MATLAB: Inner matrix dimensions must agree error message,

MATLABmatrices

Hi,
I'm getting an error message that says inner matrix dimensions must agree, but I checked both matrices and they seem fine — it's just a 2×2 matrix left-multiplying with a 2×1 matrix.
Here's the error message:
Error using  * 
Inner matrix dimensions must agree.
Error in Root_finding_practice (line 78)
    x(:,i+1) = x(:,i) - ( inv( H( x(:, i) ) ) * J( x(:,i) )  );  % Newton's method in 2 variables

And here's the piece of the code:
J = @(x)   [ cos( x(2) ), -x(1) * sin( x(2) ) ];    % Jacobian of f
 
H = @(x)   [ 0, -sin( x(2) );                       % Hessian matrix of 2nd derivatives
            -sin( x(2) ),  -cos( x(2) ) * x(1) ];
 
% An initial guess at a multivariable root:
 
x = [1; 1];
 
for i = 1:100                           % The for-loop should stop when the function tolerance is satisfied
    
    x(:,i+1) = x(:,i) - ( inv( H( x(:, i) ) ) * J( x(:,i) )  );  % Newton's method in 2 variables
What am I missing?
Thanks,

Best Answer

You can either perform element-wise multiplication with implicit expansion
x(:,i) - inv(H(x(:, i)) .* J(x(:,i)))
%              add this ^
however that will produce a 2x2 output which will cause an error when you index it into
x(:,i+1) =
or you can perform matrix multiplication
H(x(:, i)) is a 2x2 matrix, J(x(:,i)) is a 1x2 matrix. If you multiply these matricies, you either need to transpose the J term or switch the two terms
H(x(:, i)) * J(x(:,i)).'  % result is 2x1 matrix
% or
J(x(:,i)) * H(x(:, i))    % result is a 1x2 matrix
but matrix inversion using inv() requires the use of a square matrix and neither of those fulfill that requirement.
So, you've got some thinking to do regarding what the output should be and how you'll store it.
Related Question