Let P(x) be a polynomial that interpolates f(x) at x; for i = 0, 1,...,l−1,1+1, ---, n (i.e., excluding x₁) and Q(x) be another polynomial that interpolates f(x) at 2, for i = 0, 1,..., m - 1, m+ 1,,n (i.e., excluding m). Suppose l m (and of course 0≤l, m≤n). (a) Show that the linear interpolation of these two polynomials I-Im R(x) = P(x)- I-IL Im - Il τι – Tm interpolates f(x) at x₁ for i = 0, 1, 2, ..., n (i.e., R(x₁) = f (x₁) for i = 0, 1, 2, ---,n). 1 + Q(x) (b) Can you conclude that R(x) is the unique polynomial (of degree at most n) that interpolates f(x) at x; for i = 0, 1, 2,...,n? Explain in your own words.
Let P(x) be a polynomial that interpolates f(x) at x; for i = 0, 1,...,l−1,1+1, ---, n (i.e., excluding x₁) and Q(x) be another polynomial that interpolates f(x) at 2, for i = 0, 1,..., m - 1, m+ 1,,n (i.e., excluding m). Suppose l m (and of course 0≤l, m≤n). (a) Show that the linear interpolation of these two polynomials I-Im R(x) = P(x)- I-IL Im - Il τι – Tm interpolates f(x) at x₁ for i = 0, 1, 2, ..., n (i.e., R(x₁) = f (x₁) for i = 0, 1, 2, ---,n). 1 + Q(x) (b) Can you conclude that R(x) is the unique polynomial (of degree at most n) that interpolates f(x) at x; for i = 0, 1, 2,...,n? Explain in your own words.
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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