roblem $1$: Application of the Fixed$-$Point Iteration Method

In class, we have presented the non$-$dimensional frequency equation, given below, whose roots are

related to the natural frequencies of vibration of a cantilever beam.

$()=1+$ cosh$()$ cos$()=0$

The following iterative algorithm is proposed in an attempt to obtain the roots of the frequency

equation using the fixed$-$point iteration method

$$k$+1=1/($cosh$($k$))+$ cos$($k $)+$k $,=1,2,3,...$

$($In your MATLAB programming, you may replace $$ by $$ for simplicity.$)$

In the pre$-$lab, you have already developed a numerical solver for the fixed$-$point iterative algorithm

which will terminate when the relative approximate error is smaller than a user$-$specified maximum

tolerance maxtol or when the number of iterations exceeds a user$-$specified maximum number of

iterations maxitr.

a$)$ Show how the above algorithm is derived based on fixed point iteration concept. This is the template: function $[$x$,$ er$,$ n$]=$ FixedPoint$($g$,$ x$1,$ maxtol, maxitr$)$

$\%\%$ Numerical solution of f$($x$)=0$ by the fixed point iteration method

$\%$

$\%$ Wayne State University $/$ ME$2500$

$\%$ Copyrighted by: Chin An Tan and Heather L$.$ Lai

$\%$ Numerical Methods & Programming Using MATLAB: Principles and Practice

$\%$ Revised $26$ January $2016,19$ Sept $2016,27$ Jan $2017$

$\%$

$\%$ Inputs$/$Outputs:

$\%$ g $=$ iterative function

$\%$ x$1=$ initial estimate of the root

$\%$ maxtol $=$ maximum error tolerance

$\%$ maxitr $=$ maximum number of iterations allowed

$\%$ x $=$ approximate solution $($root$)$ of the equation fun$($x$)=0$

$\%$ er $=$ final absolute relative approximate error

$\%$ n $=$ number of iterations needed for convergence based on maxtol

$\%\%$ Initializations and checks

$\%$ Check for missing input arguments $($maxtol and maxitr$)$ and set defaults

if nargin $<4,$ maxitr $=25$; end

if nargin $<3,$ maxtol $=1$e$-3$; end

x $=$ x$1$; $\%$ Set the initial guess as the current estimate

er $=1$; $\%$ Initialize the approximate error to $1$

k $=0$ ; $\%$ Initialize the iteration number

$\%\%$ Implementation of fixed$-$point iteration algorithm

while er $>=$ maxtol && k $<$ maxitr

k$=$k$+1$;

xold $=$ x;

x$=$g$($x$)$;

er$=$abs$(($x$-$xold$)/$x$)$;

$\%$ Output the values of k$,$ x and er at each iteration

fprintf$(\text{'}$iter $=\%$i$,\backslash$t x $=\%$e$,\backslash$t er $=\%$e

$\text{'},$ k$,$x$,$er$)$;

end

n$=$k;

$\%\%$ Check results for no convergence

if n $==$ maxitr && er $>=$ maxtol

fprintf$(\text{'}$

$\text{'})$

warning$(\text{'}$Maximum number of iterations reached before convergence'$)$

fprintf$(\text{'}$

Solution not obtained in $\%$d iterations.

$\text{'},$maxitr$)$;

end

end