5DV005 Lab5 Solved

Description

Problem 1 Consider the problem of computing the derivative f⁰(x) using the finite difference approximation

Execute the script rdifmwe1 and examine output in detail:

1. Determine the value of k where the computed value of Richardson’s fraction has executed an illegal jump.
2. Determine the range of k values for which the computed value of Richardson’s fraction convergences monotonically to 2^p for a suitable value of p.
3. Determine the range of k values for which the computed value of Richardson’s fraction converges to 2^p at the correct rate.
4. Determine the range of k values where the error estimates become more and more accurate.
5. How is the behavior of Richardson’s fraction related to the quality of Richardson’s error estimate?

Problem 2 Copy rdifmwe1.m into /work/l5p2.m and adapt it to the problem of computing f⁰(2) where f(x) = e^x sin(x)

Do not include the derivative when you call rdif.

1. Verify that the computed value of Richardson’s fraction appears to converge towards 2^p for a suitable value of p as h tends to zero.
2. Find the last value of k, where the computed value of Richardson’s fraction behaved exactly as predicted for the exact value of Richardson’s fraction.
3. Include the exact derivative when you call rdif. Find the value of k where the accuracy of Richardson’s error estimate is maximal.
4. How is the behavior of Richardson’s fraction related to the quality of Richardson’s error estimate?

Problem 3 Consider the problem of computing the derivative f⁰(x) using the finite difference approximation

Execute the script rdifmwe2 and examine output in detail:

1. Determine the value of k where the computed value of Richardson’s fraction has executed an illegal jump.
2. Determine the range of k values for which the computed value of Richardson’s fraction convergences monotonically to 2^p for a suitable value of p.
3. Determine the range of k values for which the computed value of RichRichardson’s fraction converges to 2^p at the correct rate.
4. Determine the range of k values where the error estimates become more and more accurate.
5. Is the behavior of Richardson’s fraction related to the quality of Richardson’s error estimate?

Problem 4 Copy rdifmwe2.m into /work/l5p4.m and adapt it to the problem of computing f⁰(2) where f(x) = e^x sin(x).

Do not include the derivative when you call rdif initially.

1. Verify that the computed value of Richardson’s fraction appears to converge towards 2^p for a suitable value of p as h tends to zero.
2. Find the last value of k, where the computed value of Richardson’s fraction behaved exactly as predicted for the exact value of Richardson’s fraction.
3. Include the exact derivative when you call rdif. Find the value of k where the accuracy of Richardson’s error estimate is maximal.
4. Is the behavior of Richardson’s fraction related to the quality of Richardson’s error estimate?