Find solution of under-determined system that is close to a starting point

Question

Can I provide a starting point y to one of the eigen's solver for Ax = b?

I want to get an exact solution x that is close to the starting point y.

When would solving Ax = b with Newton's method (without line search) not converge?

Thanks.

Answer 1

Solve Ax = b for a solution that minimize L2 norm of x - y.

Given a y, that means minimizing L2 norm of x - y, subjected to the constraint Ax = b.

By Lagrange multiplier I get the following block matrix equation.

The vector u are one half of the lagrange multipliers.

/ A    0  \  / x \     / b \
|         |  |   |  =  |   |
\ I   A^T /  \ u /     \ y /

Probably not the fastest way.