Accurate solution of near-colliding Prony systems via decimation and\n homotopy continuation

2014/12/31 by Dmitry Batenkov, Batenkov, Dmitry
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Computer and information sciences #FOS: Mathematics #Mathematical functions and polynomials #Numerical Analysis (math.NA) #Polynomial and algebraic computation #Symbolic Computation (cs.SC)

paper · pdf · doi:10.48550/arxiv.1501.00160

openalex publication_date 2014/12/31 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We consider polynomial systems of Prony type, appearing in many areas of\nmathematics. Their robust numerical solution is considered to be difficult,\nespecially in "near-colliding" situations. We consider a case when the\nstructure of the system is a-priori fixed. We transform the nonlinear part of\nthe Prony system into a Hankel-type polynomial system. Combining this\nrepresentation with a recently discovered "decimation" technique, we present an\nalgorithm which applies homotopy continuation to an appropriately chosen\nHankel-type system as above. In this way, we are able to solve for the\nnonlinear variables of the original system with high accuracy when the data is\nperturbed.\n

Related

• Polynomial Homotopies for Dense, Sparse and Determinantal Systems
• Local and global geometry of Prony systems and Fourier reconstruction of piecewise-smooth functions
• Adaptive strategies for solving parameterized systems using homotopy\n continuation
• Solving Polynomial Systems Equation by Equation
• Prony's method in several variables: symbolic solutions by universal interpolation