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  • Articles: DFG German National Licenses  (3)
  • Electronic Resource  (3)
  • Complementary Pivoting Methods  (2)
  • Fixed Point  (1)
  • 1
    Electronic Resource
    Electronic Resource
    A PL homotopy for finding all the roots of a polynomial (1979)
    Springer
    Mathematical programming 16 (1979), S. 37-62 
    Details
    ISSN: 1436-4646
    Keywords: Roots of Polynomials ; Fixed Point Computing Methods ; Complementary Pivoting Methods ; Piecewise Linear Homotopy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper presents a constructive method which gives, for any polynomialF(Z) of the degreen, approximate values of all the roots ofF(Z).. The point of the method is on the use of a piecewise linear function $$\bar H$$ (Z, t) which approximates a homotopyH(Z, t) betweenF(Z) and a polynomialG(Z) of the degreen withn known simple roots. It is shown that the set of solutions to $$\bar H$$ (Z, t) = 0 includesn distinct paths,m of which converges to a root ofF(Z) if and only if the root has the multiplicitym. Starting from givenn roots ofG(Z), a complementary pivot algorithm generates thosen paths.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582093
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A modification of todd's triangulationJ 3 (1978)
    Springer
    Mathematical programming 15 (1978), S. 223-227 
    Details
    ISSN: 1436-4646
    Keywords: Triangulations ; Fixed Point Computing Methods ; Complementary Pivoting Methods ; Piecewise Linear Homotopy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract When we apply the fixed point computing method to mappings which are affine in some variables, we show that, to generate a sequence which converges to a fixed point, the mesh size need not be decreased in these coordinates. This paper modifies the triangulationJ 3 with continuous refinement of mesh size to a triangulation $$\bar J_3$$ such that the mesh size of $$\bar J_3$$ in some given coordinates is constant and the mesh size in the other coordinates shrinks to zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01609022
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    Paper (German National Licenses)
    Fulltext
  • 3
    Electronic Resource
    Electronic Resource
    Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods (1982)
    Springer
    Mathematical programming 24 (1982), S. 177-215 
    Details
    ISSN: 1436-4646
    Keywords: Variable Dimension Algorithm ; Fixed Point ; Subdivided Manifold ; Nonlinear Equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by Van der Laan and Talman. We introduce a new concept ‘primal—dual pair of subdivided manifolds’ and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms. Our basic model furnishes interpretations to several existing methods: Lemke's algorithm for the linear complementarity problem, its extension to the nonlinear complementarity problem, a variable dimension algorithm on conical subdivisions and Merrill's algorithm. We shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us. A method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585103
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    Paper (German National Licenses)
    Fulltext
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