A Method of Continuation for Calculating a Brouwer Fixed Point

Kellogg, R. Bruce; Li, T.Y.; Yorke, James A.

doi:10.1016/b978-0-12-398050-2.50012-8

Cited by 16 publications

(7 citation statements)

References 5 publications

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“…y ninguna columna de la primera matriz posee un vector estrictamente interior a este simplex puesto que no hay ninguna columna con sus cuatro coordenadas estrictamente más grandes que los cuatro números del simplex. Por lo tanto el conjunto x 8 , x 9 , x 11 , x 13 es un conjunto primitivo.…”

Section: Definición 3 [18]unclassified

Algoritmos de punto fijo usando subdivisiones simpliciales

Azofeifa

1996

RMTA

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Section: Definición 3 [18]unclassified

Algoritmos de punto fijo usando subdivisiones simpliciales

Azofeifa

1996

RMTA

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“…Recently, nonsimplicial approaches were found by Kellogg, Li and Yorke [7], [8]. Smale [9] recently discovered similar ideas for finding zeroes of maps and applied them to economics.…”

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confidence: 99%

“…He eliminates all possibilities for what it can be and, thereby, obtains a contradiction. Hence, there must be a fixed point of/ In [7] the argument was modified slightly to make it constructive with probability one ; the mapping g was considered to be defined everywhere in B" except at the fixed points off. Now for any regular value y ES"~X we have g~ * (y) is the union of a curve leading from y to a subset of the fixed points of/ plus disjoint closed paths.…”

mentioning

confidence: 99%

Finding Zeroes of Maps: Homotopy Methods That are Constructive With Probability One

Chow¹,

Mallet‐Paret²,

Yorke³

1978

Mathematics of Computation

118

106

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“…A. Yorke [16] reported in 1974 on a numerical method similar to that of Scarf. They implemented the process on a UNIVAC 1108; for a 20-dimensional problem, the machine required slightly over three seconds to locate a fixed point [17].…”

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confidence: 99%

The Homotopy Continuation Method: Numerically Implementable Topological Procedures

Alexander¹,

Yorke²

1978

Transactions of the American Mathematical Society

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Abstract. The homotopy continuation method involves numerically finding the solution of a problem by starting from the solution of a known problem and continuing the solution as the known problem is homotoped to the given problem. The process is axiomatized and an algebraic topological condition is given that guarantees the method will work. A number of examples are presented that involve fixed points, zeroes of maps, singularities of vector fields, and bifurcation. As an adjunct, proofs using differential rather than algebraic techniques are given for the Borsuk-Ulam Theorem and the Rabinowitz Bifurcation Theorem.In 1963, Hirsch [13] gave a short-and by now classic-proof of the Brouwer Fixed Point Theorem by a generic argument. This theorem is usually proved by some kind of degree argument and degree genetically counts inverse images of points. Hirsch directly looked at the inverse image of a generic point. The idea that one could replace degree arguments by looking at the inverse images of points of maps was made the theme of a book by J. Milnor [24] and later by V. Guillemin and A. Pollack [12, especially Chapters 2,3] and Hirsch [14, especially Chapter 5]. We offer these books as general references for transversality and Sard's theorem.Without knowledge of Hirsch's paper, H. Scarf [31] in 1967 used much the same ideas to numerically approximate a Brouwer fixed point. One "follows" a "path" which leads from the boundary to some one or more of the fixed points. For some contemporary papers using similar methods, see [19], [23]. This has been called the Newton method. B. C. Eaves [8] in 1972 developed a slightly different approach to the same end. His idea was to homotope one of a set of standard maps to the map in question; by foUowing the fixed points of the changing maps, the fixed-point-set of the map in question could be located. It is this version we call the homotopy continuation method and

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A Method of Continuation for Calculating a Brouwer Fixed Point

Cited by 16 publications

References 5 publications

Algoritmos de punto fijo usando subdivisiones simpliciales

Algoritmos de punto fijo usando subdivisiones simpliciales

Finding Zeroes of Maps: Homotopy Methods That are Constructive With Probability One

The Homotopy Continuation Method: Numerically Implementable Topological Procedures

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