Problem $6$: Applying Concepts of Error Analysis to the Babylonian Method$/$Algorithm

One numerical method for calculating the square root of a number is the Babylonian method. In this method $\sqrt[\phantom{2}]{p}$ is calculated in iterations. The solution process starts by choosing a value $x_1$ as a first estimate of the solution. Using this value, a second, more accurate solution $x_2$ can be calculated with $x_2=\frac{x_1+\frac{p}{x_1}}{2},$ which is then used for calculating a third, still more accurate solution $x_3,$ and so on$.$ The general equation for the algorithm to estimate the solution for the square root of a number is thus $x_{i+1}=\frac{x_i+\frac{p}{x_i}}{2}.$

a$)$ Write a MATLAB user$-$defined function that calculates the square root of a number using the Babylonian method. Name the function

function $[sqr,er]=$ Babylonian$(p,$ maxitr$)$

where maxitr is the maximum number of iterations to be performed, er is a vector of percent errors, and sqr is the estimated value of the square root of $p.$ In the program, use $x_1=p$ for the first estimate of the solution. Then, by using the general equation in a loop, calculate new, more accurate solutions. Your function should define a vector of the approximate percent errors, er$,$ defined by er $r_k=100\%*|\frac{x_{\text{n}\text{o}\text{w}}-x_{\text{o}\text{l}\text{d}}}{x_{\text{n}\text{e}\text{w}}}|$ for the $k-$th iteration and plot the error versus the number of iterations. A partially completed code has been provided $($remove the "part" from the function handle before running$).$

b$)$ Use Babylonian to find the square root of $12\mathrm{.}34.$ Use the value of $20$ for maxitr. Interpret the output plot. Compare your result with the MATLAB result using sqrt$.$

c$)$ Modify your Babylonian function so that a while$-$loop checks both the error tolerance and the maximum number of iterations. Name the function

function $[sqr,er]=$ BabylonianErr$(p,$ maxtol, maxitr $)$

where the input arguments $p$ is the number whose square root is to be determined, maxtol is the maximum tolerance, and maxitr is the maximum number of iterations allowed. The output arguments sqr is the estimated value of the square root of $p,$ and er is the percent error. In the program, use $x_1=p$ for the first estimate of the solution. Then, by using the general equation in a loop, calculate new, more accurate solutions. Terminate the looping if either of the following conditions is met

The relative approximate error is less than or equal to the maximum error tolerance, maxtol

The number of iterations exceeds the maximum number of iterations allowed, maxitr.

d$)$ Use BabylonianEr $r$ to find the square root of $12\mathrm{.}34.$ Use the value maxtol $=1e-5$ and $matr=10.$