# Solve Ax=b using Cholesky decomposition

I was reading the following article about the direct stiffness method. When it comes to solving the system of equations:

The site states:

[...]There are several different methods available for evaluating a matrix equation including but not limited to Cholesky decomposition and the brute force evaluation of systems of equations[...]

After doing some research, I figured out that the most common way is the Cholesky Decomposition. The problem that I have is that we are not solving for $$u$$ in the equation $$f=Ku$$ but for $$f$$ and $$k$$. In the Image, it’s clear that for if we do not know the value of $$f_i$$, we know the value of $$u_I$$ instead.

I am not sure how to solve this for both, the unknown values in $$u$$ and $$f$$ at the same time. I am very happy for any advice

I am not sure, if I get this right, but I general you cannot solve an equation system $$Ax=b$$ if you don't know both $$x$$ and $$b$$. However, in your case, you can solve the system since you got $$f_{x_{2}}=0, f_{x_{3}}=f_{x}, f_{y_{3}}=f_{y}$$ and $$u_{x_{1}}=u_{y_{1}}=u_{y_{2}}=0$$. For example you know, that $$k_{33}u_{x_{2}}+k_{35}u_{x_{3}}+k_{36}u_{y_{3}}=f_{x_{2}}=0$$.