The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 6. LINEAR MULTISTEP METHODS 123
Using 6.56 and 6.57 in 6.39
P (t) = a 0 + a 1 Q 1 (t) + a 2 Q 2 (t) + · · · + a n Q n (t) (6.58)
n∑
= a 0 + a k Q k (t) (6.59)
= f n +
k=1
n∑
k=1
∇ k f n
k!h k Q k(t) (6.60)
Define the parameter s, −1 ≤ s ≤ 0 in the interval [t n−1 , t n ] by
From equation 6.57,
t = t n + sh (6.61)
k−1
∏
Q k (t) = (t n + sh − (t n − jh)) (6.62)
j=0
k−1
∏
= (j + s)h (6.63)
j=0
k−1
∏
= h k (s + j) (6.64)
j=0
= h k s(s + 1)(s + 2) · · · (s + k − 1) (6.65)
Recall the definition of the binomial coefficient for n, m integers,
( n n! n(n − 1)(n − 2) · · · (n − m + 1)
=
=
m)
m!(n − m)! m!
we can define, for any real number t, not necessarily integer,
( t t(t − 1)(t − 2) · · · (t − m + 1)
=
k)
k!
Using this we calculate
( ) −s
=
k
Using 6.70 in 6.60 we get
−s(−s − 1)(−s − 2) · · · (−s − k + 1)
k!
(6.66)
(6.67)
(6.68)
= (−1)k s(s + 1)(s + 2) · · · (s + k − 1) (6.69)
k!
= (−1)k
k!h k Q k(t) (6.70)
P (t) = f n +
n∑
( )
(−1) k −s
∇ k f
k n (6.71)
which is known as Newton’s Backward Difference Formula.
k=1
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge