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Math 371, Fall 2011
1.
(Interpolating Polynomials)
The function
f
(
x
) = 1
/
(
x
+ 1) is given at the four points
x
0
= 1,
x
1
= 2,
x
2
= 3,
x
3
= 4.
(a) Write the interpolating polynomial in Lagrange form.
(
x

2)(
x

3)(
x

4)
(1

2)(1

3)(1

4)
1
2
+
(
x

1)(
x

3)(
x

4)
(2

1)(2

3)(2

4)
1
3
+
(
x

1)(
x

2)(
x

4)
(3

1)(3

2)(3

4)
1
4
+
(
x

1)(
x

2)(
x

3)
(4

1)(4

2)(4

3)
1
5
.
(b) Write the interpolating polynomial in Newton form.
1
2

1
6
(
x

1) +
1
24
(
x

1)(
x

2)

1
120
(
x

1)(
x

2)(
x

3)
(c) Evaluate
f
(1
.
5) and
f
(5) using the interpolating polynomial. Which approximate value is more
accurate?
f
(1
.
5)
≈
0
.
403125
,
f
(5)
≈
0
.
133333
(d) Use the error formula to ﬁnd an upper bound on the maximum error

f

p
3

∞
= max
1
≤
x
≤
4

f
(
x
)

p
3
(
x
)

.
max
1
≤
ξ
≤
4
±
±
±
±
f
(4)
(
ξ
)
4!
±
±
±
±
= max
1
≤
ξ
≤
4
1
(
ξ
+ 1)
5
=
1
32
.
max
1
≤
x
≤
4
(
x

1)(
x

2)(
x

3)(
x

4) = 1
.
Therefore an upper bound on the error for 1
≤
x
≤
4 is
1
32
.
2.
(Choice of Interpolation Points)
This question investigates how diﬀerent interpolation points af
fect the accuracy of the interpolation. Let
f
(
x
) =
1
20
x
2
+1
.
The following two Matlab programs can be used to generate the plots of the interpolating polynomials.
The program below computes the Lagrange interpolant.
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 Fall '08
 KRASNY
 Polynomials

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