Abstract
In a previous paper, the authors suggested a procedure for obtaining all solutions to certain systems ofn equations inn complex variables. The idea was to start with a trivial system of equations to which all solutions were easily known. The trivial system was then perturbed into the given system. During the perturbation process, one followed the solution paths from each of the trivial solutions into the solutions of the given system. All solutions to the given system were thereby obtained.
This paper utilizes a different approach that eliminates the requirement of the previous paper for a leading dominating term in each equation. We add a dominating term artificially and then fade it. Also we rely on mathematically more fundamental concepts from differential topology. These advancements permit the calculation of all solutions to arbitrary polynomials and to various other systems ofn equations inn complex variables. In addition, information on the number of solutions can be obtained without calculation.
Similar content being viewed by others
References
F.H. Branin Jr., “Widely convergent method for finding multiple solutions of simultaneous nonlinear equations”,IBM Journal of Research and Development (September, 1972).
K.M. Brown and W.B. Gearhart, “Deflation techniques for the calculation of further solutions of a nonlinear system”,Numerische Mathematik 16 (1971) 334–342.
A. Charnes, C.B. Garcia and C.E. Lemke, “Constructive proofs of theorems relating toF(x) = y, with applications”,Mathematical Programming 12 (1977) 328–343.
J. Cronin, “Analytical functional mappings”,Annals of Mathematics 58 (1953) 178–181.
D. Davidenko, “On the approximate solution of a system of nonlinear equations”,Ukrainskii Matematisčeskii Žurnal 5 (1953) 196–206.
D. DeJohn and P. Henrici,Constructive aspects of the fundamental theorem of algebra (Wiley, New York, 1969).
B.C. Eaves, “Homotopies for computation of fixed points”,Mathematical Programming 3 (1972) 1–22.
B.C. Eaves and H. Scarf, “The solution of systems of piecewise linear equations”,Mathematics of Operations Research 1 (1976) 1–27.
C.B. Garcia, “Computation of solutions to nonlinear equations under homotopy invariance”,Mathematics of Operations Research 2 (1977) 25–29.
C.B. Garcia and F.J. Gould, “A theorem on homotopy paths”,Mathematics of Operations Research 3 (1978) 282–289.
C.B. Garcia and W.I. Zangwill, “Determining all solutions to certain systems of nonlinear equations”,Mathematics of Operations Research, to appear.
C.B. Garcia and W.I. Zangwill, “Global continuation methods for finding all solutions to polynomial systems of equations inn variables”, in:Symposium on extremal methods and systems analysis (Springer-Verlag, Berlin, 1978) to appear.
F.J. Gould and J.W. Tolle, “A unified approach to complementarity in optimization”,Discrete Mathematics 7 (1974) 225–271.
L. Kantorovich, “On Newton's method for functional equation”,Doklady Academii Nauk SSSR 59 (1948) 1237–1240.
M. Kojima, “A unification of the existence theorems of the nonlinear complementarity problem”,Mathematical Programming 9 (1975) 257–277.
H.W. Kuhn, “Finding roots of polynomials by pivoting”, in: S. Karamardian in collaboration with C.B. Garcia, eds.,Fixed points: theory and applications (Academic Press, New York, 1977).
H.W. Kuhn and J.G. MacKinnon, “Sandwich method for finding fixed point”,Journal of Optimization Theory and Application 17 (1975) 189–204.
C.E. Lemke, “Bimatrix equilibrium points and mathematical programming”,Management Science 11 (1965) 681–689.
C.E. Lemke and J.T. Howson Jr., “Equilibrium points of bimatrix games”,Society of Industrial and Applied Mathematics 12 (1964) 413–423.
O.H. Merrill, “Applications and extensions of an algorithm that computes fixed points of certain upper semi-continuous point-to-set mappings”, Ph.D. Dissertation, University of Michigan, Ann Arbor, MI (1972).
G.H. Meyer, “On solving nonlinear equations with a one-parameter operator imbedding”,SIAM Journal of Numerical Analysis 5 (1968) 739–742.
J. Milnor,Topology from the differentiable viewpoint (University Press of Virginia, Charlottesville, VA, 1965).
J.M. Ortega and W.C. Rheinboldt,Iterative solutions of nonlinear equations in several variables (Academic Press, New York, 1970).
H.E. Scarf, “The approximation of fixed points of continuous mappings”,Society of Industrial and Applied Mathematics 15 (1967) 1328–1343.
H.E. Scarf and T. Hansen,Computation of economic equilibria (Yale University Press, New Haven, CT, 1973).
S. Smale, “A convergent process of price adjustment and global Newton methods”,Journal of Mathematical Economics 3 (1976) 107–120.
S. Sternberg,Lectures on differential topology (Prentice-Hall, Englewood Cliffs, NJ, 1964).
M.J. Todd,The computation of fixed points and applications (Springer-Verlag, Berlin, 1976).
E. Wasserstrom, “Numerical solutions by the continuation method”,SIAM Review 15 (1973) 89–119.
M.N. Yakovlev, “The solution of systems of nonlinear equations by a method of differentiation with respect to a parameter”,USSR Computation of Mathematics and Mathematical Physics 4 (1964) 198–203.
W.I. Zangwill, “An eccentric barycentric fixed point algorithm”,Mathematics of Operations Research 2 (1977) 343–359.
Author information
Authors and Affiliations
Additional information
Work supported in part by NSF Grant No. MCS77-15509 and ARO Grant No. DAAG-29-78-G-0160.
Work supported in part by ARO Grant No. DAAG-29-78-G-0160
Rights and permissions
About this article
Cite this article
Garcia, C.B., Zangwill, W.I. Finding all solutions to polynomial systems and other systems of equations. Mathematical Programming 16, 159–176 (1979). https://doi.org/10.1007/BF01582106
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01582106