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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 vectorvalued version of Newtons Method, which will form our primary object of study. 9.1. VectorValued Iteration. Extending the scalar analysis to vectorvalued 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 vectorvalued 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 vectorvalued function g , whose entries are the partial derivatives of its individual components. Since u ?...
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This note was uploaded on 02/10/2012 for the course MATH 5485 taught by Professor Olver during the Fall '09 term at University of Central Florida.
 Fall '09
 Olver
 Algebra, Equations

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