Question

. (30 points) Determine the roots of the simultaneous nonlinear equations (x-4)" +(y-4)2-3 x2 +y2 = 16 based off of the plots of the two equations shown in the figure, use x = 2 and y = 4 as your initial guess. Thus, your answer should be the left hand side root or the intersection point on the graph 6 5 3 0 0 2 6 Determine refined estimates with (a) the Newton- Raphson method (modify the newtmult file and use es-0.0001) (b) the fsolve function. **Please make a main file to call your newtmult and fsolve in a single script** Final answers: 1.8058 3.5692

____________

% This function is a modified versio of the newtmult function obtained
% from
% “Applied Numerical Methods with MATLAB, Chapra,
% 3rd edition, 2012, McGraw-Hill.”

function [x,f,ea,iter]=newtmult(func,x0,es,maxit,varargin)
% newtmult: Newton-Raphson root zeroes nonlinear systems
% [x,f,ea,iter]=newtmult(f,J,x0,es,maxit,p1,p2,...):
% uses the Newton-Raphson method to find the roots of
% a system of nonlinear equations
% input:
% f = the passed function
% J = the passed jacobian
% x0 = initial guess
% es = desired percent relative error (default = 0.0001%)
% maxit = maximum allowable iterations (default = 50)
% p1,p2,... = additional parameters used by function
% output:
% x = vector of roots
% f = vector of functions evaluated at roots
% ea = approximate percent relative error (%)
% iter = number of iterations

if nargin<2,error('at least 2 input arguments required'),end
if nargin<3|isempty(es),es=0.0001;end
if nargin<4|isempty(maxit),maxit=50;end

iter = 0;
x=x0;

while (1)
% find function value and the jacobian
% Calculate the new x
iter = iter + 1;
% Asses the error
if "Check stopping criteria", break, end
end

Answer 1

Solution:3 Given simultaneous nonlinear equations are

and

Using initial guess x=2 and y=4.

Now computing the partial derivatives and evaluate at the initial guessing

points at x=2 and y=4., we have

af1,0 2(1-1) = 2(2-1) -4 Им = 2(y-1)(24) = 2(4-4) = 0 ду

and

0f2,0 0120 = 2y(2.4) ду :=

Therefore, determinant of Jacobian for the first iteration becomes

=--32

The values of the functions at the initial guess points are,

fi(2,4) (2-4)(4 -4)2-51 and f2(2,4)22 +42- 16 4

Since we know that,

f1.i

and

อา * k

Therefore, the values

fi(2,4) (2-4)(4 -4)2-51 and f2(2,4)22 +42- 16 4

can be substituted in the above equations * and **, we have

-1 *8 - 4 0 ー2 2-0.25-1.75 32

and

(4 *一4)-(-1 * 4) 一32 2 3 29 8 4 4 3.62