Numerical AnalysisParts A and B ONLY QuestionareRecall that the zeros of the Chebyshev polynomialsxj = cosT₁(x) = cos(n arccos(x))2j+12n(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 that1¹ 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 boundfor Lagrange interpolating polynomials L(x) estimating the value of a function f(x):|f(n+1) (c)||(n + 1)!-|(x-xo)(x - xn)|.

The zeros of the Chebyshev polynomialareto find:a) For, the zeros of .b)

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

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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