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Module 4 : Solving Linear Algebraic Equations

Section 9 : Appendix A: Behavior of Solutions of Linear Difference Equations

9 Appendix A: Behavior of Solutions of Linear Difference Equations
Consider difference equation of the form
	--------(178)
where and $\QTR{bf}{B}$ is a $n\times n$ matrix. Starting from an initial condition $\QTR{bf}{z}^{(0)}$ , we get a sequence of vectors such that for any Equations of this type are frequently encountered in numerical analysis. We would like to analyze asymptotic behavior of equations of these type without solving them explicitly. To begin with, let us consider scalar linear iteration scheme
	--------(179)
where $z^{(k)}\in R$ and $\beta$ is a real scalar. It can be seen that
	--------(180)
if and only if To generalize this notation to a multidimensional case, consider equation of type (178) where Taking motivation from the scalar case, we propose a solution to equation (178) of type
	--------(181)
where $\lambda$ is a scalar and is a vector. Substituting equation (181) in equation (178), we get
	--------(182)
	--------(183)
Since we are interested in a non-trivial solution, the above equation can be reduced to
	--------(184)