Asked by Matthew
on 18 Nov 2012

function [r,N]= newtonraphe7(fHan,x0,fTol,iterMax) iterations=0; max_no_roots=size(fHan,2); r=0; N=0; for no_roots=1:max_no_roots-1 fun_root_new=x0; flag=1; coeff_der_function=derivate(fHan); order_fun=size(fHan,2); order_der_fun=size(coeff_der_function,2); while flag==1 fun_root_old=fun_root_new; fx=0; dfx=0; nonzero=1; while nonzero==1 powers=order_fun-1; for index=1:order_fun fx=fx+fHan(index)*fun_root_old^powers; powers=powers-1; end powers=order_der_fun-1; for index=1:order_der_fun dfx=dfx+coeff_der_function(index)*fun_root_old^powers; powers=powers-1; end if dfx==0 fun_root_old=fun_root_old+1; else nonzero=0; end end iterations = iterations + 1; fun_root_new = fun_root_old - fx/dfx; if iterations >= iterMax flag=0; elseif abs(fun_root_new-fun_root_old)<=fTol flag=0; r(no_roots)=fun_root_new; N(no_roots)=fx; end end

end

I am trying to implement the Newton-Raphson algorithm to solve nonlinear algebraic equations. My function, newtonRaphE7, finds a real root of the nonlinear algebraic equation f(x) = 0, using a Newton-Raphson algorithm.

Input arguments:

- • fHan: function handle to f(x)
- • dfHan: function handle to f′(x) = df(x)/dx,
- • x0: scalar, double, representing an initial guess, x0,
- • fTol: tolerance for stopping criterion, i.e.,
`f(r)`≤ fTol, - • iterMax: maximum number allowed iterations.

Output Arguments

- • r: the final root estimate obtained by the algorithm,
- • rHist: a vector containing the sequence of estimates including the initial guess x0,
- • N: the total number of performed iterations,
- • fRoot: the absolute value of the function f(x) at r, i.e., fRoot =
`f(r)` - If the initial guess, x0 satisfies the stopping criterion, then the root r should be returned as x0, and N should be 0 since no actual iterations were performed.

In general, rHist contains the sequence of estimates x0, x1, x2, . . . , that are calculated by the algorithm. The first element of rHist should be x0, therefore the value returned in N should be equal to length(rHist)-1.
In the case N ≥ 1, the function should compute the root, rE, using the MATLAB function fzero. Treat rE as the “exact” root, and produce a plot (using semilogy) of the absolute error (i.e., `rHist − rE`) as a function of iteration (i.e., 0:N).

**I don’t know how to in cooperate the variables “dfHan”and “froot” and “rHist” into my function. Can someone help? Thanks!**

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## 2 Comments

## bym (view profile)

Direct link to this comment:http://www.mathworks.com/matlabcentral/answers/54057#comment_111855

At first blush, your code seems more complicated than the ordinary Newton Raphson implementation. I also don't see where you have implemented the function 'fzero'

## Matthew (view profile)

Direct link to this comment:http://www.mathworks.com/matlabcentral/answers/54057#comment_111870

Yeah, I originally just tried the Newton Raphson but my instructor told me that I needed to add the extra variables in there. I don't see HOW to include them in there though ...