Question are Recall that the zeros of the Chebyshev polynomials xj = = cos (a) For n = 2, find the zeros of T₂(x). (b) Find To(x) and T₁(x). T₁(x) = cos(n arccos(x)) 2j+1 : (²₁₂+¹=), 2n j=0,...,n - 1.
Question are Recall that the zeros of the Chebyshev polynomials xj = = cos (a) For n = 2, find the zeros of T₂(x). (b) Find To(x) and T₁(x). T₁(x) = cos(n arccos(x)) 2j+1 : (²₁₂+¹=), 2n j=0,...,n - 1.
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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Question
Numerical Analysis
Parts A and B ONLY
Transcribed Image Text:Question
are
Recall that the zeros of the Chebyshev polynomials
xj = cos
T₁(x) = cos(n arccos(x))
2j+1
2n
(a) For n = 2, find the zeros of T₂(x).
(b) Find To (r) and T₁(x).
(c) Using cos(2y) = 2 cos² (y) - 1, show that
1¹ π),
j= 0,...,n-1.
T₂(x) = cos(2 arccos(x)) = 2x² - 1.
|f(x) - L(x)| ≤
Hint: set y = arccos(x).
(d) Explain how Chebyshev polynomials and their zeros can be used in regards to the error term bound
for Lagrange interpolating polynomials L(x) estimating the value of a function f(x):
|f(n+1) (c)||
(n + 1)!
-|(x-xo)(x - xn)|.
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Hi,
Can you also answer Parts C and D. Please.
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