Modified divided differences (MDD) provide a good way of representing a polynomial passing through points with unequally spaced abcissas. Recurrence relations for computing coefficients in either the monomial or Chebyshev basis from the MDD coefficients, and for computing the MDD coefficients for either the differentiated or the integrated polynomial are given. The latter operation is likely to be useful if MDD are used in a method for solving stiff differential equations

Krogh, F.

English

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JPL PUBLICATION 77-46
Recurrence Relations For
Computing With Modified
Divided Differences
(NASA-CR-155177) RECURRENCE RELATIONS FOR	 N77-33911
COMPUTING WITH MODIFIED DIVIDED ;DIFFERENCES
(Jet Propulsion Lab.)	 13 p HC A02/MF A01
CSCL 12A	 Unclas
G3/64 50222
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National Aeronautics and
Space administration
Jet Propulsion Laboraiory
California Institute of Technology
Pasadena. California 91 103
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JPL PUBLICATION 77-46
Recurrence Relations For
Computing With Modified
Divided Differences
Fred T. Krogh
September 15, 1977
National Aeronautics and
Space Administration
Jet Propulsion Laboratory
California Institute of Technology
Pasadena, California 91103
r':tDCBDINO PAGH BLANK NOT IMM
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?7-46
Preface
The work described in this report was performed by the Information
Systems Division of the Jet Propulsion Laboratory.
Abstract
Modified divided differences (MDD) provide a good way of representing 	 -
a polynomial passing through points with unequally spaced abcissas.
This report gives recurrence relations for computing coefficients in
either the monomial or Chebyshev basis from the MDD coefficients, and
for computing the MDD coefficients for either the differentiated or
the integrated polynomial. The latter operation is likely to be
useful if MDD are used in a method for solving stiff differential
equations.
-iv-
-^-
77-46
CONTENTS
1. Introduction ......................................................... 1
2. Converting to Coefficients for Other
Basis Polynomials ..................................................... 2
3. Computing Coefficients for the Integrated
or Differentiated Polynomial .......................................... 6
References................................................................. 8
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,.1.	 I	 l	 !	 1	 4
77-46
1. Introduction
A modification of divided differences was first suggested by
Blanch [1] as a means of getting some of the desirable characteristics
of differences when working with unequal intervals. Modified divided
d i fferences of the form used here were introduced in [2] and developed
further in [3]. Shampine and Gordon use them in [4] and Jackson [5]
has made a careful study of related computational approaches, concluding
that the form used here is best. Jackson has also showed that the
round-off characteristics of these modified divided differences are
excellent.
The polynomial of lowest degree passing through the points
(t i , w(t i )), i = n, n-1, ..., n-q+l can be expressed in terms of the
MDD coefficients i (n) according to the formula
q-1
(1)	 Pq-l,n(t) = Pq-l,n(tn + h h T) = 
1iD 
^ i (n) ci,n(T)
Where we use notation similar to that in [3].
-1-
f I I	 I^ f T -^-
77-46
(2) hi = t i - ti-1
T = (t-tn)/hn
Ci(n)	 hn + hn-1 + ... + hn-i = t n -to-i-1	 Ci-l(n-l) + h 
a i ( n ) = hn/&i(n)
SO (n) = 1
a i+l (n) = (q (n) /ci(n- 1))ai(n)
^O( n ) = w(tn)
^ i+l (n) _ ^ i (n) - si(n)^i(n-1)
c0,n = 1
c l'n = 't0 (n)T = T
c i,n	 ta i-1 (n)T + ^i-2(n)/^i-1(n)] ci-1,n, i Z 2
Here we think of the ^ i (n) as coefficients of a polynomial expressed in terms
of the basis polynomials c i,n (T). In [3], the orientation was focused more
on the ^ i (n) which are the modified divided differences of w computed at t=tn.
2. Converting to Coefficients for Other Basis Polynomials
Salzer [6] gives a general recurrence for converting coefficients for
one set of polynomial basis functions to coefficients for another set of poly-
nomial basis functions. In Salzer's notation the c i (T) (Henceforth we no
longer explicitly include n when referring to the various parameters defined
in eq. (?)) satisfy the recurrence.
(3) cm+l(T) + (a(m) + b(m)T) CM(T) + c ( m )
 cm-1(T) = 0
where	 a(0) = 0, a(m) = ^m-1/gym' m ? 1
b(m) = -am
c(m) = 0
-2-
- --	
__
	
1
(4)
a00)	
^
a(k1_ )	 +(^A	 ) a(k)0	 qk2	 g _ k_	 k _ 23g 	 0
(k+l)	 (k)	 (k)
y	 am	 - aq-k-2`'m-1 +(Cq-k-3Aq-k-2) a m , m = 1, 2, ..., k
(k+l)	 (k)
•	 ak+l	 - 'q-k-2 a 
where k = 0, 1, ..., q-2, and C_ 1
 = 0. The desired coefficients are then
am	 , m=0,1,..., q-1. Since a0 = 1, 
amq-1)	 amg 1 2) for 	m > 0. Substitu-
ting (a '.k)/aq_k_2) for amk) , (4) simplifies to
M+l
0)(5) a17. aq-2^q-1
(k+l)	 (k)
^ 1	= 
aq-k-3^q-k-2 + 81
(k+l)	 (k)	 (k)
am+l	 = 'q-k-3am	
+ am+l , m= 1, 2, ..., k
(k+1)	 (k)
^k+2	
= 'q-k-3 ak+l
(q-2)
where k =0, 1, ..., q-3. The desired coefficients are then given by 
+ 0' am
m= 1, 2, ..., q-1. The operations in (5) require only a single vector of
(k)
storage since 
^m 
and aq+m-k-2 ' k=0, 1, ... can all occupy the same location.
Denoting this location by ym , we nave
(b)
	 VPq-1 = "q-2`)q-1
'Wq+m-k-2
	 `xq-k-3 + q+m-k-2 `q+m-k-1, m=0,1, ..., k
_ 	 k=0, 1, ..., q-3
`q-1
	
`q-k-3"q -1
-3-
71-46
One gets a similar recurrence from 9q. (5.11) of [3] by replacing the itera-
tion indices k = 1, 2,	 i; i=q-1, ..., 1 with i=q-1, ..., k; k=1, 2, ..., q-1,
then replacing i with q+k-j-1 and then interchanging the order of iteration
indices once again.
For the case of converting to the Chebyshev basis, we have A(m) = 0,
B(0) = -1, B(m) = -2, m > 0, C(m) = 1. Substituting as before, and then seLcing
(8
k) 
/2a
	 = a0k), (a
m
k)/aq-k-2^ = 4mk), rn > 0, one obtains
(0)
(7) a0	 - 2aq-20q-1
(k+ 1)	 (k)
	 (k)	 (0)
a0	
- 
a
q-k-3 
[2m
q-k-2 + al
	
+ a0	 = (2aq-3^q_2 + a 0	if k=0)
{ k+1 )	 (k)	 ^{k)
	 {k)
am	
- (l/2)aq-k-3
 [am-1 + am+1 ] + a^m	m=1, 2, ..., k-1
(k+1)	 (k)	 (k)
ak	 (112)aq-k-3 a k-1 + a 
a k+1 1) - (1/2)aq-k-3 a k)k
(q-1)
where k =0, 1, ..., q-3, and the desired coefficients are a m
	given by
( q- 1 )	 (q-2)
(8) a0	 = ^ 0
 + (112) a1
( q - 1 )	 (q-2)	 ^(q-2)
am	
- (112) [am-1
	
+ am+l ], m=1, 2, ..., q-3
q-2	 q-3
( q
-1)	 (q-2)
aq-1
	 = (112) aq-2
.4-
V	 I	 I
77-46
As before, the --alculations in (7) can be done in a single vector
of storage as follows (initially ' gym = gym)
( g )	 V q-1 = 2aq-2^ q-1
^ q-2 = 2aq-3yq-2 + ^ q-1
^q-1	 (1/2)aq_Pq-1
^q-k-2 = aq-k-3 [2^q-k-2 + ^q-k ] + ^q-k-1
^ q-k-2+m	 (1 J2)aq-k-3 1^ q-k-2+m + ^ q-k+ml + ^ q-k-l+m
m= 1, 2, ..., k-1
k=l, 2, ..., q-3
^q-2	 (1/2)aq-k-3' q-2 + ^q-1
^ q-1	 (112 )aq-k-3^ q-1
The final Chebyshev coefficients are then obtained, using (8) as follows
(10)	 ^O = ^O + (1/2)Vpl
^m = 0/2)[Vm + ^ m+2 ], m=1, ..., q-1
^q-2 = (1/2)^q-2
^q - i = (1/2)^q-1
If one is using the method outlined in [3] or the code in [4], the
a's are available at the time of integration. Clearly some time can be saved
in the above calculations if z is redefined to be (t-t n )/2h, since this has
the effect of setting bm = -2am and as a result the multiplications by 112 are
-5-
77-46
eliminated. (For this case, simply replace a i by 2a i
 in (9), and replace all
1/2's in (10) with 1. For t = t-t n , divide by h everyplace we just indicated
a multiplication by 2.)
If one wants to approximate thesolution of a differential equation
by a Chebyshev polynomial, and thus wants to get the coefficients for the in-
tegrated polynomial, one can save alittle computation by computing the integra-
ted coefficients after the calculations indicated in (7) or (9). Thus for
3 s n < q-4 one has for 3n , the n-th coefficient of ;he integrated polynomial
( 11 )	 (q-1)(1/200 n-I
an =	 (q-2)
(1/400 n-2
(q-1)
- an+l ) after (8)
(q-2)
- an+2 ) after (7)
3. Computing Coefficients for the Integrated or Differentiated Polynomial
When integrating stiff equations, the best integration method is
different depending on how far the most active component is from its local
equilibrium point. If a procedure such as that in [3] is used, changes in
the method involve either getting coefficients for the integrated polynomial
when leaving a transient region, or for the differentiated polynomial when a
transient is encountered. We have also encountered a desire for such a
transformation in another context.
The best general algorithm we have been able to find for the case
of variable stepsize involves running through the recurrences in (6), repla-
cing 
^k with 4'k-1/k, k=q, ..., 1 for integration (or with (k+1 N+1 , k=0, It
..., q-2 for differentiation). For integration, ^0 is set to the constant of
-6-
i
I _ -	 !	 #
11-46
integration. Then simply run th? recurrence (6) in reverse order
(12) ^q'_l = ^q,_1Jaq'-k-3
Wq'+m-k-2 = ( ^q'+m-k-1 - ^q'+m-k-2)/c'q'-k-3' m= k, ..., 0	 k=q'-29 ..., 0
W-1 = Vq'_1/aq'-k-3
where q' = q+l, for integration, and q' =q-1 for differentiation. In the case
of a constant stepsize, one can accomplish this goal more efficiently by re-
peated differencing of the corrector equation
q-1
k
(13) oYn = h 1: Y*Vy,n
k=Q
Thus
q-j	 k+j-1
(14) V^yn = !'	 yn	 y' n	 j=1, 2, ..., q9
k=Q
and the reverse operation simply involves treating eq. (14) as a triangular
k
system to be solved for the V y'n.
-7-
T ___1 -	 %
77-46
REFERENCES
1. Gertrude Blanch, "On Modified Divided Differences," Math. of Com .
8, 1954, pp. 1-11, 67-75.
2. Fred T. Krogh, "A Variable Step Variable Order Multistep Method for
the Numerical Solution of Ordinary Differential Equations," Information
Processing 68 (Proceedings of the IFIP Congress 1968.), pp. 194-199,
North Holland Publishing Co., Amsterdam, 1961.
3. Fred T. Krogh, "Changing Stepsize in the Intergration of Differential
Equations Using Modified Divided Differences," Proceedings of the
r,,.nference on the Numerical Solution of Ordinary Differential Equations,
',ber 1972, Lecture Notes in Mathematics Vol. 362, pp. 22-71,
springer-Verlag, New York.
4. L.F. Shampine & M.K. Gordon, "Computer Solution of Ordinary Differ-
ential Equations, The Initial Value Problem," W.H. Freeman and Co.,
San Francisco, 1975.
5.. L.W. Jackson, "The Computation of Coefficients of Variable-Step Adams
Methods," Univ. of Toronto Department of Computer Science Technical
Report No. 94, 1976.
6. Herbert E. Salzer, "A Recurrence Scheme for Converting from One
Orthogonal Expansion into Another," Comm. ACM 16, 1973, pp. 705-707.
—8—	 NASA-1PLaCom! L A Ca,J


Recurrence relations for computing with modified divided differences

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