Finding the Jacobian of non square inverse matrix

Question

I am trying to use the Newton method to a non square system. The jacobian matrix is not square and I cannot inverse it. Can anyone help? How can I calculate this 'inverse'?.

Answer 1

Apparently you can use the pseudo-inverse of the Jacobian to implement the Newton-Raphson method : http://en.wikipedia.org/wiki/Inverse_kinematics#The_Jacobian_inverse_technique

Answer 2

About the Nihl's post, the so called "pseudo Newton method" - when we use the pseudo-inverse of the derivative -

We consider a system of $n$ equations in $p$ unknowns with $n\not= p$.

If $n<p$, for instance $f(x,y)=0$, then the pseudo Newton method gives a particular point on the curve $f(x,y)=0$.

If $n>p$, for instance $f_1(x,y)=f_2(x,y)=f_3(x,y)=0$, then it is more complicated. In general, there are no solutions. Assume that the $3$ curves intersect in $3$ nearby points $A,B,C$ (we are in this case when the intersection exists but, during the previous calculations, we did approximations). Eventually, the pseudo Newton method gives a point $P$ that is pretty much in the triangle $ABC$. Yet, if we put some weight on the curve $f_1$ (change $f_1$ with $5f_1$ for instance), then $P$ comes closer to the curve $f_1=0$.