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.


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?



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



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