Power Magnetic Devices. Scott D. Sudhoff

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СКАЧАТЬ is in the interior of Ω, it can be shown that

(1.3-3)

and that

(1.3-4)

Note that the statement F(x^*) ≥ 0 means that F(x^*) is positive semi‐definite, which is to say that y^TF(x^*)y ≥ 0 ∀ y ∈ ℝⁿ. This second condition verifies that a minimum rather than a maximum has been found. It is important to understand that the conditions (1.3-3) and (1.3-4) are necessary but not sufficient conditions for x^* to be a mimimizer, unless f is a convex function. If f is convex, (1.3-3) and (1.3-4) are necessary and sufficient conditions for x^* to be a global minimizer.

      At this point, we are posed to set forth Newton’s method of finding function minimizers. This method is iterative and is based on a kth estimate of the solution denoted x[k]. Then an update formula is applied to generate a (hopefully) improved estimate x[k + 1] of the minimizer. The update formula is derived by first approximating f as a Taylor series about the current estimated solution x[k]. In particular,

(1.3-5)

where H denotes higher‐order terms. Neglecting these higher‐order terms and finding the gradient of f(x) based on (1.3-5), we obtain

(1.3-6)

From the necessary condition (1.3-3), we next take the right‐hand side of (1.3-6) and equate it with zero; then we replace x, which will be our improved estimate, with x[k + 1]. Manipulating the resulting expression yields

(1.3-7)

      which is Newton’s method.

      Clearly, Newton’s method requires f ⊂ C², which is to say that f is in the set of twice differentiable functions. Note that the selection of the initial solution, x[1], can have a significant impact on which (if any) local solution is found. Also note that method is equally likely to yield a local maximizer as a local minimizer.

      Example 1.3A

      Let us apply Newton’s method to find the minimizer of the function

(1.3A-1)

This example is an arbitrary mathematical function; we will consider how to construct f(x) so as to serve a design purpose in Section 1.9. Inspection of (1.3A-1) reveals that the global minimum is at x₁ = 2 and x₂ = ln (2). However, let us apply Newton’s method to find the minimum. Our first step is to obtain the gradient and the Hessian. From (1.3A-1), we have

      (1.3A-2)

      and

      (1.3A-3)

Table 1.1 Newton’s Method Results

k	x[k]	f(x[k])	∇f(x[k])	F(x[k])
1		43
2		14.2
3		9.25
10		8.00

Let us arbitrarily take our initial estimate of the solution to be . Table 1.1 lists СКАЧАТЬ