How quickly minimizing $M*x-v$ (numerically) ?
Let $v$ in $R^m$ and let $M$ be a matrix from $R^n$ to $R^m$, with $m>n$ big numbers.
I want to compute a vector $x$ in $R^n$ such that the norm of $M*x-v$ is minimal.
One way is to compute the projection $w$ of $v$ on the image of $M$.
For so, we can compute the projection $p$ on the image of $M$, as follows :
MTGS=M.transpose().gram_schmidt() # it's orthogonalization, not orthonormalization l=MTGS.rank() U= for i in range(l): v=MTGS[i] u=v/(v.norm()) L=list(u) U.append(L) N=matrix(m,l,U) p=N.transpose()*N
This vector $x$ minimizes the norm of $M*x-v$, but this method is very expensive in time, because it computes $p$ and $w$, while I just need $x$.
Is there another method, less expensive in time, for computing $x$ ?
Remark : I'm ok with numerical methods.