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  • 1
    Electronic Resource
    Electronic Resource
    Exact order of convergence of the secant method (1993)
    Springer
    Journal of optimization theory and applications 78 (1993), S. 541-551 
    Details
    ISSN: 1573-2878
    Keywords: Order of convergence ; Newton's method ; secant method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the exact order of convergence of the secant method when applied to the problem of finding a zero of a nonlinear function defined from ℝ into ℝ. Under the standard assumptions for which Newton's method has the exact Q-order of convergencep, wherep is some positive integer, we establish that the secant method has the Q-order and the exact R-order of convergence $$S(p) = (1/2)[1 + \sqrt {1 + 4(p - 1)]} $$ . We prove also that, forp=2 andp=3, the secant method has the exact Q-order of convergenceS(p). Moreover, we present a counterexample to show that, forp≥4, it may not have an exact Q-order of convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939881
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  • 2
    Electronic Resource
    Electronic Resource
    Preconditioners for distance matrix algorithms (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Journal of Computational Chemistry 15 (1994), S. 227-232 
    Details
    ISSN: 0192-8651
    Keywords: Computational Chemistry and Molecular Modeling ; Biochemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Computer Science
    Notes: A recent gradient algorithm in nonlinear optimization uses a novel idea that avoids line searches. This so-called spectral gradient algorithm works well when the spectrum of the Hessian of the function to be minimized has a small range or is clustered. In this article, we find a general preconditioning method for this algorithm. The preconditioning method is applied to the stress function, which arises in many applications of distance geometry, from statistics to finding molecular conformations. The Hessian of stress is shown to have a nice block structure. This structure yields a preconditioner which decreases the amount of computation needed to minimize stress by the spectral gradient algorithm. © 1994 by John Wiley & Sons, Inc.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/jcc.540150212
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Molecular conformations from distance matrices (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Journal of Computational Chemistry 14 (1993), S. 114-120 
    Details
    ISSN: 0192-8651
    Keywords: Computational Chemistry and Molecular Modeling ; Biochemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Computer Science
    Notes: Two algorithms are introduced that show exceptional promise in finding molecular conformations using distance geometry on nuclear magnetic resonance data. The first algorithm is a gradient version of the majorization algorithm from multidimensional scaling. The main contribution is a large decrease in CPU time. The second algorithm is an iterative algorithm between possible conformations obtained from the first algorithm and permissible data points near the configuration. These ideas are similar to alternating least squares or alternating projections on convex sets. The iterations significantly improve the conformation from the first algorithm when applied to the small peptide E. coli STh enterotoxin. © 1993 John Wiley & Sons, Inc.
    Additional Material: 4 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/jcc.540140115
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    Paper (German National Licenses)
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