# Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

Problem

Solving a non-linear system of equations.

Number of variable is same as number of equations.

When I fix a set variables (say $$\vec{y}$$) and keep another set free (say $$\vec{x}$$), the system becomes an under-determined, dense, and linear system of the subset of variables ($$A(\vec{y})\vec{x} = \vec{b}(\vec{y})$$, $$A(\vec{y})$$ is a dense matrix, and $$\vec{b}(\vec{y})$$ is a dense vector). Let's call this sub-system as system 1.

When I fix $$\vec{x}$$ and keep $$\vec{y}$$ free, the system becomes an over-determined, sparse, and non-linear system of the subset of variables ($$F(\vec{x}, \vec{y}) = 0$$). The Jacobian $$J(\vec{x}, \vec{y})$$ has closed form. Let's call this sub-system as system 2.

About half of the equations in system 2 are equality constraints that are linear in terms of $$\vec{y}$$. Each of the constraints are quite sparse and involves only about 5% of all variables.

Can I solve with the following?

Algorithm 1

1. Initialize $$\vec{x} = \vec{x}_0$$, and $$\vec{y} = \vec{y}_0$$
2. Fix $$\vec{y}_{n - 1}$$, solve $$A(\vec{y}_{n-1})\vec{x}_{n} = \vec{b}(\vec{y}_{n-1})$$ for one of all the possible $$\vec{x}_{n}$$ because this system is under-determined.
3. Fix $$\vec{x}_{n - 1}$$. Perform one iteration of Newton's method for solving $$F(\vec{x}_{n-1}, \vec{y}_{n}) = 0$$ for $$\vec{y}_{n}$$.
4. If not converged, go to step 2.

Algorithm 2

If I replace step 2 by an iteration of Newton's method for solving system 1, then I guess the steps become a block Newton's method.

Question

But I don't know if these two algorithms can work because system 1 is under-determined and system 2 is over-determined.

Can this work?

Thanks.

Related

An approximate block Newton method for coupled iterations of nonlinear solvers: Theory and conjugate heat transfer applications

https://www.sciencedirect.com/science/article/pii/S0021999109004379