*Bounty: 100*

*Bounty: 100*

**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**

- Initialize $vec{x} = vec{x}_0$, and $vec{y} = vec{y}_0$
- 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*. - Fix $vec{x}_{n – 1}$. Perform one iteration of Newton’s method for solving $F(vec{x}_{n-1}, vec{y}_{n}) = 0$ for $vec{y}_{n}$.
- 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

Solving over-determined non-linear system by Newton’s method with Moore-Penrose inverse is fine: https://arxiv.org/pdf/1703.07810.pdf