Final Exam, Math 151A/2, Winter 2001, UCLA, 03/21/2001, 8am11am
I.
(a) Let
x
0
,
x
1
, ...,
x
n
be
n
+1 distinct points in [
a, b
], with
x
0
=
a
and
x
n
=
b
, and
f
∈
C
n
+1
[
a, b
].
Let
P
(
x
) =
P
0
,
1
,...,n
(
x
) be the Lagrange polynomial interpolating the points
x
0
,
x
1
, ...,
x
n
, such
that
P
(
x
i
) =
f
(
x
i
), for all
i
= 0
,
1
, ..., n
.
Express
f
(
x
) in terms of
P
(
x
) and a remainder term (the error formula).
(b) Consider the case
n
= 2 and the data
x
x
0
= 0
x
1
= 1
x
2
= 2
f
(
x
) = ln(
x
+ 1)
0
ln 2
ln 3
(i) Find polynomials
L
i
(
x
),
i
= 0
,
1
,
2, such that
L
i
(
x
i
) = 1 and
L
i
(
x
j
) = 0 if
i
negationslash
=
j
.
(ii) Deduce the Lagrange polynomial
P
(
x
) =
P
0
,
1
,
2
(
x
) interpolating this data points.
(iii) Write the error formula and find a bound for the error

f
(0
.
5)
−
P
(0
.
5)

.
II.
(a) Let
i, j
be two distinct integers in
{
0
,
1
, ..., n
}
. Express
P
0
,
1
,...,n
(
x
) in terms of
P
0
,
1
,...,i
−
1
,i
+1
,...,n
(
x
)
and of
P
0
,
1
,...,j
−
1
,j
+1
,...,n
(
x
) (Neville’s method).
(b) Suppose
x
j
=
j
for
j
= 0, 1, 2, 3 and it is known that
P
0
,
1
(
x
) =
x
+ 1
,
P
1
,
2
(
x
) = 3
x
−
1
,
and
P
1
,
2
,
3
(1
.
5) = 4
.
Find
P
0
,
1
,
2
,
3
(1
.
5).
III.
For a function
f
the forward divideddifferences are given by
x
0
= 0
f
[
x
0
] =
f
[
x
0
, x
1
] =
x
1
= 1
f
[
x
1
] =
f
[
x
0
, x
1
, x
2
] = 4
f
[
x