Complexity Analysis of Algorithms in Algebraic Computation
Candidate: Vikram Sharma
Advisor: Chee Yap

Abstract

Numerical computations with real algebraic numbers require algorithms for approximating and isolating real roots of polynomials. A classical choice for root approximation is Newton's method. For an analytic function on a Banach space, Smale introduced the concept of approximate zeros, i.e., points from which Newton's method for the function converges quadratically. To identify these approximate zeros he gave computationally verifiable convergence criteria called point estimates. However, in developing these results Smale assumed that Newton's method is computed exactly. For a system of $n$ homogeneous polynomials in $n+1$ variables, Malajovich developed point estimates for a different definition of approximate zero, assuming that all operations in Newton's method are computed with fixed precision. In the first half of this dissertation, we develop point estimates for these two different definitions of approximate zeros of an analytic function on a Banach space, but assume the strong bigfloat computational model of Brent, i.e., where all operations involve bigfloats with varying precision. In this model, we derive a uniform complexity bound for approximating a root of a zero-dimensional system of $n$ integer polynomials in $n$ variables. We also derive a non-asymptotic bound, in terms of the condition number of the system, on the precision required to implement the robust Newton method

The second part of the dissertation analyses the worst-case complexity of two algorithms for isolating real roots of a square-free polynomial with real coefficients: The Descartes method and Akritas' continued fractions algorithm. The analysis of both algorithms is based upon amortization bounds such as the Davenport-Mahler bound. For the Descartes method, we give a unified framework that encompasses both the power basis and the Bernstein basis variant of the method; we derive an $O(n(L+\log n))$ bound on the size of the recursion tree obtained by applying the method to a square-free polynomial of degree n with integer coefficients of bit-length $L$, the bound is tight for $L=\Omega(\log n)$; based upon this result we readily obtain the best known bit-complexity bound of $\wt{O}(n^4L2) $ for the Descartes method, where $\wt{O}$ means we ignore logarithmic factors. Similar worst case bounds on the bit-complexity of Akritas' algorithm were not known in the literature. We provide the first such bound, $\wt{O}(n^{12}L3)$, for a square-free integer polynomial of degree $n$ and coefficients of bit-length $L$.