# Solve linear system with Newton-Raphson method

Is it possible to solve a linear matrix system $$A x = b$$ using the Newton-Raphson method? If yes, how can this be done? More special, how is the derivative build?

• Newton’s method for this would very likely involve solving the system as a sub-step, so likely you wouldn’t see this help much. On the other hand, you could use some form of gradient descent effectively (indeed, one can view conjugate gradients in this way, and that works quite well in many situations). – cdipaolo Nov 5 '19 at 21:40
• @cdipaolo Plain gradient descent for linear systems is known as "Richardson iteration". – Federico Poloni Nov 15 '19 at 13:22
• @FedericoPoloni Ah I hadn't seen that before. Thanks for the comment! – cdipaolo Nov 15 '19 at 21:29
• No problem! For a quick comment: as you can probably expect, plain gradient descent is not too effective in practice. As far as I know, it is only used as a smoother in multigrid, and as a last resource for some ridiculously huge linear systems such as quantum chemistry where you only have $A$ in black-box form and you lack the space to run Arnoldi. – Federico Poloni Nov 16 '19 at 9:33

This is because each step of Newton's method involves solving a linear system with the Jacobian of the nonlinear function. In this case the Jacobian just equals $$A$$.
In other words: this is a little circular because it requires you to solve the system $$Ax=b$$ in the first place, which doesn't help.
• And the linear system you have to solve in each step of the Newton method is $Ax=b$ :-) – Wolfgang Bangerth Nov 5 '19 at 22:20