Help me with Matlab code on Newton-Raphson method to solve a system of nonlinear equations.

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Mabud Sarkar 2022-4-30

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评论： Mabud Sarkar 2022-5-1

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I want to approximate the zeros of the following nonlinear system, using N-R method:

f(x,y)=x+y^9/3+x^{243}/9+y^{2187}/27=0;

g(x,y)=y+x^{27}/3+y^{243}/9+x^{6561}/27=0.

I have tried with the following Matlab code. But I am getting error. Can you please help me out ?

close all; clc,clear

% Newton Raphson solution of two nonlinear algebraic equations

xy = [-1 -1]; % initial guesses

iter=0;

maxiter=100;

xy_N = 0.5;

TOL = 0.05;

error1 = [1, 1];

f = zeros(iter, 2);

error = zeros(iter, 2);

% begin iteration

while xy_N>TOL

iter=iter+1;

x = xy(1);

y = xy(2);

% calculate the functions

f(iter,:) = [x+y^9/3+x^{243}/9+y^{2187}/27; y+x^{27}/3+y^{243}/9+x^{6561}/27];

% calculate the Jacobian

J= [1+27*x^{242}, 3*y^8+81*y^{2186}; 9*x^{26}+24*x^{6560}, 1+27*y^{242}];

xy_new = xy - ((J)\f(iter,:)')';

error1=abs(xy_new-xy);

error(iter,:)=error1;

% Calculate norm:

xy_N = norm(f(iter,:));

XYN(iter) = xy_N;

xy=xy_new;

%iter=iter+1;

if (iter > maxiter)

fprintf('Did not converge! \n', maxiter);

break

end

if iter<maxiter

f(iter+1:end,:)=[];

end

% Print results

fprintf('Number of iterations is: %f \n ', iter)

fprintf('Final values of [x, y] = %f %f \n', xy(end, :));

fprintf('Final values of EQN: %f %f \n', f(end, :));

fprintf('Final values of ERROR: %f %f \n', error(end, :));

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Answer 1

Ahmad 2022-4-30

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https://ww2.mathworks.cn/matlabcentral/answers/1708755-help-me-with-matlab-code-on-newton-raphson-method-to-solve-a-system-of-nonlinear-equations#answer_954405

why you use { } ?

this is for cell type

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Mabud Sarkar 2022-4-30

Thanks. I have changed the notation and used the following expression:

close all; clc,clear

% Newton Raphson solution of two nonlinear algebraic equations

xy = [-1 -1]; % initial guesses

iter=0;

maxiter=100;

xy_N = 0.5;

TOL = 0.05;

error1 = [1, 1];

f = zeros(iter, 2);

error = zeros(iter, 2);

% begin iteration

while xy_N>TOL

iter=iter+1;

x = xy(1);

y = xy(2);

% calculate the functions

f(iter,:) = [x+(1/3)*y^9+(1/9)*x^243+(1/27)*y^2187; y+(1/3)*x^27+(1/9)*y^243+(1/27)*x^6561];

% calculate the Jacobian

J= [1+27*x^242, 3*y^8+81*y^2186; 9*x^26+24*x^6560, 1+27*y^242];

xy_new = xy - ((J)\f(iter,:)')';

error1=abs(xy_new-xy);

error(iter,:)=error1;

% Calculate norm:

xy_N = norm(f(iter,:));

XYN(iter) = xy_N;

xy=xy_new;

%iter=iter+1;

if (iter > maxiter)

fprintf('Did not converge! \n', maxiter);

break

end

if iter<maxiter

f(iter+1:end,:)=[];

end

% Print results

fprintf('Number of iterations is: %f \n ', iter)

fprintf('Final values of [x, y] = %f %f \n', xy(end, :));

fprintf('Final values of EQN: %f %f \n', f(end, :));

fprintf('Final values of ERROR: %f %f \n', error(end, :));

But it is giving error. The following display:

Warning: Matrix is singular to working precision. Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN.

Number of iterations is: 4.000000

Final values of [x, y] = NaN NaN

Final values of EQN: NaN NaN

Final values of ERROR: NaN NaN

Can you please help me to fix the error ?

Alex Sha 2022-5-1

[x,y]=[0,0] is the exact and only solution

Mabud Sarkar 2022-5-1

No there are more 3 zeros nearest to (-1,1), (1,-1) and (-1,-1). Please see the answer of @Cesareo here

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Help me with Matlab code on Newton-Raphson method to solve a system of nonlinear equations.

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Help me with Matlab code on Newton-Raphson method to solve a system of nonlinear equations.

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