Geometria Analitica Con Elementi Di Algebra Lineare Seconda by Marco Abate

By Marco Abate

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Thus we deduce from expressions (44), (46) and (50) that the condition (51) ^ 3 is satisfied by every ordinary iteration of the algorithm. It follows that condition (28) holds only if the reduction in 47 M. J. D. POWELL F(x) obtained by the kth iteration is bounded by the i n ­ equality F(x ( k ) + _6(k)) - F(x ( k ) ) < - 0 . 075 r\ e . (52) Now if F(x) is bounded below the condition (52) is satisfied only a finite number of times. Therefore only a finite number of ordinary iterations set A^+l) = || 6 ^ / || or A (k + 1) = 2 ||_g(k) || ^ w h .

F(x' r ), we deduce the limit lim F(x ( k )) = F(x*) . k-^oo 49 (58) M. J. D. (TI) = min F(x), l|x-x*ll>,n, x^S. (59) The value of F (r\) must exceed F ( x ' ) , so expression (58) implies that there exists an integer cr(r|) > a- such that F(x ( k ) ) < F ( T ! ) for all k > o-(-n) . Therefore, b e c a u s e the smallness of v{ is arbitrary, the sequence x' ' converges to x* • Theorem 4 is proved. Theorem 4 s u g g e s t s , correctly, that it is usual in practice for the points x^\ generated by the algorithm, to converge to a point x , which is the position of a local minimum of F(x) .

To complete the proof it remains to show that the superlinear convergence is not damaged by the "special iterations" of the algorithm. I (k+1) v -x II f ~ M | | x v (k) *i|2 -x II /ncx , (95) which implies that a special iteration cannot increase the error of the estimate of x > by more than the constant factor \l 2M"/d . Therefore, because every special iteration is followed by an ordinary iteration, the ratio || x* p ' x" ||/||x* ' - x* II tends to zero as k tends to infinity, for any fixed integer p > 2.

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