By J.F. Traub

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**Example text**

Our sixth-order method (with ν 3) improves on a fifth-order method of Jarratt [70]. Generalizations Generalizations to methods using higher derivatives are possible. 2 (take k = m = 1 , and ν = η + 1) . 2 is given in Brent [75], we omit proofs here, and adopt an informal style of presentation. Other possible generalizations are mentioned in Section 7. 60 f^^\ O P T I M A L Z E R O - F I N D I N G M E T H O D S USING D E R I V A T I V E S 2· MOTIVATION We first consider methods using one evaluation of f , and two of f , per iteration.

The main results of the paper deal with the complexity of solving certain nonlinear operator equations f(x) = 0. Upper bounds are established by a new procedure for obtain ing starting points for Newton's method. The procedure finds points where the conditions of the Newton-Kantorovich Theorem are satisfied. It is believed that the principle of the procedure can be used for other iterative schemes, where Newton-Kantorovich-like theorems are available, for f satis fying various kinds of conditions.

1 If P(x) = a + bx + cx^ + dx^ satisfies P(0) = P'(0) = P'(2/3) = 0 , then P(l) = 0 . 1, we may show that (for α = 2/3) f(Xj^) - Q(Xj^) 61 = 0(6^^) , R I C H A R D P. B R E N T where ^ = ^0 - W is the approximation given by Newton's method, and δ = Ιν^όΐ = I^N-^ol · Now Xj^ - x^ = 0(6^) , and f'(x) - Q'(x) = 0(6^) for χ near Xj^ , so |f(x^)| = |f(x^) - QCx^l for some ξ between x^^ and x^ . Thus |f(xp| = 0(6^^) + OCO^-O^) = 0(6"^) , and x^ - 3. Ζ = 0(|f(xp|) = OCO'^) . = 0(|XQ - Ζ Γ ) . A SIXTH-ORDER METHOD To obtain a sixth-order method using one more derivative evaluation than the fourth-order method described above, we need distinct, nonzero parameters, and , such that P(0) = P'(0) = P'(a^) = Ρ·(α2) = 0 implies P(l) = 0 , for all fifth-degree polynomials P(x) = a + bx + ...