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A new method to compute second derivatives
Keywords:
algorithm, finite difference, numerical approximationAbstract
In this article we consider the problem of computing approximations to the second derivatives of functions of n variables using finite differences. We show how to derive different formulas and how to comput the errors of those approximations as functions of the increment h, both for first and second derivatives. Based upon those results we describe the methods of Gill and Murray and the one of gradient difference. On the other hand we introduce a new algorithm which use conjugate directions methods for minimizing functions without derivatives and the corresponding numerical comparisons with the other two methods. Finally, numerical experiences are given and the corresponding conclusions are discussed.
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References
[1] Fletcher,R. (1987), Practical Methods of Optimization. Wiley, New York
[2] Gill, P.E, Murray, W. and Pitfield, R.A. (1972). The implementation of two modified Newton algorithms for unconstrined
optimization. NPL Report NAC11.
[3] Gill,P.E , Murray,W., Saunders,M.A. and Wright, M.H(1980), Computing the Finite-Difference Approximations to
Derivatives for Numerical Optimization.Technical Report, Departament of Operations Research-Standford University.
[4] Gill,P.E , Murray,W., Saunders,M.A. and Wright, M.H(1991) Numerical Linear Algebra and Optimization,
vol.1.Addison Wesley.
[5] Gill,P.E , Murray,W., Saunders,M.A. and Wright, M.H(1993) Practical Optimization. Academic Press.
[6] Powell, M.J.D.(1964) An efficient method for finding the minimun of a function of several variables without calculating
derivatives. Computer Journal, vol.7, pp 155-162.
[7] Press,(1992), Numerical Recipes, Academic Press.
[8] G.W.Stewart(1967) A modification of Davidon’s minimization method to accept difference approximations to
derivates,J.Ass.Comput.Mach.,14,72-83.
[9] G.R.Walsh(1975), Methods of Optimization, Wiley, London.
[2] Gill, P.E, Murray, W. and Pitfield, R.A. (1972). The implementation of two modified Newton algorithms for unconstrined
optimization. NPL Report NAC11.
[3] Gill,P.E , Murray,W., Saunders,M.A. and Wright, M.H(1980), Computing the Finite-Difference Approximations to
Derivatives for Numerical Optimization.Technical Report, Departament of Operations Research-Standford University.
[4] Gill,P.E , Murray,W., Saunders,M.A. and Wright, M.H(1991) Numerical Linear Algebra and Optimization,
vol.1.Addison Wesley.
[5] Gill,P.E , Murray,W., Saunders,M.A. and Wright, M.H(1993) Practical Optimization. Academic Press.
[6] Powell, M.J.D.(1964) An efficient method for finding the minimun of a function of several variables without calculating
derivatives. Computer Journal, vol.7, pp 155-162.
[7] Press,(1992), Numerical Recipes, Academic Press.
[8] G.W.Stewart(1967) A modification of Davidon’s minimization method to accept difference approximations to
derivates,J.Ass.Comput.Mach.,14,72-83.
[9] G.R.Walsh(1975), Methods of Optimization, Wiley, London.
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Published
2002-05-01
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[1]
“A new method to compute second derivatives”, JCS&T, vol. 1, no. 06, p. 9 p., May 2002, Accessed: Jun. 15, 2025. [Online]. Available: https://journal.info.unlp.edu.ar/JCST/article/view/965
