Abstract
ABSTRACT The main goal of this manuscript is to introduce a discrete dynamical system defined by symmetric matrices and a real parameter. By construction, we rediscovery the Power Iteration Method from the Projected Gradient Method. Convergence of the discrete dynamical system solution is established. Finally, we consider two applications, the first one consists in find a solution of non linear equation problem and the other one consists in verifies the optimality conditions when we solve quadratic optimization problems over linear equality constraints.
Highlights
Discrete dynamical system appears as a tool in order to understand differential equations from numerically view point (for more details, see Galob (2007) and chapter 6 in Loneli & Rumbos (2003))
The equation 1 is not exclusive for differential equations, for example it appears in order to find fixed points for contractive operators (remember, F is contractive if F(x) − F(y) ≤ λ x − y, with λ ∈ (0, 1) and x, y ∈ dom(F))
The reader can verify that for B = 0 1, the se[10] quences generated by Power Iteration Method diverge for any stated point x, because does not have a dominant eigenvalue, but B = I + (1/3)B is positive definite and it has a dominant eigenvalue
Summary
Discrete dynamical system appears as a tool in order to understand differential equations from numerically view point (for more details, see Galob (2007) and chapter 6 in Loneli & Rumbos (2003)). The equation 1 is not exclusive for differential equations, for example it appears in order to find fixed points for contractive operators (remember, F is contractive if F(x) − F(y) ≤ λ x − y , with λ ∈ (0, 1) and x, y ∈ dom(F)). For details about contractive operators, see classical books in functional analysis or general topology or fixed point theorems as for instance Brezis (1983), Istra ̧tescu (1981), Kelley (1955). We consider the following operator Tλ : S → S defined by.
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