1. Let fj = f(r;), where r; are nonuniformly spaced grid points ri < r2 < ...< In. Let the spacing between these points be hj = rj+1- Tj for j = 1, 2, ...,n – 1. (a) Write down the quadratic Lagrange polynomial that interpolates the data (r,-1: fj-1). (#j. f;) and (#j+1, fj+1). (b) By differentiating the polynomial found in (a), derive an approximation f for the derivative f'(r;). You must write your final answer in terms of fj-1, fj, fj+1, hj-1 and hj. (c) Use the polynomial interpolation error theorem to show that the absolute value of the error in the derivative approximation from part (b) is bounded by CMh? where |S"| < M for some positive constant M, h = max h; and C is a constant that you must determine.
1. Let fj = f(r;), where r; are nonuniformly spaced grid points ri < r2 < ...< In. Let the spacing between these points be hj = rj+1- Tj for j = 1, 2, ...,n – 1. (a) Write down the quadratic Lagrange polynomial that interpolates the data (r,-1: fj-1). (#j. f;) and (#j+1, fj+1). (b) By differentiating the polynomial found in (a), derive an approximation f for the derivative f'(r;). You must write your final answer in terms of fj-1, fj, fj+1, hj-1 and hj. (c) Use the polynomial interpolation error theorem to show that the absolute value of the error in the derivative approximation from part (b) is bounded by CMh? where |S"| < M for some positive constant M, h = max h; and C is a constant that you must determine.
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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Transcribed Image Text:1. Let f; = f(r;), where r; are nonuniformly spaced grid points ri < r2 < ..< In. Let the
spacing between these points be hj = rj+1- Tj for j = 1, 2, ...,n – 1.
(a) Write down the quadratic Lagrange polynomial that interpolates the data (r,-1, fj-1),
(*j. f;) and (#;+1; fj+1).
(b) By differentiating the polynomial found in (a), derive an approximation f for the
derivative f'(r;).
You must write your final answer in terms of fj-1, fj; fj+1, hj-1 and hj.
(c) Use the polynomial interpolation error theorem to show that the absolute value of
the error in the derivative approximation from part (b) is bounded by CMh? where
|S"| < M for some positive constant M, h = max h; and C is a constant that you
must determine.
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