(chap:errorConstant:exercise)=
# Chapter 5 Exercises
You should try the following exercise questions first, then check with the answers.
:::{exercise}
:label: chap04-ex-4.1
Use a Gregory-Newton interpolation formula to derive the error term
for the following integration formulae:
1. $\displaystyle y_{{j}} ~=~ y_{j-1} + \tfrac{h}{12}(5f_{{j}} + 8f_{j-1} - f_{{j-2}})$
Hint: G-N backward $0$ to $-1$
2. $\displaystyle y_{j+1} ~=~ y_j + \tfrac{h}{2}(f_{j+1} + f_j)$
:::
:::{solution} chap04-ex-4.1
:class: dropdown
1. $\frac{-1}{24}h^4f'''(z)$,
1. $\frac{-1}{12}h^3f''(z)$
- {download}`handwritten solution for Exercise 4.1 `
- Video explanation of Q1
- video explanation of Q2
:::
:::{exercise}
:label: chap04-ex-4.2
Use the method described in the text (see {prf:ref}`chap04-example-3`) to find the
order and the error constants of the following formulae:
1. $\displaystyle y_{j+1} ~=~ y_j
+ \tfrac{h}{12}(5f_{j+1} + 8f_j - f_{j-1})$
2. $\displaystyle y_{j+1} ~=~ y_{j-1}
+ \tfrac{h}{3}(f_{j+1} + 4f_j + f_{j-1})$
3. $\displaystyle y_{j+1} ~=~ y_j + \tfrac{h}{24}(55f_j - 59f_{j-1}
+ 37f_{j-2} - 9f_{j-3})$
4. $\displaystyle y_{j+1} ~=~ y_j + \tfrac{h}{720}(1901f_j - 2774f_{j-1}
+ 2616f_{j-2} - 1274f_{j-3} + 251f_{j-4})$
You can use the Matlab or Python program given in {numref}`chap:errorConstant:constants:code`.
:::
:::{solution} chap04-ex-4.2
:class: dropdown
1. $\frac{-1}{24}h^4f'''(z)$,
1. $\frac{-1}{90}h^5f^{(iv)}(z)$，
1. $\frac{251}{720}h^5f^{(iv)}(z)$,
1. $\frac{95}{288}h^6f^{(v)}(z)$
Video explanantion of 1 & 2
:::
:::{exercise}
:label: chap04-ex-4.3
Derive a Milne's Device local error estimate for the Milne-Simpson
Predictor-Corrector method:
$$\begin{aligned}
y^{(p)}_{j+1} ~&=~ y_{j-3} + \tfrac{h}{3}(8f_j - 4f_{j-1} + 8f_{j-2})
+ \tfrac{28}{90}h^5y^{(v)}(z_1)
\\
y^{(c)}_{j+1} ~&=~ y_{j-1} + \tfrac{h}{3}(f_{j+1} + 4f_j + f_{j-1})
- \tfrac{1}{90}h^5y^{(v)}(z_2)
\end{aligned}$$
:::
:::{solution} chap04-ex-4.3
:class: dropdown
$$
\begin{aligned}
\epsilon
& = \frac{C_n^{(c)}}{C_n^{(p)}-C_n^{(c)}} \left( y_{j+1}^{(c)} - y_{j+1}^{(p)} \right) \\
& = \frac{- \tfrac{1}{90}}{\tfrac{28}{90}-\left(- \tfrac{1}{90}\right)} \left( y_{j+1}^{(c)} - y_{j+1}^{(p)} \right) \\
& = \frac{-1}{29} \left( y_{j+1}^{(c)} - y_{j+1}^{(p)} \right)
\end{aligned}
$$
:::
:::{exercise}
:label: chap04-ex-4.4
Modify [](ch03:program2) for Adams--Bashforth method (Chapter 2)
to implement Milne-Simpson Predictor-Corrector method given in {ref}`chap04-ex-4.3`. Use your
program to find solution for the initial--value--problem
$$\begin{aligned}
&& y' ~&=~ -y + x + 1
& y(0) &= 1,
& &0 \leq x \leq 1 &&
\end{aligned}$$
with $h \!=\! 0.1$. Use the Milne's Device error
formula, derived in question 3 above to find an error estimate at
$\,x \!=\! 1.0\,$. Compare the error with the actual error and that
of the Adams--Bashforth--Moulton method calculated in the lecture
notes, {prf:ref}`chap04-example-5`, and comment.
:::
:::{solution} chap04-ex-4.4
:class: dropdown
At $x = 1.0$:
The error in ABM method using Milne's Device error formula is
$\approx 1.2\times10^{-7}$, the actual error is
$\approx 1.2\times10^{-6}$;
The error in MS method using Milne's Device error formula is
$\approx 5.5\times10^{-8}$, the actual error is
$\approx 3.0\times10^{-7}$;
Video explanation
:::