Algorithms for Minimization Without DerivativesCourier Corporation, 1.1.2002 - 195 sivua This outstanding text for graduate students and researchers proposes improvements to existing algorithms, extends their related mathematical theories, and offers details on new algorithms for approximating local and global minima. None of the algorithms requires an evaluation of derivatives; all depend entirely on sequential function evaluation, a highly practical scenario in the frequent event of difficult-to-evaluate derivatives. Topics include the use of successive interpolation for finding simple zeros of a function and its derivatives; an algorithm with guaranteed convergence for finding a minimum of a function of one variation; global minimization given an upper bound on the second derivative; and a new algorithm for minimizing a function of several variables without calculating derivatives. Many numerical examples augment the text, along with a complete analysis of rate of convergence for most algorithms and error bounds that allow for the effect of rounding errors. |
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a₁ a₂ ALGOL 60 ALGOL procedures ALGOL W approximation BEGIN bisection Chapter Chebyquad Comp computed condition number d-unimodal defined definition Dekker's algorithm DIAGRAM effect of rounding eigenvalues equation example f is unimodal f(x₁ Fibonacci search finding a zero Fletcher function evaluations required function f given in Section gives global minimum golden section search Golub Hessian matrix INTEGER VALUE interval inverse quadratic interpolation iteration Lemma linear interpolation linear searches Lipschitz Lipschitz continuous LONG REAL ARRAY LONG REAL PROCEDURE macheps machine precision Math matrix method minimization nonlinear number of function numerical results order at least order of convergence points polynomial Powell Powell's problem procedure glomin procedure zero quadratic convergence rounding errors satisfies second derivative singular value decomposition step successive linear interpolation superlinear convergence TEST Theorem 5.1 tion tolerance upper bound variables weak order x₁ Xn+1 Xn+g zero2 μ₁