Condition number bounds for problems with integer coefficients

TL;DR

This paper addresses how to bound the condition numbers of fundamental numerical problems when inputs have integer coefficients, and how those bounds relate to a polynomial-time notion of complexity. It introduces height-based tools and projective homogenization to derive explicit bounds that depend only on the problem dimension and the bit-size of the data for linear solving, least squares, non-symmetric eigenvalues, univariate polynomials, and polynomial systems, while also providing convergence rates for QR without shift and Graeffe iteration. The results show that, under integer input, the conditioning grows only polynomially with data size, enabling polynomial-time complexity estimates and sharpening the connection between numerical conditioning and classical computational complexity. The methods unify several problems under a height-theoretic framework and offer concrete, verifiable bounds with potential extensions to other degenerate-locus measures and higher-dimensional systems.

Abstract

An apriori bound for the condition number associated to each of the following problems is given: general linear equation solving, minimum squares, non-symmetric eigenvalue problems, solving univariate polynomials, solving systems of multivariate polynomials. It is assumed that the input has integer coefficients and is not on the degenerate locus of the respective problem (i.e. the condition number is finite). Then condition numbers are bounded in terms of the dimension and of the bit-size of the input. In the same setting, bounds are given for the speed of convergence of the following iterative algorithms: QR without shift for the symmetric eigenvalue problem, and Graeffe iteration for univariate polynomials.

Theorems & Definitions (31)

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