This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: AIMS Lecture Notes 2006 Peter J. Olver 9. Numerical Solution of Algebraic Systems In this part, we discuss basic iterative methods for solving systems of algebraic equa- tions. By far the most common is a vector-valued version of Newtons Method, which will form our primary object of study. 9.1. VectorValued Iteration. Extending the scalar analysis to vector-valued iterative systems is not especially dif- ficult. We will build on our experience with linear iterative systems. We begin by fixing a norm kk on R n . Since we will also be computing the associated matrix norm k A k , as defined in Theorem 7.13, it may be more convenient for computations to adopt either the 1 or the norms rather than the standard Euclidean norm. We begin by defining the vector-valued counterpart of the basic linear convergence condition (2.21). Definition 9.1. A function g : R n R n is a contraction at a point u ? R n if there exists a constant 0 < 1 such that k g ( u )- g ( u ? ) k k u- u ? k (9 . 1) for all u sufficiently close to u ? , i.e., k u- u ? k < for some fixed > 0. Remark : The notion of a contraction depends on the underlying choice of matrix norm. Indeed, the linear function g ( u ) = A u if and only if k A k < 1, which implies that A is a convergent matrix. While every convergent matrix satisfies k A k < 1 in some matrix norm, and hence defines a contraction relative to that norm, it may very well have k A k > 1 in a particular norm, violating the contaction condition; see (7.31) for an explicit example. Theorem 9.2. If u ? = g ( u ? ) is a fixed point for the discrete dynamical system (2.1) and g is a contraction at u ? , then u ? is an asymptotically stable fixed point. Proof : The proof is a copy of the last part of the proof of Theorem 2.6. We write k u ( k +1)- u ? k = k g ( u ( k ) )- g ( u ? ) k k u ( k )- u ? k , 3/15/06 141 c 2006 Peter J. Olver using the assumed estimate (9.1). Iterating this basic inequality immediately demonstrates that k u ( k )- u ? k k k u (0)- u ? k for k = 0 , 1 , 2 , 3 , . . . . (9 . 2) Since < 1, the right hand side tends to 0 as k , and hence u ( k ) u ? . Q.E.D. In most interesting situations, the function g is differentiable, and so can be approxi- mated by its first order Taylor polynomial g ( u ) g ( u ? ) + g ( u ? ) ( u- u ? ) = u ? + g ( u ? ) ( u- u ? ) . (9 . 3) Here g ( u ) = g 1 u 1 g 1 u 2 . . . g 1 u n g 2 u 1 g 2 u 2 . . . g 2 u n . . . . . . . . . . . . g n u 1 g n u 2 . . . g n u n , (9 . 4) denotes the n n Jacobian matrix of the vector-valued function g , whose entries are the partial derivatives of its individual components. Since u ?...
View Full Document