Abstract
A Matlab implementation, multiplicity, of a numerical algorithm for computing the multiplicity structure of a nonlinear system at an isolated zero is presented. The software incorporates a newly developed equation-by-equation strategy that significantly improves the efficiency of the closedness subspace algorithm and substantially reduces the storage requirement. The equation-by-equation strategy is actually based on a variable-by-variable closedness subspace approach. As a result, the algorithm and software can handle much larger nonlinear systems and higher multiplicities than their predecessors, as shown in computational experiments on the included test suite of benchmark problems.
Supplemental Material
Available for Download
Software for An algorithm and software for computing multiplicity structures at zeros of nonlinear systems
- Bates, D., Hauenstein, J., Peterson, C., and Sommese, A. 2009. A numerical local dimension test for points on the solution set of a system of polynomial equations. SIAM J. Num. Anal. 47, 3608--3623. Google ScholarDigital Library
- Bates, D., Hauenstein, J., Sommese, A., and Wampler, C. 2010. Bertini: Software for numerical algebraic geometry. www.nd.edu/∼ sommese/bertini.Google Scholar
- Bates, D., Peterson, C., and Sommese, A. 2006. A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set. J. Complexity 22, 475--489. Google ScholarDigital Library
- Dayton, B. H., Li, T.-Y., and Zeng, Z. 2011. Multiple zeros of nonlinear systems. Math. Comput. 80, 2143--2168.Google ScholarCross Ref
- Dayton, B. H. and Zeng, Z. 2005. Computing the multiplicity structure in solving polynomial systems. In Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC). ACM, New York, 116--123. Google ScholarDigital Library
- Decker, D. W., Keller, H. B., and Kelly, C. T. 1983. Convergence rate for Newton's method at singular points. SIAM J. Numer. Anal. 20, 296--314.Google ScholarCross Ref
- Duo, J., Shi, J., and Wang, Y. 2008. Structure of the solution set for a class of semilinear elliptic equations with asymptotic linear nonlinearity. Nonlinear Anal. 69, 8, 2369--2378.Google ScholarCross Ref
- Fulton, W. 1998. Intersection Theory. Springer-Verlag, Berlin.Google Scholar
- Göttsche, L. 1994. Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties. Springer-Verlag, Berlin.Google Scholar
- Griffin, Z., Hauenstein, J., Peterson, C., and Sommese, A. 2011. Numerical computation of the Hilbert function of a zero-scheme. http://www.math.nd.edu/∼ sommeset/preprints.Google Scholar
- Hauenstein, J. D. 2010. Algebraic computations using Macaulay dual spaces. http://www4.ncsu.edu/∼ jhauenst/preprints/index.html.Google Scholar
- Hauenstein, J. D. and Wampler, C. W. 2010. Isosingular sets and deflation. http://www4.ncsu.edu/∼ jhauenst/preprints/index.html.Google Scholar
- Kahan, W. 1972. Conserving confluence curbs ill-condition. Tech. rep. 6, Computer Science, University of California, Berkeley.Google Scholar
- Leykin, A., Verschelde, J., and Zhao, A. 2006. Newton's method with deflation for isolated singularities of polynomial systems. Theor. Comp. Sci. 359, 111--122. Google ScholarDigital Library
- Macaulay, F. 1916. The Algebraic Theory of Modular Systems. CUP, Cambridge, UK.Google Scholar
- Morgan, A. 1987. Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems. Prentice Hall Inc., Englewood Cliffs, NJ.Google Scholar
- Morgan, A. P. and Sommese, A. J. 1989. Coefficient-parameter polynomial continuation. Appl. Math. Comput. 29, 2, 123--160. (Erratum: ibid., vol. 51, p. 207 (1992)). Google ScholarDigital Library
- Moritsugu, S. and Kuriyama, K. 1999. On multiple zeros of systems of algebraic equations. In Proceedings of the International Symposium on Symbolic and Algebraic Computation. (ISSAC). ACM, New York, 23--30. Google ScholarDigital Library
- Ojika, T. 1987. Modified deflation algorithm for the solution of singular problems.J. Math. Anal. Appl. 123, 199--221.Google ScholarCross Ref
- Stetter, H. J. 2004. Numerical Polynomial Algebra. EngineeringPro Collection, SIAM, Philadelphia, PA. Google ScholarDigital Library
- Sturmfels, B. 2002. Solving systems of polynomial equations. Regional Conference Series in Mathematics, vol. 97. AMS, Providence, RI.Google ScholarCross Ref
- Verschelde, J. 1996. Homotopy continuation methods for solving polynomial systems. Ph.D. thesis, Katholieke Universiteit Leuven.Google Scholar
- Ward, J. R. 2005. Rotation numbers and global bifurcation in systems of ordinary differential equations. Adv. Nonlinear Stud. 5, 375--392.Google ScholarCross Ref
- Ward, J. R. 2007. Existence, multiplicity, and bifurcation in systems of ordinary differential equations. Electronic J. Diff. Eqns. 15, 399--415.Google Scholar
- Zeng, Z. 2003. A method for computing multiple roots of inexact polynomials. In Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISAAC). ACM Press, New York, 266--272. Google ScholarDigital Library
- Zeng, Z. 2008. Apatools: A Maple and Matlab toolbox for approximate polynomial algebra. In Software for Algebraic Geometry, M. Stillman, N. Takayama, and J. Verschelde, Eds., Vol. 148, Springer, New York, 149--167.Google Scholar
- Zeng, Z. 2009. The closedness subspace method for computing the multiplicity structure of a polynomial system. InInteractions of Classical and Numerical Algebraic Geometry, D. J. Bates, G. Besana, S. D. Rocco, and C. W. Wampler, Eds., Contemporary Mathematics, vol. 496, American Mathematics Society, Providence, RI, 347--362.Google Scholar
Index Terms
- Algorithm 931: An algorithm and software for computing multiplicity structures at zeros of nonlinear systems
Recommendations
Algorithm 835: MultRoot---a Matlab package for computing polynomial roots and multiplicities
MultRoot is a collection of Matlab modules for accurate computation of polynomial roots, especially roots with non-trivial multiplicities. As a blackbox-type software, MultRoot requires the polynomial coefficients as the only input, and outputs the ...
Numerical factorization of multivariate complex polynomials
Algebraic and numerical algorithmOne can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however ...
Exact polynomial factorization by approximate high degree algebraic numbers
SNC '09: Proceedings of the 2009 conference on Symbolic numeric computationFor factoring polynomials in two variables with rational coefficients, an algorithm using transcendental evaluation was presented by Hulst and Lenstra. In their algorithm, transcendence measure was computed. However, a constant c is necessary to compute ...
Comments