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Introduction

In this chapter we discuss ways of solving the generalized eigenvalue problem given by (1.91) numerically using the node number ordering (see section 1.7.1). The reason for using this formulation of the problem is that the numerical solution is more easily obtained--solution of block-tri-diagonal systems of equations is a standard problem in numerical analysis. Furthermore, this ordering allows us in a convenient way to construct an implementation capable of handling the general G group problem since the block structure is maintained independent of the total number of groups one wishes to have. With the mathematical properties of the system of equations in mind^2.1 two numerical methods come to mind

.

The power method.

.

The method of fractional iteration.

These two methods build the foundation upon which the implementation, treated in the next chapter, is build. It is these two methods we describe in this chapter.

For convenience we repeat the form of the generalized eigenvalue problem

$$\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex} \h... ...hspace{0.15ex} \hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$$ (2.1)

where in our case $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}$ is a block-diagonal matrix and $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ is a block-tri-diagonal matrix.

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