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Homework 8 due Thursday December 2 before 13:30 in corre
sponding group folder on door of oﬃce CM110
1. Convince yourself that the polynomial of degree
≤
n
which agrees with a function
f
at distinct nodes
x
0
, x
1
, . . . , x
n
may be written as
p
n
(
x
) =
f
(
x
n
) + (
x
−
x
n
)
f
[
x
n
, x
n

1
] + (
x
−
x
n
)(
x
−
x
n

1
)
f
[
x
n
, x
n

1
, x
n

2
]
+
. . .
+ (
x
−
x
n
)(
x
−
x
n

1
)
. . .
(
x
−
x
1
)
f
[
x
n
, x
n

1
, . . . , x
0
]
For equally spaced nodes
x
k
=
x
0
+
kh
, (
k
= 0
,
1
, . . . , n
), backward diﬀerences are
∇
f
n
=
f
(
x
n
)
−
f
(
x
n

1
), and for
k
≥
0
∇
k
+1
f
n
=
∇
k
f
n
− ∇
k
f
n

1
.
From the divideddiﬀerence formula above derive Newton’s backward diﬀerence for
mula, in the form
p
n
(
x
n
+
th
)=
f
(
x
n
) +
t
∇
f
n
+
t
(
t
+ 1)
2
∇
2
f
n
+
. . .
+
t
(
t
+ 1)
. . .
(
t
+
n
−
1)
n
!
∇
n
f
n
.
[
Note that for interpolation, rather than extrapolation,
t
is negative
.]
———————————————————————————————
It has been shown in lectures that the polynomial of degree
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This note was uploaded on 02/09/2011 for the course MATH 2051 taught by Professor Dr.a.wacher during the Fall '11 term at Durham.
 Fall '11
 Dr.A.Wacher
 Numerical Analysis

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