We return now to the n ow to the nonlinear equations problem: given F: R” – R”, find XHERM such that

We return now to the n ow to the nonlinear equations problem: given F: R” – R”, find XHERM such that F(x,) = 0, (6.5.1) ction we show how Newton’s method for (6.5.1) can be combined Whal methods for unconstrained optimization to produce global In this section we show how N with globa methods for (6.5.1). The Newton step for (6.5.1) is xt = x – J(x.)- F(x), (6.5.2) where Jx.) is the Jacobian matrix of F at xe. From Section 5.2 we know that (6.5.2) is locally q-quadratically convergent to xx, but not necessarily globally convergent. Now assume xc is not close to any solution xx of (6.5.1). How would one decide then whether to accept x + as the next iterate? A reasonable answer is that || F(x) || should be less than || F(x)|| for some norm ||· |, a convenient choice being the l, norm || F(x) || 2 = F(x)”F(x). Requiring that our step result in a decrease of || F(x)||2 is the same thing we would require if we were trying to find a minimum of the function || F(x) ||2. we have in effect turned our attention to the corresponding mini- mization problem: min f(x) = {F(x)”F(x), (6.5.3) XERN where Lui

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