| 1. |
- Burdakov, Oleg, 1953-
(författare)
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Conjugate direction methods for solving systems of equations and finding saddle points. Part 1
- 1982
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Ingår i: Engineering Cybernetics. - 0013-788X. ; 20:3, s. 13-19
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Tidskriftsartikel (refereegranskat)abstract
- This article is devoted to the development of methods of pseudo-orthogonal directions for solving systems of nonlinear equations in which the mapping is uniformly monotonic. Rapidly convergent methods that do not involve evaluation of derivatives are constructed. They constitute a generalization of such methods of unconditional minimization as the method of parallel displacements, Zangwill's method, and Powell's method. The methods developed can be applied, in particular, to finding saddle points of uniformly convex-concave functions.
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| 2. |
- Burdakov, Oleg, 1953-
(författare)
-
Conjugate direction methods for solving systems of equations and finding saddle points. Part 2
- 1982
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Ingår i: Engineering Cybernetics. - 0013-788X. ; 20:4, s. 23-31
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Tidskriftsartikel (refereegranskat)abstract
- This article is the second part of a paper devoted to the development of methods of pseudo-orthogonal directions for solving systems of nonlinear equations (for part I see Eng. Cybern. 1982, Vol. 20, No. 3, p. 13-19). An approach is developed for studying the reate of convergence of such methods. Local quadratic convergence of the proposed methods to the solution of the system of equations is proven. This implies the quadratic convergence of some methods of unconstrained minimization: the method of parallel displacements, Zangull's method, and Powell's method.
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| 3. |
- Burdakov, Oleg, 1953-
(författare)
-
Methods of the secant type for systems of equations with symmetric Jacobian matrix
- 1983
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Ingår i: Numerical Functional Analysis and Optimization. - : Taylor & Francis. - 0163-0563 .- 1532-2467. ; 6:2, s. 183-195
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Tidskriftsartikel (refereegranskat)abstract
- Symmetric methods (SS methods) of the secant type are proposed for systems of equations with symmetric Jacobian matrix. The SSI and SS2 methods generate sequences of symmetric matrices J and H which approximate the Jacobian matrix and inverse one, respectively. Rank-two quasi-Newton formulas for updating J and H are derived. The structure of the approximations J and H is better than the structure of the corresponding approximations in the traditional secant method because the SS methods take into account symmetry of the Jacobian matrix. Furthermore, the new methods retain the main properties of the traditional secant method, namely, J and H-1are consistent approximations to the Jacobian matrix; the SS methods converge superlinearly; the sequential (n + 1)-point SS methods have the R-order at least equal to the positive root of tn+1-tn-1=0.
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| 4. |
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| 5. |
- Burdakov, Oleg, 1953-
(författare)
-
Stable versions of the secants method for solving systems of equations
- 1983
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Ingår i: Computational Mathematics and Mathematical Physics. - : Elsevier. - 0965-5425 .- 1555-6662. ; 23:5, s. 1-10
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Tidskriftsartikel (refereegranskat)abstract
- An interpretation of quasi-Newton methods of solving sets of equations is given, and provides the basis of four versions of the secantsmethod, stable with respect to a linear dependence of the directions of motion. The first version involves an approximation of the matrix of first derivatives (the Jacobi matrix), and the second an approximation of the inverse Jacobi matrix. The other two versions are aimed at solving sets of equations with symmetric Jacobi matrix. The stability of the versions is proved.
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