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  • 1
    Electronic Resource
    Electronic Resource
    A class of direct methods for linear systems (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 361-376 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05 ; CR:5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A class of methods of direct type for solving determined or underdetermined, full rank or deficient rank linear systems is presented and theoretically analyzed. The class can be considered as a generalization of the methods of Brent and Brown as restricted to linear systems and implicitly contains orthogonal,LU andLL T factorization methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01391414
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    The local convergence of ABS methods for nonlinear algebraic equations (1987)
    Springer
    Numerische Mathematik 51 (1987), S. 429-439 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we consider an extension to nonlinear algebraic systems of the class of algorithms recently proposed by Abaffy, Broyden and Spedicato for general linear systems. We analyze the convergence properties, showing that under the usual assumptions on the function and some mild assumptions on the free parameters available in the class, the algorithm is locally convergent and has a superlinear rate of convergence (per major iteration, which is computationally comparable to a single Newton's step). Some particular algorithms satisfying the conditions on the free parameters are considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397545
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  • 3
    Electronic Resource
    Electronic Resource
    Optimal conditioning of self-scaling variable Metric algorithms (1976)
    Springer
    Mathematical programming 10 (1976), S. 70-90 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Variable Metric Methods are “Newton—Raphson-like” algorithms for unconstrained minimization in which the inverse Hessian is replaced by an approximation, inferred from previous gradients and updated at each iteration. During the past decade various approaches have been used to derive general classes of such algorithms having the common properties of being Conjugate Directions methods and having “quadratic termination”. Observed differences in actual performance of such methods motivated recent attempts to identify variable metric algorithms having additional properties that may be significant in practical situations (e.g. nonquadratic functions, inaccurate linesearch, etc.). The SSVM algorithms, introduced by this first author, are such methods that among their other properties, they automatically compensate for poor scaling of the objective function. This paper presents some new theoretical results identifying a subclass of SSVM algorithms that have the additional property of minimizing a sharp bound on the condition number of the inverse Hessian approximation at each iteration. Reducing this condition number is important for decreasing the roundoff error. The theoretical properties of this subclass are explored and two of its special cases are tested numerically in comparison with other SSVM algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580654
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Solution of linear least squares via the ABS algorithm (1993)
    Springer
    Mathematical programming 58 (1993), S. 111-136 
    Details
    ISSN: 1436-4646
    Keywords: ABS algorithms ; linear least squares ; overdetermined linear systems ; QR factorization ; Gram–Schmidt algorithm ; numerical experiments
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The ABS class for linear and nonlinear systems has been recently introduced by Abaffy, Broyden, Galantai and Spedicato. Here we consider various ways of applying these algorithms to the determination of the minimal euclidean norm solution of over-determined linear systems in the least squares sense. Extensive numerical experiments show that the proposed algorithms are efficient and that one of them usually gives better accuracy than standard implementations of the QR orthogonalization algorithm with Householder reflections.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581261
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  • 5
    Electronic Resource
    Electronic Resource
    A class of scaled direct methods for linear systems (1990)
    Springer
    Annals of the Institute of Statistical Mathematics 42 (1990), S. 187-201 
    Details
    ISSN: 1572-9052
    Keywords: Linear systems ; direct methods ; scaling of equations ; conjugate direction methods ; biorthogonalization methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A generalization of the class of direct methods for linear systems recently introduced by Abaffy, Broyden and Spedicato is obtained by applying these algorithms to a scaled system. The resulting class contains an essentially free parameter at each step, giving a unified approach to finitely terminating methods for linear systems. Various properties of the generalized class are presented. Particular attention is paid to the subclasses that contain the classic Hestenes-Stiefel method and the Hegedus-Bodocs biorthogonalization methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00050789
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  • 6
    Book
    Book
    Algorithms for continous optimization: the state of the art. Proceedings of the NATO ASI, Il Ciocco, Barga, Italy, September 5-18, 1993; 434 (1994)
    Dordrecht, Netherlands :Kluwer,
    Details
    Title: Algorithms for continous optimization: the state of the art. Proceedings of the NATO ASI, Il Ciocco, Barga, Italy, September 5-18, 1993; 434
    Contributer: Spedicato, Emilio
    Publisher: Dordrecht, Netherlands :Kluwer,
    Year of publication: 1994
    Pages: 565 S.
    Series Statement: NATO ASI series C: Mathematical and physical sciences 434
    Type of Medium: Book
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  • 7
    Book
    Book
    ABS Projection Algorithms: Mathematical techniques for linear and nonlinear equations (1989)
    Chichester :Ellis Horwood,
    Details
    Title: ABS Projection Algorithms: Mathematical techniques for linear and nonlinear equations
    Author: Abaffy, Jozsef
    Contributer: Spedicato, Emilio
    Publisher: Chichester :Ellis Horwood,
    Year of publication: 1989
    Pages: 220 S.
    Series Statement: Mathematics and its applications
    Type of Medium: Book
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  • 8
    Book
    Book
    Computer algorithms for solving linear algebraic equations : The state of the art; Vol. 77 (1991)
    Berlin u.a. :Springer,
    Details
    Title: Computer algorithms for solving linear algebraic equations : The state of the art; Vol. 77
    Contributer: Spedicato, Emilio
    Publisher: Berlin u.a. :Springer,
    Year of publication: 1991
    Pages: 352 S.
    Series Statement: NATO ASI series F: computer and systems sciences Vol. 77
    Type of Medium: Book
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