An efficient derivative-free method for solving nonlinear equations

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An efficient derivative-free method for solving nonlinear equations

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ACM Transactions on Mathematical Software (TOMS) archive
Volume 11 , Issue 3 (September 1985) table of contents

Pages: 250 - 262

Year of Publication: 1985

ISSN:0098-3500

Author

D. Le

University of New South Wales, Energy Systems Analysis Group, CSIRO Division of Energy Technology, Lucas Heights Research Laboratories, Private Mail Bag 7, Sutherland, New South Wales, 2232, Australia

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 7, Downloads (12 Months): 63, Citation Count: 5

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ABSTRACT

An algorithm is presented for finding a root of a real function. The algorithm combines bisection with second and third order methods using derivatives estimated from objective function values. Globaql convergence is ensured and the number of function evaluations is bounded by four times the number needed by bisection. Numerical comparisons with existing algorithms indicate the superiority of the new algorithm in all classes of problems.

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

	1	ANDERSON, N., AND BJORCK, A. A new high order method of regula falsi type for computing a root of an equation. BIT 13 (1973), 253-264.
	2	BRENT, R. P. An algorithm with guaranteed convergence for finding a zero of a function. Comput. J. 14 (1971), 422-425.
	3	BRENT, R.P. Algorithms for Minimisation Without Derivatives. Prentice Hall, Englewood Cliffs, N.J. 1973.
	4	J. C. P. Bus , T. J. Dekker, Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function, ACM Transactions on Mathematical Software (TOMS), v.1 n.4, p.330-345, Dec. 1975 [doi>10.1145/355656.355659]
	5	Germund Dahlquist , Ake Bjorck, Numerical Methods, Dover Publications, Incorporated, 2003
	6	DEKKER, T.J. Finding a zero by means of successive linear interpolation. In Constructive Aspects of the Fundamental Theorem ofAlgebra, B. Dejon and P. Henrici, Eds. Wiley Interscience, New York, (1969), 37-48.
	7	DOWELL, M., AND JARRATT, P. A modified regula falsi method for computing the root of an equation. BIT 11 (1971), 168-174.
	8	DOWELL, M., AND JARRATr, P. The 'Pegasus' method for computing the root of an equation. BIT 12 {1972), 503-508.
	9	GONNET, G.H. On the structure of zero finders. BIT 17 (1977), 170-183.
	10	LE, D. Three new rapidly convergent algorithms for finding a zero of a function. SIAM J. Sc~ Star. Comp. 6, 1 (1985), 193-208.
	11	RALSTON, A., AND RABINOWlTZ, P., First Course in Numerical Analysis. McGraw-Hill, New York, (1978).
	12	SWIFT, A., AND LINDFIELD, G.R. Comparison of a continuation method with Brent's method for the numerical solution of a single nonlinear equation. Comput. J. 21 (1978) 359-362.
	13	TRAUB, J.F. Iterative Methods for the Solution of Equations. Prentice Hall, Englewood Cliffs, N. J., (1964).

CITED BY 5

	W. H. Enright , J. D. Pryce, Corrigenda: “Two FORTRAN Packages for Assessing Initial Value Methods”, ACM Transactions on Mathematical Software (TOMS), v.15 n.3, p.287, Sept. 1989
	G. E. Alefeld , F. A. Potra , Yixun Shi, Algorithm 748: enclosing zeros of continuous functions, ACM Transactions on Mathematical Software (TOMS), v.21 n.3, p.327-344, Sept. 1995
	D. Le, Corrigenda: “An Efficient Derivative-Free Method for Solving Nonlinear Equations”, ACM Transactions on Mathematical Software (TOMS), v.15 n.3, p.287, Sept. 1989
	A. Corana , M. and C. Martini and S. Ridella, Corrigenda: “Minimizing Multimodal Functions of Continuous Variables with the ‘Simulated Annealing’ Algorithm”, ACM Transactions on Mathematical Software (TOMS), v.15 n.3, p.287, Sept. 1989
	M.A. Salgueiro da Silva, A novel method for robust minimisation of univariate functions with quadratic convergence, Journal of Computational and Applied Mathematics, v.200 n.1, p.168-177, March, 2007