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1. Let f(x) = e". (a) Compute the centered difference approximation of f'(1/2), i.e. DgF (1/2), for h = 0.1/2", n = 0, 1, ... , 10, and verify its quadratic rate of convergence. (b) Determine approximately the optimal value ho which gives the minimum total error (the sum of the discretization error plus the round-off error) and verify this numerically. (c) Construct and compute a fourth order approximation to f'(1/2) by applying Richardson extrapolation to Df (1/2). Verify the rate of convergence numer- ically. What is the optimal ho in this case?

1. Let f(x) = e". (a) Compute the centered difference approximation of f'(1/2), i.e. DgF (1/2), for h = 0.1/2", n = 0, 1, ... , 10, and verify its quadratic rate of convergence. (b) Determine approximately the optimal value ho which gives the minimum total error (the sum of the discretization error plus the round-off error) and verify this numerically. (c) Construct and compute a fourth order approximation to f'(1/2) by applying Richardson extrapolation to Df (1/2). Verify the rate of convergence numer- ically. What is the optimal ho in this case?

Advanced Engineering Mathematics
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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1. Let f(x) = e". (a) Compute the centered difference approximation of f'(1/2), i.e. Df(1/2), for h = 0.1/2", n = 0, 1, . .. , 10, and verify its quadratic rate of convergence. (b) Determine approximately the optimal value ho which gives the minimum total error (the sum of the discretization error plus the round-off error) and verify this numerically. (c) Construct and compute a fourth order approximation to f (1/2) by applying Richardson extrapolation to Df(1/2). Verify the rate of convergence numer- ically. What is the optimal ho in this case?
Transcribed Image Text:1. Let f(x) = e". (a) Compute the centered difference approximation of f'(1/2), i.e. Df(1/2), for h = 0.1/2", n = 0, 1, . .. , 10, and verify its quadratic rate of convergence. (b) Determine approximately the optimal value ho which gives the minimum total error (the sum of the discretization error plus the round-off error) and verify this numerically. (c) Construct and compute a fourth order approximation to f (1/2) by applying Richardson extrapolation to Df(1/2). Verify the rate of convergence numer- ically. What is the optimal ho in this case?
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