Write a function with the header [s] = myNewton 3 (f1, f2, f3, x, eps) which sol

Question

Write a function with the header [s] = myNewton 3 (f1, f2, f3, x, eps) which solves a system of three non-linear equations (given by function handles f1, f2, and f3) given an initial guess vector x, and returns the root vector s. f1, f2 and f3 should be handles to functions of x (and x will be a column vector in R^3). Use a convergence criteria on the incrementor (delX) such that the algorithm keeps going as long as abs (delX) > eps. CHECK YOUR ALGORITHM FOR INFINITE LOOPS! Test case: f_1 (x, y, z) = x^5 + y^3 z^4 + 1 f_2 (x, y, z) = x^2 yz f_3 (x, y, z) = z^4 - 1 Try your function with these initial guess vectors: x = {-100 0 -100} x = {-100 0 100} >> f1 = @ (x) x(1).^5 + x(2).^3*x(3).^4 + 1; >> f2 = @ (x) x(1).^2*x(2)*x(3); >> f3 = @ (x) x(3).^4 - 1; >> x = [-100; 0; -100]; >> s = myNewton3 (f1, f2, f3, x, 1e - 7) s = -1 0 1 >> x = [-100; 0; 100]; >> s = myNewton3 (f1, f2, f3, x, 1e - 7) s = -1 0 1

Explanation / Answer

function [x,iter]=newton(x0,f,fp) % newton-raphson algorithm

N = 100; eps = 1.e-7; % define max. no. iterations and error

f1 = @(x) x(1).^5 + x(2).^3*x(3).^4 + 1;

f2 = @(x) x(1).^2*x(2)*x(3);

f3 = @(x) x(3).^4-1;

x = [-100; 0; -100];

Jacobian J

function [x,iter] = newtonm(x0,f,J)
% Newton-Raphson method applied to a
% system of linear equations f(x) = 0,
% given the jacobian function J, with
% J = del(f1,f2,...,fn)/del(x1,x2,...,xn)
% x = [x1;x2;...;xn], f = [f1;f2;...;fn]
% x0 is an initial guess of the solution
N = 100; % define max. number of iterations
epsilon = 1e-7; % define tolerance
maxval = 10000.0; % define value for divergence
xx = x0; % load initial guess
while (N>0)
JJ = feval(J,xx);
if abs(det(JJ))<epsilon
error('newtonm - Jacobian is singular - try new x0');
abort;
end;
xn = xx - inv(JJ)*feval(f,xx);

if abs(feval(f,xn))<epsilon

x=xn;

iter = 100-N;

return;

if abs(feval(f,xx))>maxval

iter = 100-N;

disp(['iterations = ',num2str(iter)]);

error('Solution diverges');

abort;

end;

N = N - 1;

xx = xn;

end;

error('No convergence after 100 iterations.');

abort; % end function

function [f] = f2(x)

% f2(x) = 0, with x = [x(1);x(2)]

% represents a system of 2 non-linear equations

f1(x,y,z) = x^5 + y^3 z^4 + 1;

f2(x,y,z) = x^2 y z;

f3(x,y,z) = z^4 - 1;

f = [f1;f2;f3];

% end function

function [J] = jacob3x3(x)

% Evaluates the Jacobian of a 3x3

% system of non-linear equations

J(1,1) = 2*x(1);

J(1,2) = 2*x(2);

J(1,3) = x(2);

J(2,2) = x(1);

J(2,1) = x(1);

J(2,2) = x(1);

J(2,3) = x(1);

J(3,1) = x(1);

J(3,2) = x(1);

J(3,3) = x(1);

% end function

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