Granularity issues for solving polynomial systems via globally convergent algorithms on a hypercube

Polynomial systems of equations frequently arise in many applications such as solid modelling, robotics, computer vision, chemistry, chemical engineering, and mechanical engineering. Locally convergent iterative methods such as quasi-Newton methods may diverge or fail to find all meaningful solutions of a polynomial system. Recently a homotopy algorithm has been proposed for polynomial systems that is guaranteed globally convergent (always converges from an arbitrary starting point) with probability one, finds all solutions to the polynomial system, and has a large amount of inherent parallelism. There are several ways the homotopy algorithms can be decomposed to run on a hypercube. The granularity of a decomposition has a profound effect on the performance of the algorithm. The results of decompositions with two different granularities are presented. The experiments were conducted on an iPSC-16 hypercube using actual industrial problems.

[1]  Layne T. Watson,et al.  Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms , 1987, TOMS.

[2]  Richard H. Byrd,et al.  Using parallel function evaluations to improve hessian approximation for unconstrained optimization , 1988 .

[3]  L. Watson,et al.  HOMPACK: a suite of codes for globally convergent homotopy algorithms. Technical report No. 85-34 , 1985 .

[4]  R. Schnabel,et al.  A view of unconstrained optimization , 1989 .

[5]  Layne T. Watson,et al.  A globally convergent parallel algorithm for zeros of polynomial systems , 1989 .

[6]  Thomas A. Rice,et al.  A parallel algorithm for finding the roots of a polynomial , 1982, ICPP.

[7]  G. Golub,et al.  Linear least squares solutions by householder transformations , 1965 .

[8]  Werner C. Rheinboldt,et al.  Algorithm 596: a program for a locally parameterized , 1983, TOMS.

[9]  Layne T. Watson,et al.  The Granularity of Parallel Homotopy Algorithms for Polynomial Systems of Equations , 1988 .

[10]  Daniel Gajski,et al.  Iterative algorithms for tridiagonal matrices on a WSI-multiprocessor , 1982, ICPP.

[11]  R. White.  A nonlinear parallel algorithm with application to the Stefan problem , 1986 .

[12]  A. Morgan.  Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems , 1987 .

[13]  Robert B. Schnabel,et al.  Concurrent Function Evaluations in Local and Global Optimization ; CU-CS-345-86 , 1987 .

[14]  Layne T. Watson,et al.  A parallel algorithm for simple roots for polynomials , 1984 .

[15]  L. Watson.  Numerical linear algebra aspects of globally convergent homotopy methods , 1986 .

[16]  A. Morgan.  A transformation to avoid solutions at infinity for polynomial systems , 1986 .

[17]  J. Yorke,et al.  Finding zeroes of maps: homotopy methods that are constructive with probability one , 1978 .

[18]  D. Heller.  A Survey of Parallel Algorithms in Numerical Linear Algebra. , 1978 .

[19]  E. Allgower,et al.  Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations , 1980 .

[20]  Layne T. Watson,et al.  Message length effects for solving polynomial systems on a hypercube , 1989, Parallel Comput..

[21]  Alexander P. Morgan.  A homotopy for solving polynomial systems , 1986 .

[22]  L. Watson.  A globally convergent algorithm for computing fixed points of C2 maps , 1979 .

[23]  R. Schnabel,et al.  Solving Systems of Nonlinear Equations by Tensor Methods. , 1986 .

[24]  Layne T. Watson,et al.  Algorithm 555: Chow-Yorke Algorithm for Fixed Points or Zeros of C2 Maps [C5] , 1980, TOMS.

[25]  Milan Kubicek,et al.  Algorithm 502: Dependence of Solution of Nonlinear Systems on a Parameter [C5] , 1976, TOMS.

[26]  D. A. Reed,et al.  A model of asynchronous iterative algorithms for solving large, sparse, linear systems , 1984 .

[27]  Ming-Yang Chern,et al.  A Fast Algorithm for Concurrent LU Decomposition and Matrix Inversion , 1983, ICPP.

[28]  S. Lakshmivarahan,et al.  Parallel Algorithms for Solving Certain Classes of Linear Recurrences , 1985, FSTTCS.

[29]  H. Schwandt.  Newton-like interval methods for large nonlinear systems of equations on vector computers , 1985 .

[30]  R. White.  Parallel algorithms for nonlinear problems , 1986 .

[31]  L. Shampine,et al.  Computer solution of ordinary differential equations : the initial value problem , 1975 .