Other Other methods methods to compute the gradient are base on adjoint equations and on automatic differentiation. In this case, numerical deriv derivative ativess can provid providee an accurate accurate evalua evaluatio tion n of the gradien gradient. In most practical situations, indeed, the formula involved in the computation is extremely complicated. If some situations, this is not possible. If the functi function on is more compl complic icate ated, d, we can perform perform the computation with a symbolic computing system (such as Maple or Mathematica). The practical computation computation may be b e performe p erformed d ”by ”by hand” with with paper paper and pencil. In simple simple cases, cases, we can provide provide the exact gradient. Consider the situation where we want to solve an optimization problem with a method method which which requires requires the gradien gradientt of the cost function function. Before getting into the details, we briefly motivate motivate the need for approxiapproximate numerical derivatives. In this document, we analyse the computation of the numerical derivative of a given function. The scripts are available under the CeCiLL licence: /licences/Licence_ ences/Licence_CeCILL_V2-en.txt CeCILL_V2-en.txt The Scilab scripts are provided on the Forge, inside the project, under the scripts sub-directory. The LATEX sources are provided under the terms of the Creative Commons Attribution ShareAlike 3.0 Unported License: /licenses/by-sa/3.0 licenses/by-sa/3.0 index.php/p/doc dex.php/p/docnumder/ numder/ The LATEX sources are available on the Scilab Forge. This This document document is an open-sourc open-sourcee project. Introd trodu uctio tion Intr Introdu oduct ctio ion n 52 8 Not Notes es and ref refere erence ncessĬopyright c 2008-2009 - Michael Baudin This file must be used under the terms of the Creative Commons AttributionShareAlike 3.0 Unported License: 51 7.3 Com Computi putings ngs the the coeffici coefficien ents ts in Sci Scilab lab. 49 7.2 Automa Automaticall tically y computi computings ngs the coefficien coefficients ts. ħ Auto Automati maticall cally y computing computing the the coefficien coefficients tsħ.1 The coeffici coefficien ents ts of finite finite differen difference ce formu formulas las. Perfor Pe rformanc mancee of fini finite te diffe differenc rences es. Vary arying ing order order to chec check k accurac accuracy y. Taki aking ng into into accoun accountt bounds on parame parameters ters. Computing Computi ng deriv derivativ atives es with with more accuracy accuracy.
Nested Nes ted deriv derivativ atives es with with Scila Scilab b . Computi Com puting ng higher higher degree degree deriv derivati ative vess. Derivativ Deriv atives es of a vectorial vectorial functio function n with Scilab. Derivativ Deriv atives es of a multiv multivariate ariate function function in Scilab Scilab. Numerical Numeri cal deriv derivativ atives es of multiv multivariate ariate function functionss. Multivari Multiv ariate ate func function tionss. Ĥ Fini Finite te difference differencess of multiv multivaria ariate te functions functions A collection collection of finite difference formulas formulas.
Accuracy Accurac y of finite difference formulas formulas.
A three three points points formula formula for for the second second deriv derivati ative ve. Somee finite differen Som difference ce formula formulass for the first derivati derivative ve. Centere Cen tered d form formula ula wit with h 4 poin points ts. Centere Cen tered d form formula ula wit with h 2 poin points ts. Numerical Numeri cal experimen experiments ts with with the robust robust forwa forward rd formula formula. Floating Floati ng point point implemen implementation tation of the forwa forward rd formula formula. 64 Vari arious ous resu results lts for sin sin(2 (2 ). 2 Fin Finite ite diff differen erences ces. 2.1.1 Tayl aylor’s or’s form formula ula for univ univariate ariate functio functions ns 2.1.2 2.1. Ģ A surp surpri risi sing ng res resul ultt In the the thir third d part, part, we pres presen entt the the derivative function function, its featur features es and its performances. We presen presentt severa severall formula formulass and their their associa associated ted optima optimall steps. In the second part, we analyse the method to use the optimal step to compute derivatives with finite differences on floating point system systems. In the first part, we present a result which is surprising when we are not familiar with floating point numbers. Numerical Derivatives in Scilab Micha¨el el Baudin May 2009 Abstract This document document present present the use of numerical numerical derivativ derivatives es in Scilab.
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