reasons for error in lu decomposition

Question

This is a very general question. Let's assume i have 3 pairs of point correspondences $(p_i, q_i)$ with real coordinates an i want to compute the transformation matrix that transforms the point $p_i$ to its corresponding point $q_i$. I use LU decomposition to do this. If i modify the coordinates a bit mostly the changes in the transformation matrix are minimal. But sometimes change of coordinates lead to a totaly different transformation matrix. What can cause this behaviour?

Answer 1

Let $P=[p_1,p_2,p_3],Q=[q_1,q_2,q_3]$; we assume that $P$ is invertible. Then $AP=Q$ implies $A=QP^{-1}$.

The calculation of $A$ is unstable when

the calculation of $P^{-1}$ is unstable, that is when $cond(P)=||P||||P^{-1}||$ is a large number. Indeed, if $cond(P)=10^k$, then we lose $k$ significant digits.

Assume that $||Q-\tilde{Q}||,||P^{-1}-\tilde{P^{-1}}||\approx 10^{-p}$ where $\tilde{Q},\tilde{P^{-1}}$ are the obtained values of $Q,P^{-1}$; then $||\tilde{Q}\tilde{P^{-1}}-A||\leq 10^{-p}(||P^{-1}||+||Q||)$ (problem when $||P^{-1}||$ is large).