Polynominal Approximation of Functions of Matrices and Its Application the the Solution of a General System of Linear Equations

Frequently, during the process of solving a mathematical model numerically, we end up with a need to operate on a vector v by an operator which can be expressed as f A while A is N x N matrix. Except for very simple matrices, it is impractical to construct the matrix fA explicity. Usually an approximation to it is used. In the present research, we develop an algorithm which uses a polynomial approximation to fA. It is reduced to a problem of approximating fz by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f z in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described.