Description

Contents
1 The time and the place 1
2 Introduction 1
3 The problems 1

1 The time and the place
The lab session will take place on
2 Introduction
This week’s problems do not require much in terms of programming. However, however careful inspection and analysis of the output is required. Use any excess time to complete problems from previous weeks.
3 The problems
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 2p for a suitable value of p.
3. Determine the range of k values for which the computed value of Richardson’s fraction converges to 2p 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) = ex 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 2p 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 2p for a suitable value of p.
3. Determine the range of k values for which the computed value of RichRichardson’s fraction converges to 2p 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) = ex 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 2p 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?