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Implementation issues for high-order algorithms

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Abstract

Newton-Raphson method, which dates back to 1669–1670, is widely used to solve systems of equations and unconstrained optimization problems. Nowton-Raphson consists in linearizing the system of equations and provides quadratic local convergence order. Quite soon after Newton and Raphson introduced their iterative process. Halley in 1694 proposed a higher-order method providing cubic asymptotic convergence order. Chebyshev in 1838 proposed another high-order variant. In “High-order Newton-penalty algorithms” [J.-P. Dussault, J. Comput. Appl. Math. 182, No. 1, 117–133 (2005; Zbl 1077.65061)], by interpreting Newton’s iteration as a linear extrapolation, formulae were proposed to compute higher-order extrapolations generalizing Newton-Raphson’s and Chebyshev’s methods. In this paper, we provide details using an automatic differentiation (AD) tool to implement those high-order extrapolations. We present a complexity analysis allowing to predict the efficiency of those high-order strategies.

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... We use this complexity bound to assess the overall complexity of our HoC algorithm. Actually, as described in [4], obtaining ∇F costs n times the cost of F for a vector valued function F : R n → R n . But when one considers the scalar quantity ∇F (x)u, it can be expressed as: ∇(F (x) · u), i.e. derivative of a new scalar valued function Φ : x → F (x) · u. ...
... Therefore, in this example, c will be equivalent to n times cost of F i which is the cost of four multiplications, one addition, and four subtractions. When one refers to [4], flops of these different operations are described for a Sparc system. For more recent computers, we can check on Intel Optimization Reference Manual [11] for example. ...
... , additional costs are given by: [4] 25c + 30n Solving the linear system 2n 2 − n Vector sum and dot product 2n T otal 2n 2 + 31n + 25c ...
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The 1669–1670 Newton–Raphson's method is still used to solve equations systems and unconstrained optimization problems. Since this method, some other algorithms inspired by Newton's have been proposed: in 1839, Chebyshev developed a high-order cubical convergence algorithm, and in 1967, Shamanskii proposed an acceleration of Newton's method. By considering Newton-type methods as displacement directions, we introduce in this article, new high-order algorithms extending these famous methods. We provide convergence order results and per iteration complexity analysis to predict the efficiency of such iterative processes. Preliminary examples confirm the applicability of our analysis.
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