Debugging Newton's Method code in two variables,

Hi,

I wrote a simple code for Newton's Method in two variables but am having some trouble debugging it. Here's the message I get:

Index exceeds matrix dimensions.

Error in Root_finding_practice>@(x)[cos(x(2)),-x(1)*sin(x(2));x(2)*cos(x(1)),sin(x(1))]

Error in Root_finding_practice (line 34)

x(i+1) = x(i) - ( inv( J( x(i) ) ) * f( x(i) ) );

The function file code is:

function F = nonlinear_equations(x)
 
F(1) = x(1) * cos( x(2) );
 
F(2) = x(2) * sin( x(1) );
 
end

and the script file code is:

f = @(x) nonlinear_equations;
 
% Jacobian
 
J = @(x) [ cos( x(2) ), -x(1)*sin(x(2));
    x(2) * cos(x(1)), sin(x(1)) ];
 
x = [ 1, 1 ];
 
for i = 1:1000   % it should be stopped when tolerance is reached
        
    x(i+1) = x(i) - (  inv( J( x(i) ) ) * f( x(i) ) );
       
    if( abs( f( x(i+1) ) ) < 0.0001 )   % tolerance
            
            disp(double(x(i+1)));
            
            break;  
    end
       
end 

What am I missing? I suspect it's the way I've defined the Jacobian anonymous function ...

Thanks,

Answer 1

[Ameer Hamza]

Check this code

f = @(x) nonlinear_equations(x);

% Jacobian
J = @(x) [cos( x(2) ), -x(1)*sin(x(2));
    x(2)*cos(x(1)), sin(x(1))];
x = [1; 1];

for i = 1:1000   % it should be stopped when tolerance is reached
    
    x(:,i+1) = x(:,i) - inv(J(x(:,i)))*f(x(:,i));
    
    if( abs(f(x(:, i+1))) < 0.0001)   % tolerance
        disp(double(x(:, i+1)));
        break;
    end
end

function F = nonlinear_equations(x)
F = zeros(2, 1);
F(1) = x(1) * cos( x(2) );
F(2) = x(2) * sin( x(1) );
end

[Noob]

Hi Ameer,

Thanks for your answer; how come you added the F = zeros(2,1) to the function file code? Is it necessary?

Now, I'm getting this error, after implementing your suggestions:

Not enough input arguments.

Error in nonlinear_equations (line 6)

F(1) = x(1) * cos( x(2) );

Error in Root_finding_practice>@(x)nonlinear_equations

Error in Root_finding_practice (line 36)

x(:,i+1) = x(:,i) - ( inv( J( x(:,i) ) ) * f( x(:,i) ) );

[Ameer Hamza]

Yes, F = zeros(2, 1); is needed to make sure that the function returns a column vector. Otherwise the matrix multiplication will fail. The alternative is

function F = nonlinear_equations(x)
F(1) = x(1) * cos( x(2) );
F(2) = x(2) * sin( x(1) );
F = F(:);
end

Both are doing the same thing.

About the error; can you exactly paste the code which gives this error.

[Noob]

I see; sure, here's the code:

The function file first:

function F = nonlinear_equations(x)
 
% F = zeros(2, 1);
 
F(1) = x(1) * cos( x(2) );
 
F(2) = x(2) * sin( x(1) );
 
F = F(:);
 
end

And the script file:

% root-finding practice in two variables:
 
f = @(x) nonlinear_equations;
 
% Jacobian
 
J = @(x) [ cos( x(2) ), -x(1)*sin(x(2)); 
           x(2) * cos(x(1)), sin(x(1)) ];
 
x = [1; 1];
 
for i = 1:1000   % it should be stopped when tolerance is reached
        
    x(:,i+1) = x(:,i) - ( inv( J( x(:,i) ) ) * f( x(:,i) )  );
       
    if( abs( f( x(:,i+1) ) ) < 0.0001 )   % tolerance
            
            disp(double(x(:,i+1)));
            
            break;  
    end
       
end 

Thanks,

[Ameer Hamza]

I also changed the line to

f = @(x) nonlinear_equations(x);

Alternative is

f = @nonlinear_equations;

[Noob]

Hi Ameer,

Thanks so much! It works now, and I think I prefer your alternative f = @nonlinear_equations, rather than creating what looks like a redundant anonymous function.

Have a great day / night.

[Noob]

Hi Ameer,

Just a quick question, if you don't mind: how can I evaluate the function now? The extra variable is throwing me off a bit:

>> z = nonlinear_equations(-0.000000006238568,0.809548393600924)

Error using nonlinear_equations

Too many input arguments.

Thanks,

[Noob]

Oh, I see; I just have to define the inputs as a vector first.

Thanks again, Ameer!

[Ameer Hamza]

Yes, input need to be passed as vector.

I am glad to be of help :)