×

An efficient start system for multi-homogeneous polynomial continuation. (English) Zbl 0791.65033

When solving a system of polynomial equations using homotopy continuation, the number of solution paths can often be significantly reduced by casting the equations in multi-homogeneous form. This requires a multi-homogeneous start system having a full set of nonsingular solutions that can be easily computed. We give a general form of such a start system that works for any multi-homogeneous structure and that can be used efficiently in continuation.

MSC:

65H10 Numerical computation of solutions to systems of equations
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
12Y05 Computational aspects of field theory and polynomials (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Brunovský, P., Meravý, P. (1984): Solving systems of equations by bounded and real homotopy. Numer. Math.43, 397-418 · Zbl 0543.65030 · doi:10.1007/BF01390182
[2] Chow, S.N., Mallet-Paret, J., Yorke, J.A. (1979): A homotopy for locating all zeros of a system of polynomials. In: H.-O. Peitgen, H.O. Walther, eds., Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Mathematics730, pp. 228-237. Springer, Berlin Heidelberg New York · Zbl 0427.65034
[3] Garcia, C.B., Zangwill, W.I. (1977): Global continuation methods for finding all solutions to polynomial systems of equations inN variables, Center for Math. Studies in Business and Economics Report7755. University of Chicago, Chicago · Zbl 0428.90085
[4] Li, T.Y. (1983): On Chow, Mallet-Paret and Yorke homotopy for solving system of polynomials. Bull. Inst. Math. Acad. Sin.11, 433-437 · Zbl 0538.65030
[5] Morgan, A. (1986): A homotopy for solving polynomial systems. Appl. Math. Comput.18, 87-92 · Zbl 0597.65046 · doi:10.1016/0096-3003(86)90030-5
[6] Morgan, A., Sommese, A. (1987): A homotopy for solving general polynomial systems that respectsm-homogeneous structures. Appl. Math. Comput.24, 101-113 · Zbl 0635.65057 · doi:10.1016/0096-3003(87)90063-4
[7] Morgan, A., Sommese, A. (1989): Coefficient-parameter polynomial continuation. Appl. Math. Comput.29, 123-160 · Zbl 0664.65049 · doi:10.1016/0096-3003(89)90099-4
[8] Primrose, E.J.F., Freudenstein, F. (1963): Geared five-bar motion: Part 2?Arbitrary cotnmensurate gear ratio. Trans. ASME, Series E. J. Appl. Mech.85, 170-175
[9] Verschelde, J., Beckers, M., Haegemans, A. (1991): A new start system for solving deficient polynomial systems using continuation. Appl. Math. Comput.44, 225-239 · Zbl 0743.65051 · doi:10.1016/0096-3003(91)90059-V
[10] Verschelde, J., Cools, R. (1993): Symbolic homotopy construction. Appl. Alg. Engr. Comm. Comput. (to appear) · Zbl 0804.65058
[11] Verschelde, J., Haegemans, A. (1991): The BQ-algorithm for the construction of am-homogeneous start system. Department Computer Science Report TW 147. Katholieke Universiteit Leuven, Belgium · Zbl 0743.65051
[12] Wampler, C., Morgan, A., Sommese, A. (1990): Numerical continuation methods for solving polynomial systems arising in kinematies. J. Mech. Design112, 59-68 · doi:10.1115/1.2912579
[13] Wampler, C., Morgan, A., Sommese, A. (1992): Complete solution of the nine-point path synthesis problem for four-bar linkages. J. Mech. Design114, 153-159 · doi:10.1115/1.2916909
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.