# Newton Raphson Iteration method in Matlab

I am currently trying to learn how to use ${\tt Matlab}$ properly but having a bit of trouble, I want to write a code that, say i had a differentiable function, will take as input $f$, $f'$, $x_0$ and, and return output $x_K$ for some $K$ in natural numbers $N$ such that $$x_k := x_{k-1} - \frac{f(x_{k-1})}{f'(x_{k-1})} \text{ for } k = 1,2,..,k \text{ and } |f(x_K)| < \epsilon.$$ I'm guessing $f$ and $f'$ should be passed as strings, and I should use the ${\tt Matlab}$ command line. Could someone please show me how to write a code for an M-file to run this iteration?

Also, would I have to write a driver code that will call the code above and will take as inputs the functions f,f0 (e.g., as strings or function handles, if you prefer) and x0.

EDIT: This is what I have written but I get errors when I try and run it,

function [] = newton1Driver1()
%% function [] = newton1Driver1()

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

f = inline("exp(x)-2");

df = inline("exp(x)");

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[solution,residual] = newton1(f,df,1);

error = solution-fsolve(f,1)

%% plot the log of the error vs. the steps

plot(log(error));

%% We write the output to a file, so it can be read by typesetting software

filename = sprintf("Output/newton1-1-%1.4f.csv",solution)

file = fopen(filename,"w")

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

headers = ["step $k$,","solution $x_k$,","residual $\\abs{f(x_k)}$,","error

$\\abs{x_k-\\hat x}$"]

out = [1:length(solution);solution;abs(residual);error]

fprintf(file,"%i,%1.8e,%1.4e,%1.4e\n",out)

fprintf(file,"\n");

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

fclose(filename)

-

MATLAB allows for the creation of function handles. This can be done a few ways. One approach is to explicitly define the functions:

function myNewtonCode()

K = 25;
x = zeros(K,1);
x(1) = x_init;

for k=2:K
x(k) = x(k-1)-f(x(k-1))/fp(x(k-1));
end

function y=f(x)
y = 45*x^2+22*x-13;
end

function yp=fp(x)
% Derivative of f
yp = 90*x+22;


Replace the ... with the actual form of $f$, whatever that is.

You could also do this to call it inline from the MATLAB command window

function x = testCode(x_init,f,fp)
% Filename: testCode.m

K = 25;
x = zeros(K,1);
x(1) = x_init;

for k=2:K
x(k) = x(k-1)-f(x(k-1))/fp(x(k-1));
end


And call it with x = testCode(1,@f,@fp);. Note that you have to create separate function files for $f$ and $f'$.

A less clunky way of doing this is through anonymous functions:

f = @(x)...
fp = @(x)...
x = testCode(1,f,fp);


Notice that when you do it with anonymous functions vs. static files on the disk, you drop the @ symbol.

In general, you should never need to pass functions as strings. In doing so, you'd have to call into eval, which is bad practice and could corrupt variables unintentionally.

-