Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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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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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