Arrow matrix

Structure of Arrow Matrix

A matrix with special structure is the arrow matrix (see the figure). In this matrix there is a large, banded portion, with a much smaller number of columns at one side and rows at the bottom. This matrix arises when solving sets of fluid flow problems with free surfaces, when using continuation (homotopy) methods, or parameter estimation methods. The matrix problem is sub-divided as

We then solve

A x = f

and, using the same LU decomposition of A,

A y = B

These problems are sparse and the special methods are applied to them. Next solve the dense matrix problem

(D - C y ) v = ­ C x + g

and evaluate

u = x - y v

The solution is then (u^T, v^T).