Computing Isolated Singular Solutions of Polynomial Systems: Case of Breadth One

We present a symbolic-numeric method to refine an approximate isolated singular solution $\hat{\mathbf{x}}=(\hat{x}_{1}, \ldots, \hat{x}_{n})$ of a polynomial system $F=\{f_1, \ldots, f_n\}$, when the Jacobian matrix of $F$ evaluated at $\hat{\mathbf{x}}$ has corank one approximately. Our new approach is based on the regularized Newton iteration and the computation of differential conditions satisfied at the approximate singular solution. The size of matrices involved in our algorithm is bounded by $n \times n$. The algorithm converges quadratically if $\hat{\mathbf{x}}$ is close to the isolated exact singular solution.

[1]  Reinhard Menzel.  On solving nonlinear least-squares problems in case of rankdeficient Jacobians , 2005, Computing.

[2]  B. Mourrain.  Isolated points, duality and residues , 1997 .

[3]  Lihong Zhi,et al.  Computing the multiplicity structure from geometric involutive form , 2008, ISSAC '08.

[4]  Grégoire Lecerf.  Quadratic Newton Iteration for Systems with Multiplicity , 2002, Found. Comput. Math..

[5]  Hans J. Stetter,et al.  Numerical polynomial algebra , 2004 .

[6]  Barry H. Dayton.  Numerical local rings and local solution of nonlinear systems , 2007, SNC '07.

[7]  Zhonggang Zeng,et al.  Computing the multiplicity structure in solving polynomial systems , 2005, ISSAC.

[8]  T. Ojika,et al.  Deflation algorithm for the multiple roots of a system of nonlinear equations , 1983 .

[9]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[10]  A. Griewank.  On Solving Nonlinear Equations with Simple Singularities or Nearly Singular Solutions , 1985 .

[11]  Lihong Zhi,et al.  Computing the multiplicity structure of an isolated singular solution: Case of breadth one , 2012, J. Symb. Comput..

[12]  Marc Giusti,et al.  On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One , 2007, Found. Comput. Math..

[13]  C. Kelley,et al.  Newton’s Method at Singular Points. I , 1980 .

[14]  Nan Li.  An improved method for evaluating Max Noether conditions: case of breadth one , 2012, SNC '11.

[15]  Maria Grazia Marinari,et al.  Gröbner bases of ideals defined by functionals with an application to ideals of projective points , 1993, Applicable Algebra in Engineering, Communication and Computing.

[16]  M. R. Osborne,et al.  Newton’s Method for Singular Problems when the Dimension of the Null Space is $>1$ , 1981 .

[17]  G. Reddien.  On Newton’s Method for Singular Problems , 1978 .

[18]  L. Kantorovich,et al.  Functional analysis in normed spaces , 1952 .

[19]  Yun-Qiu Shen,et al.  Newton's method for singular nonlinear equations using approximate left and right nullspaces of the Jacobian , 2005 .

[20]  Zhonggang Zeng,et al.  Multiple zeros of nonlinear systems , 2011, Math. Comput..

[21]  L. B. Rall.  Convergence of the newton process to multiple solutions , 1966 .

[22]  G. Reddien.  Newton's method and high order singularities , 1979 .

[23]  A. Griewank.  Analysis and modification of Newton's method at singularities , 1980 .

[24]  Xiaojun Chen,et al.  Convergence of Newton's Method for Singular Smooth and Nonsmooth Equations Using Adaptive Outer Inverses , 1997, SIAM J. Optim..

[25]  M. R. Osborne,et al.  Analysis of Newton’s Method at Irregular Singularities , 1983 .

[26]  Siegfried M. Rump,et al.  Verified error bounds for multiple roots of systems of nonlinear equations , 2010, Numerical Algorithms.

[27]  Marc Giusti,et al.  On Location and Approximation of Clusters of Zeros of Analytic Functions , 2005, Found. Comput. Math..

[28]  C. Kelley,et al.  Convergence Acceleration for Newton’s Method at Singular Points , 1982 .

[29]  Gene H. Golub,et al.  Matrix computations , 1983 .

[30]  Maria Grazia Marinari,et al.  ON MULTIPLICITIES IN POLYNOMIAL SYSTEM SOLVING , 1996 .

[31]  Takeo Ojika,et al.  Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations , 1987 .

[32]  H. Michael Möller,et al.  Multivariate Polynomial System Solving Using Intersections of Eigenspaces , 2001, J. Symb. Comput..

[33]  Anton Leykin,et al.  Newton's method with deflation for isolated singularities of polynomial systems , 2006, Theor. Comput. Sci..

[34]  Robert M. Corless,et al.  A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots , 1997, ISSAC.

[35]  H. M. Möller,et al.  Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems , 1995 .