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Spectral residual method without gradient information for solving largescale nonlinear systems: Theory and experiments
, 2004
"... Abstract. A fully derivativefree spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for ..."
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Abstract. A fully derivativefree spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than wellknown NewtonKrylov methods for largescale problems is also presented. 1.
Two Classes of Multisecant Methods for Nonlinear Acceleration ∗
, 2007
"... Many applications in science and engineering lead to models which require solving largescale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasiNewton methods that implicitly update the approximate Jacob ..."
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Many applications in science and engineering lead to models which require solving largescale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasiNewton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allows to take into account a variable number of secant equations at each iteration. The first is the Broydenlike class, of which Broyden’s family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear EirolaNevanlinnatype methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the selfconsistent field (SCF) iteration, is accelerated by various strategies termed ‘mixing’. 1
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"... Stock market trading rule discovery using technical charting heuristics ..."
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Stock market trading rule discovery using technical charting heuristics
Documenta Math. 301 Broyden Updating, the Good and the Bad!
"... bounded deterioration, superlinear convergence So far so good! We had an updating procedure (the ’full ’ secant method) that seemed to work provided that certain conditions of linear independence were satisfied, but the problem was that it did not work very well. In fact it proved to be quite numeri ..."
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bounded deterioration, superlinear convergence So far so good! We had an updating procedure (the ’full ’ secant method) that seemed to work provided that certain conditions of linear independence were satisfied, but the problem was that it did not work very well. In fact it proved to be quite numerically unstable. Charles Broyden in On the discovery of the ‘good Broyden ’ method [6]. The idea of secant updating As Joanna Maria Papakonstantinou recounted in her comprehensive historical survey [29], regula falsi and other variants of the secant method for solving one equation in one variable go back to the Babylonian and Egyptian civilizations nearly 4000 years ago. They may be viewed just as a poor man’s version of what is now known as Newton’s method, though we should also credit Al Tusi [20]. During antiquity the very concept of derivatives was in all likelihood