Interpolate the function f(x) = 2* over the interval [1,1] by a polynomial (a) Construct the interpolating polynomial p2 (x) of degree n = 2 in the Newton form using uniformly spaced nodes. (b) Find an upper bound for the error |ƒ(x) — p₂(x)| at x = ½. (c) Find an upper bound for the error |f(x) - p₂(x)| on [-1,1]. Note that max |f(x) p₂(x)| ≤ x= [a,b] h³ M3 9√3 2 max f(x) pn(x)| ≤ x= [a,b] where h = (ba)/2 and |f"(x)| ≤ M3 for x = [a, b]. (d) How many equally-spaced nodes are required to interpolate the function f(x) to within € = 0.01? Note that for n > 2, 1 max | f(n+¹) (§) (n + 1)! € [a,b] hn+¹n! 4 where h = (b -a)/n.
Interpolate the function f(x) = 2* over the interval [1,1] by a polynomial (a) Construct the interpolating polynomial p2 (x) of degree n = 2 in the Newton form using uniformly spaced nodes. (b) Find an upper bound for the error |ƒ(x) — p₂(x)| at x = ½. (c) Find an upper bound for the error |f(x) - p₂(x)| on [-1,1]. Note that max |f(x) p₂(x)| ≤ x= [a,b] h³ M3 9√3 2 max f(x) pn(x)| ≤ x= [a,b] where h = (ba)/2 and |f"(x)| ≤ M3 for x = [a, b]. (d) How many equally-spaced nodes are required to interpolate the function f(x) to within € = 0.01? Note that for n > 2, 1 max | f(n+¹) (§) (n + 1)! € [a,b] hn+¹n! 4 where h = (b -a)/n.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3. Interpolate the function f(x) = 2ª over the interval [−1, 1] by a polynomial
(a) Construct the interpolating polynomial p2(x) of degree n = 2 in the Newton form using uniformly
spaced nodes.
1
(b) Find an upper bound for the error |ƒ(x) — p2(x)| at x = 2
(c) Find an upper bound for the error |ƒ(x) — p2(x)| on [−1,1]. Note that
max f(x) p2(x)| ≤
x= [a,b]
h³ M3
9√3
9
max | f(x) — pn(x)| ≤
x= [a,b]
where h = (b − a)/2 and |ƒ""'(x)| ≤ M3 for x € [a, b].
(d) How many equally-spaced nodes are required to interpolate the function f(x) to within € = 0.01?
Note that for n > 2,
1
max | f(n+¹) (§) |
(n + 1)! E[a,b]
hn + 1n!
4
where h = (b − a)/n.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91f76606-a4d9-42f0-be0d-7b1366a5593f%2Ff8aa3345-e4d6-4925-ac70-bed59086da2f%2Ffksmu4p_processed.png&w=3840&q=75)
Transcribed Image Text:3. Interpolate the function f(x) = 2ª over the interval [−1, 1] by a polynomial
(a) Construct the interpolating polynomial p2(x) of degree n = 2 in the Newton form using uniformly
spaced nodes.
1
(b) Find an upper bound for the error |ƒ(x) — p2(x)| at x = 2
(c) Find an upper bound for the error |ƒ(x) — p2(x)| on [−1,1]. Note that
max f(x) p2(x)| ≤
x= [a,b]
h³ M3
9√3
9
max | f(x) — pn(x)| ≤
x= [a,b]
where h = (b − a)/2 and |ƒ""'(x)| ≤ M3 for x € [a, b].
(d) How many equally-spaced nodes are required to interpolate the function f(x) to within € = 0.01?
Note that for n > 2,
1
max | f(n+¹) (§) |
(n + 1)! E[a,b]
hn + 1n!
4
where h = (b − a)/n.
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