readme.md

PreCol - Application in Preconditioning

1. Partial Coloring and Preconditioning
2. Coloring Algorithms  

Given a large sparse nonsingular $n\times n$ Jacobian matrix $J$, we are considering the solution to the following system of linear equations,

J x = b,

in which $x$ and $b$ are $n\times 1$ vectors. Iterative solvers are considered to be among the effective solution techniques~\cite{ilu2003}. %These solvers are matrix-free which makes AD as a suitable method of differentiation.

Iterative techniques are typically used in combination with the preconditioning techniques~\cite{precond1,ilu2003}. Rather than solving the previous system, we can solve the preconditioned system

 M^{-1} J x= M^{-1} b,

where the $n \times n$ matrix $M$ serves as a preconditioner that approximates the coefficient matrix,

 M approximates J

Some preconditioning techniques like ILU preconditioning can generate a preconditioner which has nonzero at some places in which the Jacobian matrix $J$ has zero elements. These nonzero elements are called fill-in.