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Minimizaton Notes

Minimizaton Notes - AIMS Lecture Notes 2006 Peter J Olver 9...

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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 Newton’s Method, which will form our primary object of study. 9.1. Vector–Valued 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 k · k 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 σ < 1 such that k g ( u ) - g ( u ? ) k ≤ σ k u - u ? k (9 . 1) for all u sufficiently close to u ? , i.e., k u - u ? k < δ for some fixed δ > 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 ≤ σ k u ( k ) - u ? k , 3/15/06 141 c 2006 Peter J. Olver

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using the assumed estimate (9.1). Iterating this basic inequality immediately demonstrates that k u ( k ) - u ? k ≤ σ k k u (0) - u ? k for k = 0 , 1 , 2 , 3 , . . . . (9 . 2) Since σ < 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 0 ( u ? ) ( u - u ? ) = u ? + g 0 ( u ? ) ( u - u ? ) . (9 . 3) Here g 0 ( 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 ? is fixed, the the right hand side of (9.3) is an affine function of u . Moreover, u ? remains a fixed point of the affine approximation. Proposition 7.25 tells us that iteration of the affine function will converge to the fixed point if and only if its coefficient matrix, namely g 0 ( u ?
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Minimizaton Notes - AIMS Lecture Notes 2006 Peter J Olver 9...

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