Algorithms in Algebraic Geometry pp 79-97 | Cite as

# Higher-Order Deflation for Polynomial Systems With Isolated Singular Solutions

## Abstract

Given an approximation to a multiple isolated solution of a system of polynomial equations, we provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton’s method. Using first-order derivatives of the polynomials in the system, our first-order deflation method creates an augmented system that has the multiple isolated solution of the original system as a regular solution.

In this paper we consider two approaches to computing the “multiplicity structure” at a singular isolated solution. An idea coming from one of them gives rise to our new higher-order deflation method. Using higher-order partial derivatives of the original polynomials, the new algorithm reduces the multiplicity faster than our first method for systems which require several first-order deflation steps. In particular: the number of higher-order deflation steps is bounded by the number of variables.

## Key words

Deflation isolated singular solutions Newton’s method multiplicity polynomial systems reconditioning symbolic-numeric computations## AMS(MOS) subject classifications

Primary 65H10 Secondary 14Q99 68W30## Preview

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