By Birkhoff G.D.
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54) where 0 < 0 < 1. e. f(xo) ~ f(x,) + (xo - x,)f'(x,). This will make f(xo) zero if xo - x, =- f(x,)/f'(x,). In other words, if x, is an approximation to Xo, x, - f(x,)/ f'(x,) should be a better one. 55) Note that if x, converges we expect its limit to be a zero of f if f' does not vanish there. In fact, the iteration will converge to a multiple zero as will be seen later. 55). 50) if F(x) =x - I(x)/I'(x). 8e tells us that Newton's method converges to a simple zero of if 111"/f,21 < 1 in a neighbourhood of the zero.
55) Note that if x, converges we expect its limit to be a zero of f if f' does not vanish there. In fact, the iteration will converge to a multiple zero as will be seen later. 55). 50) if F(x) =x - I(x)/I'(x). 8e tells us that Newton's method converges to a simple zero of if 111"/f,21 < 1 in a neighbourhood of the zero. Since f is small near zero, the basic assertion is that the method will converge if x I is close enough to the zero. However, it must not be concluded that, if Xl is closer to one zero than another, the iteration will necessarily converge to the nearby zero.
24). For instance, if we choose M (f, g) = L i= 1 f(xi)g*(X i) for some fixed Xi we can easily verify that the properties are valid and so we 2 may deduce that II 1 1 «», II or L~ 1 I/(Xi) 1 cn4>n(Xj)1 is a minimum when L:= L:= M en = (I, 4>n) = L i= 1 f(Xi)4>:(Xi ) · It is this kind of problem which arises in fitting data at a discrete number of points by the method of least squares. Note that it is frequently a computational advantage to employ orthonormal polynomials for least squares rather than expansions in non-orthogonal functions because the matrices tend to be diagonally dominant even when round-off error is present.