Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

On Paths Generated by Fixed Point Algorithms

R. Saigal
Mathematics of Operations Research
Vol. 1, No. 4 (Nov., 1976), pp. 359-380
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/3689213
Page Count: 22
  • Download ($30.00)
  • Cite this Item
On Paths Generated by Fixed Point Algorithms
Preview not available

Abstract

This paper considers algorithms that compute fixed points (or more generally solve equations) which are based on complementary pivoting. These algorithms, starting with a map $f^{0}$ and its fixed point x0, deform $f^{t}\ \text{to}\ f^{\infty}$ = f as t goes from 0 to ∞, and follow the path xt of fixed points of $f^{t}$. In this paper we study these paths. In particular, we treat some special applications of these algorithms from a global point of view, and thus look at the behavior and properties of these paths sufficiently far from the solution. Our methods are motivated by methods of global analysis. We show that in many implementations of these algorithms the path can be specified. These include the cases when the mapping is linear and when it is smooth and monotone. The results of the linear analysis are used to study the local behavior of these paths (i.e., properties and behavior sufficiently close to the solution) for smooth mappings. An important implication of this study is that the paths can be modified and thus the work done by these algorithms can be controlled. We also show, for smooth mappings, how our results can be implemented to increase the efficiency of some standard algorithms.

Page Thumbnails

  • Thumbnail: Page 
359
    359
  • Thumbnail: Page 
360
    360
  • Thumbnail: Page 
361
    361
  • Thumbnail: Page 
362
    362
  • Thumbnail: Page 
363
    363
  • Thumbnail: Page 
364
    364
  • Thumbnail: Page 
365
    365
  • Thumbnail: Page 
366
    366
  • Thumbnail: Page 
367
    367
  • Thumbnail: Page 
368
    368
  • Thumbnail: Page 
369
    369
  • Thumbnail: Page 
370
    370
  • Thumbnail: Page 
371
    371
  • Thumbnail: Page 
372
    372
  • Thumbnail: Page 
373
    373
  • Thumbnail: Page 
374
    374
  • Thumbnail: Page 
375
    375
  • Thumbnail: Page 
376
    376
  • Thumbnail: Page 
377
    377
  • Thumbnail: Page 
378
    378
  • Thumbnail: Page 
379
    379
  • Thumbnail: Page 
380
    380