## Description

**Problem 1 **Consider the problem of computing the derivative *f*^{0}(*x*) using the finite difference approximation

Execute the script rdifmwe1 and examine output in detail:

- Determine the value of
*k*where the computed value of Richardsonâ€™s fraction has executed an illegal jump. - Determine the range of
*k*values for which the computed value of Richardsonâ€™s fraction convergences monotonically to 2for a suitable value of^{p }*p*. - Determine the range of
*k*values for which the computed value of Richardsonâ€™s fraction converges to 2at the correct rate.^{p } - Determine the range of
*k*values where the error estimates become more and more accurate. - How is the behavior of Richardsonâ€™s fraction related to the quality of Richardsonâ€™s error estimate?

**Problem 2 **Copy rdifmwe1.m into /work/l5p2.m and adapt it to the problem of computing *f*^{0}(2) where *f*(*x*) = *e ^{x }*sin(

*x*)

Do *not *include the derivative when you call rdif.

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

**Problem 3 **Consider the problem of computing the derivative *f*^{0}(*x*) using the finite difference approximation

Execute the script rdifmwe2 and examine output in detail:

- Determine the value of
*k*where the computed value of Richardsonâ€™s fraction has executed an illegal jump. - Determine the range of
*k*values for which the computed value of Richardsonâ€™s fraction convergences monotonically to 2for a suitable value of^{p }*p*. - Determine the range of
*k*values for which the computed value of RichRichardsonâ€™s fraction converges to 2at the correct rate.^{p } - Determine the range of
*k*values where the error estimates become more and more accurate. - Is the behavior of Richardsonâ€™s fraction related to the quality of Richardsonâ€™s error estimate?

**Problem 4 **Copy rdifmwe2.m into /work/l5p4.m and adapt it to the problem of computing *f*^{0}(2) where *f*(*x*) = *e ^{x }*sin(

*x*)

*.*

Do *not *include the derivative when you call rdif initially.

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