8.8 Quiz 4

Please submit solutions to the following via moodle by 23:59 on Wednesday 14th December. Each question is worth [2] marks. For Q4.3-Q4.5, please see Week09Looping.pdf.

Q4.1Two integrals. Calculate

$$I=\int_{0}^{\infty}e^{-y}\cos xy\,dy$$

and hence find

$$J=\int_{0}^{1}\!\!\!\int_{0}^{\infty}e^{-y}\cos xy\,dydx.$$

One of the following is true.

(A) $I=1/(x+x^{2})$ and $J$ diverges. $\;\;\;\;$ (B) $I=x/(1+x^{2})$ and $J={{1}\over{2}}\log 2$.

(C) $I=1/(1+x^{2})$ and $J=\pi/4$. $\;\;\;\;$ (D) $I$ diverges; $\;\;\;\;$ (E) $I=1/(1+x^{2})$ and $J=\log 2$.

Q4.2Stationary points. The stationary points of $f(x,y)=\frac{1}{6}x^{3}+y^{2}+xy$ are classified in the following.

(A) There are four stationary points.

(B) There is a saddle at $(0,0)$ and a local minimum at $(1,-1/2)$.

(C) There is a saddle at $(0,0)$ and a local maximum at $(1,-1/2)$.

(D) There is a local maximum at $(0,0)$ and a local minimum at $(1,-1/2)$.

(E) There is an inflexion at $(0,0)$ and a local minimum at $(1,1/2)$.

Q4.3Newton-Raphson method (i). One procedure for which a while loop is useful for implementation is the Newton-Raphson method for finding the roots of the differentiable function $f(x)$. This algorithm iterates the following recursion

$\displaystyle x_{n}=x_{n-1}-\frac{f(x_{n-1})}{f^{\prime}(x_{n-1})},$

whilst the difference between successive evaluations is greater than a specified small value. Consider using the Newton-Raphson method to find one of the roots of the function:

$\displaystyle f(x)$

$\displaystyle=$

$\displaystyle x^{4}-4x^{2}-\frac{36}{7}$

Take ${\color{blue}\colorlet{pgfstrokecolor}{.}x_{0}=-4}$ as the your starting point and use

$\displaystyle|x_{n}-x_{n+1}|\leq 1\times 10^{-5}$

as the convergence criterion.

Give the value of ${\color{blue}\colorlet{pgfstrokecolor}{.}x_{n}}$ at convergence to 2 decimal places.

Q4.4Newton-Raphson method (ii). Determine the number of iterations needed to reach convergence i.e. the first ${\color{blue}\colorlet{pgfstrokecolor}{.}n}$ for which

$\displaystyle|x_{n}-x_{n+1}|\leq 1\times 10^{-5}.$

Q4.5Hénon Map. The Hénon map is defined by the recursion equations

	$\displaystyle x_{n+1}$	$\displaystyle=$	$\displaystyle y_{n}+1-ax_{n}^{2}$
	$\displaystyle y_{n+1}$	$\displaystyle=$	$\displaystyle bx_{n}$

Write some R code to produce sequences ${\color{blue}\colorlet{pgfstrokecolor}{.}x_{n},y_{n}}$ as above, up to ${\color{blue}\colorlet{pgfstrokecolor}{.}n=1000}$, starting at ${\color{blue}\colorlet{pgfstrokecolor}{.}x_{0}=y_{0}=0}$, for parameters ${\color{blue}\colorlet{pgfstrokecolor}{.}a,b}$ as follows.

(i) ${\color{blue}\colorlet{pgfstrokecolor}{.}a=1.15,b=0.45}$, (iii) ${\color{blue}\colorlet{pgfstrokecolor}{.}a=1.4,b=0.3}$

(ii) ${\color{blue}\colorlet{pgfstrokecolor}{.}a=1.05,b=0.5}$, (iv) ${\color{blue}\colorlet{pgfstrokecolor}{.}a=1.2,b=0.2}$

(Please submit this to Paul Levy.)