - 1 sx I 1. In (5) and (6) the same notation of F ( x )  and F ( x )  is em- 
ployed for the tenth degree approximation as for the infinite series in (1). 
Comparison of (5)  and (6) with the true values given by the infinite series shows 
4 5 
the relative e r r o r s  to be within approximately 0.6 x for (5) and 1.1 x 10 -26 
for (6). 
Three expansions will be derived. The first will be for 0 S x  5 1 only, 
corresponding to real 0, o r  the circular case. The second will be for - 1 i x 5 0 
only, corresponding to imaginary 8, or  the hyperbolic case. Letting x = -x', 
0 5 x's  1, and then dropping the prime, we have E( - x )  and only + signs in 
the right members of (l), (4)-(6). The third case will be for - 1 5 x 5 1, s o  
that the identical approximation formulas for P ( x )  , i = 4 , 5 ,  will be used for 
positive or negative x. This third, o r  universal case requires less program- 
ming than the separate circular or  hyperbolic cases, but in return for the doubled 
range in x ,  the series falls off less rapidly, the coefficients a in (8) below 
being around 2 times the coefficients a in (7) below. 
1 
1 
r 
r 
r 
The approximations for all three cases will be left in the form of series 
of Chebyshev polynomials adjusted to  the interval for x, without rearrange- 
ment of those ser ies  into the equivalent polynomials in x. The advantage will 
3 
L 
be threefold: 
a) We avoid the larger coefficients that occur in the polynomial 
form. 
b) We are  able to see at a glance the e r ro r  in stopping at any 
particular term of the Chebyshev series, so that a single 
expansion in terms of Chebyshev polynomials meets varying 
needs inaccuracy. 
c) The series itself, taken to any number of terms, is calculated 
directly by a simple recurrence scheme that bypasses the need 
for  calculating the Chebyshev polynomials themselves (see 
[3 I ,  pp. 76-78).  
For 0 5 x 5 1  we express f (x )  = F ( x ) ,  F ( x ) ,  F ( - x )  or F (-x) as 4 5 4 5 
a Chebyshev series in the form 
+ a T*(x) f ( x )  = -a T*(x) + a T*(x) + a 2 T i ( x )  + - - - 1 2 0 0  1 1  n n  
-1 
where T*(x) = cos re ,  8 = cos (2x-1) and, of course, the coefficients a 
differ in each of these four cases. The index n is determined so that 
Jan+l n+l 
practice is just about 1 a 
r r 
T* ( x )  + - - -  + a 10 T* 10 (x) \ !S \an+l \+  - - - + \ a lo \  (which inactual 
I) is less than the desired truncation e r ror .  
n+l 
For - 1 S x s, 1 we express f (  x )  = F4(x) or F (x )  as a Chebyshev 5 
series in the form 
f ( x )  = -a  1 T (x)  + a T (x)  + a T ( x )  + - - - + a T ( x )  2 0 0  1 1  2 2  n n  
4 
-1 
where T ( x )  = cos re, B=cos x ,  and, as above, the a 
cases and n is the stopping point f o r  the desired approximation. 
differ in these two 
r r 
For (7) and (8) we let bn+l = b 
- - -  , b from 
= 0 and then find successively b 
n+2 n '  
n-1' 0 
b 
b = (4x - 2)br+l - br+2 + ar r 
for (7 ) ,  and from 
r+l- 'r+2 + b = 2xb r 
for (8). For both (7) and (8) we have 
1 
2 0  f ( x )  = - ( b  - b2) 
To obtain (7) for F ( f x )  and F ( *x)  we replace xr in (5) and (6 ) ,  4 5 
in terms of the Chebyshev polynomials Tz( x )  , given by the formula 
r 
1 xr = - 
2 k=l 
after which the coefficients a a re  found by a direct calculation. r 
r 
To obtain (8) for F ( x )  and F (x)  we replace x in  (5) and ( 6 ) ,  in 4 5 
terms of the Chebyshev polynomials %( x )  , given by the formulas 
for r odd, and 
5 
for  r even, and proceed with a similar calculation for the coefficients a . r 
Although for our present purposes r 4 10 , because of the possible further 
applications of (12), (13a) and (13b) to many other computational problems, 
we give the numerical values of the coefficients of both T*(x) and Tk(x) , 
up to r = 12 , in the Appendix. 
k 
Following are the coefficients a r = 0 , 1 , - - - , 10 , for (7) and r y  
(8). The user  is reminded that the actual constant term in each series, namely 
1 1 - a T*( x)  and - a T (x)  , is half the number a 
2 0 0  2 0 0  0 
for  r = O .  
occurring in (9) and (10) 
Table I: Coefficients for f (x)  = F4(x) , for (7), when 0 S x  4 1 
0 0.08196 28745 37748 87356 66405 84 
1 -0.00068 21719 17055 76698 55965 44 
2 0.00000 30489 82441 27043 79448 46 
3 -0.00000 00084 82185 58214 78661 10 
4 0.00000 00000 16087 46174 06625 02 
5 -0.00000 00000 00022 12548 05356 18 
6 0.00000 00000 00000 02307 18615 75 
7 -0,00000 00000 00000 00001 88667 51 
8 0.00000 00000 00000 00000 00124 22 
9 -0.00000 00000 00000 00000 00000 07 
10  0.00000 00000 00000 00000 00000 00 
6 
Table D: Coefficients for  f(x) = F (x) , for (7), when 0 sx 5 1 5 
r - 
a r -
0 0.01647 03051 97075 80345 15031 486 
1 -0.00009 78401 56475 20776 49071 272 
2 0.00000 03398 04170 58306 91915 001 
3 -0.00000 00007 72908 39503 69125 458 
4 0.00000 00000 01239 76717 25031 307 
5 -0.00000 00000 00001 47720 42685 158 
6 0.00000 00000 00000 00135 88043 633 
7 -0.00000 00000 00000 00000 09939 834 
8 0.00000 00000 00000 00000 00005 920 
9 -0.00000 00000 00000 00000 00000 003 
10 0.00000 00000 00000 00000 00000 000 
Table III: Coefficients for f (x)  = F4( -x), for  (7), when 0 5 x 5  1 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
0.08474 09967 93308 59723 83476 42 
0.00070 69753 31110 55728 26359 39 
0.00000 31523 27765 34872 63273 09 
0.00000 00087 43155 31301 41308 34 
0.00000 00000 16535 55162 10400 77 
0,00000 00000 00022 68558 95845 40 
0.00000 00000 00000 02360 57322 66 
0.00000 00000 00000 00001 92681 56 
0.00000 00000 00000 00000 00126 66 
0.00000 00000 00000 00000 00000 07 
0.00000 00000 00000 00000 00000 00 
7 
Table N: 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Coefficients for f (x)  = F ( -x ) ,  for  (7), when O < x <  1 5 
0.01686 71619 09789 45664 69213 580 
0.00010 05960 28916 41442 62983 631 
0.00000 03491 99069 69242 77799 145 
0.00000 00007 92982 80386 21772 479 
0.00000 00000 01269 63963 63803 429 
0.00000 00000 00001 51015 17022 159 
0.00000 00000 00000 00138 69027 193 
0.00000 00000 00000 00000 10130 979 
0.00000 00000 00000 00000 00006 027 
0.00000 00000 00000 00000 00000 003 
0.00000 00000 00000 00000 00000 000 
Table V: Coefficients for f (x)  = F ( x ) ,  for (8), when -1 S x S l  4 
a r -r 
0 0.08335 81364 86421 56668 31392 69 
1 -0.00138 90955 75952 37019 96258 59 
2 0.00001 24018 37511 04702 40212 23 
3 -0,00000 00688 96882 71921 48319 62 
4 0.00000 00002 60968 42395 82630 39 
- 
5 -0.00000 00000 00716 93868 34908 41 
6 0.00000 00000 00001 49361 23559 98 
7 -0.00000 00000 00000 00244 05323 67 
8 0.00000 00000 00000 00000 32112 17 
9 -0.00000 00000 00000 00000 00034 75 
10 0.00000 00000 00000 00000 00000 03 
8 
Table VI: Coefficients for f (  x) = F5( x), for (8), when - 1 S x s 1 
n =  0 1 2 3 
e for F 3.3(-2) 6.0(-4) 6.6(-6) 5.0(-8) 
e for F5 2.4(-2) 3.3(-4) 3.0(-6) 1.9(-8) 
n 4 
n 
r - 
4 5 
2.8(-10) 1.1(-12) 
9.2(-11) 3.4(-13) 
a r -
0 0.01666 94225 19033 65120 33819 817 
1 -0.00019 84314 87971 93972 29051 717 
2 0.00000 13779 46257 73635 77918 635 
3 -0.00000 00062 63266 07292 44576 958 
4 0.00000 00000 20074 33194 85438 040 
5 -0.00000 00000 00047 79567 24612 936 
6 0.00000 00000 00000 08785 92625 128 
7 -0.00000 00000 00000 00012 84487 856 
8 0.00000 00000 00000 00000 01529 149 
9 -0.00000 00000 00000 00000 00001 511 
10 0.00000 00000 00000 00000 00000 001 
To determine at a glance the relative e r ro r  in using any of these tables 
1 1 / 2  ao) ,  which is always 
in connection with (9) or  (10) and (l l) ,  starting with a 
terms beyond a 
and neglecting all n 
simply look at the ratio a n i l  
0.008 for any of the F or  F series re- less than a n+l ]OY.04 or a n+l / 4 5 
spectively . 
To see the improvement in the number of required terms for any desired 
accuracy, we also may glance at the following schedule of the upper bounds for 
the relative e r rors ,  say e 
F or F respectively, through the terms in xn: 
in using the uneconomized series (5) or  (6) for 
n ’  
4 5 
9 
n =  6 
e for F 3.7(-15)  
e for F, 9.9( -16)  
n 4 
n 3 
This schedule is based upon the worst choice of 1 x I = 1 ,  so that whenever the 
largest value of 1x1 does not exceed some 6 < 1 ,  the e may be improved 
to Bn+'e . There is no such corresponding advantage in the use of the econo- 
mized formulas for the smaller values of 1 x I, since they are designed 
primarily to minimize the maximal e r ro r  over the entire range of x. 
n 
n 
7 8 9 10 
9.9(-18) 2 .1( -20)  3 .9( -23)  6 .0(-26)  
2.3(-18) 4.6(-21) 7.7(-24)  1-1( -26)  
10 
APPENDIX 
Powers of x in Terms of Chebgshev Polynomials 
r $ E T:(x): 
A. x in Terms of 
* 
1 = To 
2 1  x = -(3T:+4T:+ 8 T*) 2 
x3 = - ( lOTt+ 1 
x4 = -(35T*+56T:+28Ti+ 1 8T*+Tq*) 
x 5 = -(126T*+  210T:+ 12OTl+ 45T3*+ 1OTq*+ T;) 
x6 = - (462T*+ 792T:+495T*+220T;+ 66Tq*+ 12Tg*+ T;) 
x7 = - (1716T*+ 3003T1*+2002T2*+ 1001T:+ 364Tq*+ 9lTg*+ 14T*+ T;) 
x8 = - ( 6435 To* + 11440 T; + 8008 T l  + 4368 T* + 1820 Tq* + 560 Tg* 
15T:+ 6T*+T*) 
32 2 3  
12 8 0 3 
5 12 0 
2048 0 2 
8192 0 6 
32768 3 
+ 120 T* t 16 T; + T l )  6 
131072 0 2 x9 = 
(24310 T* t 43758T: + 31824 T* + 18564 T;+ 8568 Tq* 
+ 3060 T: + 816 T* + 153 T* + 18 T*+ T*) 6 7 8 9  
10 - (92378 T* + 167960 T:+ 125970 T* + 77520 T l +  38760 T i  
524288 0 2 x -  
+ 15504T*+ 4845T*+ 1140T;+ 19OT;+ 2OTg*+ T:o) 
( 352716 T* t 646646 T: + 497420 T; + 319770 T; + 170544 Tq* 
5 6 
11 - 
209 7152 0 x -  
+ 74613 T*+ 26334 T* + 7315 T; + 1540 T; -C 231 T; 
+ 22 T:o + T1:) 
5 6 
12 - ( 1352078 T* + 2496144 T: + 1961256 T l +  1307504 T; 
8388608 0 
x -  
+ 735471 T* + 346104 T l  + 134596 T l  + 42504 T; 
+ 10626T*+ 2024Tg*+ 276T1*0+ 24T1*l+ Tlz) 
* 4 
8 
11 
r B. x in Terms of Tk %(x): 
1 = To 
x =  
2 1  x = 2(T0+T2)  
3 1  x = 4(3T1+T3) 
4 1  x = - (3To+4T2+T4)  8 
1 x5 = -(10T1+5T +T5) 
16 3 
6 1 x = -(10T + 1 5 T 2 + 6 T  +T6) 
32 0 4 
7 1 x = -(35T1+21T3+7T + T  ) 
64 5 7  
8 1 x = -(35To+56T2+28T + 8 T  +Ts) 
12 8 4 6 
9 1 x = -(126Tl+84T3+36T + 9 T 7 + T  ) 
256 5 9 
1 xlO= -(126T0+210T2+120T + 4 5 T  + lOT + T  ) 
5 12 4 6 8 10 
1 p, - (462T1+ 330T + 165T5+ 55T + l l T g +  T ) 1024 3 7 11 
2048 0 4 6 8 12 
p= - (462T + 792T2+495T +220T + 6 6 T  +12TIO+T ) 
12 
References 
L11 
1 
[SI 
Pines, S .  ; 'Mean Conic State Transition Matrix," unpublished memo- 
randum to Philco Corp., Western Development Lab., Palo Alto, Calif. , 
Order No. WDL-C-2431, November 1965. 
Pines, S.  and Fang, T.C. ; "A Uniform Solution of the Euler-Lagrange 
Equations During Coast for  the Central Force Field," unpublished re- 
port of Analytical Mechanics Associates, Inc., November 1965. 
Anon. ; Modern Computing Methods, published by Mathematics Division 
of the National Physical Laboratory, 2nd ed. , Philosophical Library, 
Inc., New York, 1961. 
13 


Chebyshev approximations for the Stumpff series of orders four and five Interim report

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