## Description

**Problem 1 **Copy rintmwe1 into /lab6/work/l6p1.m and adapt it to the problem of computing the integral

numerically using the trapezoidal rule.

- What evidence do you find to support support the conjecture that there exist an asymptotic error expansion of the form

*T *âˆ’ *A _{h }*=

*Î±h*+

^{p }*Î²h*+

^{q }*O*(

*h*)

^{r}*,Â Â Â Â Â Â Â Â Â Â*0

*< p < q < r.*

- Based on the numerical evidence, what is a reasonable value of
*p*? - Based on the numerical evidence, what is a reasonable value of
*q*? - What is the smallest value of
*k*for which the integral can be computed with a relative error less than*Ï„*= 10^{âˆ’6}?

**You must explain why your error estimate is reliable!**

- Compute the exact value of the integral and include this information in l6p2.
- Is the behavior of Richardsonâ€™s fraction related to the quality of Richardsonâ€™s error estimate?

**Problem 2Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **rintmwe1.m into /work/l6p2.m and adapt it to the problem of com-

puting the integral

using the trapezoidal rule as your approximation *A _{h}*.

- What evidence do you find to support support the conjecture that there exist an asymptotic error expansion of the form

*T *âˆ’ *A _{h }*=

*Î±h*+

^{p }*Î²h*+

^{q }*O*(

*h*)

^{r}*,Â Â Â Â Â Â Â Â Â Â*0

*< p < q < r.*

- Based on the numerical evidence, what is a reasonable value of
*p*? - Based on the numerical evidence, what is a reasonable value of
*q*? - What is the smallest value of
*k*for which the integral can be computed with a relative error less than*Ï„*= 10^{âˆ’6}?

**You must explain why your error estimate is reliable**

- Compute the exact value of the integral and include this information in l6p1.
**Hint:**It is quite easy to compute the integral if you make a drawing of the graph first. - Is the behavior of Richardsonâ€™s fraction related to the quality of Richardsonâ€™s error estimate?

**Problem 3Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â **rintmwe1.m into /work/l6p3.m and adapt it to the problem of com-

puting the integral

using the trapezoidal rule as your approximation *A _{h}*.

- What evidence do you find to support support the conjecture that there exist an asymptotic error expansion of the form

*T *âˆ’ *A _{h }*=

*Î±h*+

^{p }*Î²h*+

^{q }*O*(

*h*)

^{r}*,Â Â Â Â Â Â Â Â Â Â*0

*< p < q < r.*

- Based on the numerical evidence, what is a reasonable value of
*p*? 3. Based on the numerical evidence, what is a reasonable value of*q*? - Why is Richardsonâ€™s fraction not close to 2
for small values of^{p }*k*? - Why is Richardsonâ€™s fraction not close to 2
for very large values of^{p }*k*?. - What is the smallest value of
*k*for which the integral can be computed with a relative error less than*Ï„*= 10^{âˆ’6}?

**You must explain why your error estimate is reliable!**

**Problem 4 **rdifmwe1 into /work/l6p4.m and adapt it to the problem of computing the the target *T *= *f*^{0}(*x*), where *f *is you favorite differentiable function and *x *is your favorite real number using the mysterious rule

where *A _{h }*is your favorite rule for computing

*f*

^{0}(

*x*) which obeys an asymptotic error expansion of the form

*T *âˆ’ *A _{h }*=

*Î±h*+

^{p }*Î²h*+

^{q }*O*(

*h*)

^{r}*,Â Â Â Â Â Â Â Â Â Â*0

*< p < q < r.*

- What evidence can you uncover that suggests that
*M*obeys an asymptotic error expansion of the form_{h }

*T *âˆ’ *M _{h }*= Â¯

*Î±h*+

^{q }*Î²h*

^{Â¯ r }+

*O*(

*h*)

^{s}*,Â Â Â Â Â Â Â Â Â*0

*< q < r < s.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â*(1)

- Based on your numerical evidence, what is a reasonable value of
*q*? 3. Based on your numerical evidence, what is a reasonable value of*r*? - Include the exact value of the derivative of
*f*in the script. - Examine the relationship between Richardsonâ€™s fraction and the quality of the error estimate.

# 1Â Â Â Â Â Â Â Â Â Â Â Â Concluding remarks

- You will find that quality of the error estimate improves even after the computed value of Richardsonâ€™s fractions have start to deviate from the expected pattern. This happens from time to time, but it is not something you can count on.
- If you return an approximation without an error estimate or an error bound, then you work is incomplete.
- If you return an error bound or an error estimate without explaining why it is reliable, then your work is incomplete.