On a condition number of general random polynomial systems

  title={On a condition number of general random polynomial systems},
  author={Hoi H. Nguyen},
  journal={Math. Comput.},
  • H. Nguyen
  • Published 24 June 2015
  • Mathematics
  • Math. Comput.
Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us to study these numbers for a broad family of probability distributions. Our work also extends to certain perturbed systems. 

Probabilistic Condition Number Estimates for Real Polynomial Systems II: Structure and Smoothed Analysis

This work provides explicit estimates for condition numbers of structured random real polynomial systems, and extends these estimates to smoothed analysis setting, for sensitivity of real zeros to perturbations of their coefficients.

Smoothed analysis for the condition number of structured real polynomial systems

This work provides explicit estimates for condition numbers of structured random real polynomial systems, and extends these estimates to the smoothed analysis setting.

Probabilistic Condition Number Estimates for Real Polynomial Systems I: A Broader Family of Distributions

The sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients is considered and new probabilistic estimates are established that allow a much broader family of measures than considered earlier.

Sparsity, Randomness and Convexity in Applied Algebraic Geometry

The main theorem provides quantitative versions of some known algebraic facts, and also refines earlier quantitative results, on the condition number of polynomial systems ‘on average’, which is a vital invariant of poynomial systems which controls their computational complexity.



Eigenvalues and condition numbers of random matrices

Given a random matrix, what condition number should be expected? This paper presents a proof that for real or complex $n \times n$ matrices with elements from a standard normal distribution, the ex...

Complexity of Bezout's Theorem V: Polynomial Time

Invertibility of random matrices: norm of the inverse

Let A be an n × n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A-1 does not exceed Cn3 / 2

Small Ball Probability, Inverse Theorems, and Applications

Let ξ be a real random variable with mean zero and variance one and A ={a 1; …; a n } be a multi-set in R d . The random sum $$S_A : = a_1 \xi _1 + \cdots + a_n \xi _n$$ where ξ i are iid

Smallest singular value of a random rectangular matrix

We prove an optimal estimate of the smallest singular value of a random sub‐Gaussian matrix, valid for all dimensions. For an N × n matrix A with independent and identically distributed sub‐Gaussian

The Littlewood-Offord problem and invertibility of random matrices

Complexity of Bezout’s Theorem II Volumes and Probabilities

In this paper we study volume estimates in the space of systems of n homegeneous polynomial equations of fixed degrees d i with respect to a natural Hermitian structure on the space of such systems

Smooth analysis of the condition number and the least singular value

A general estimate is given for the condition number and least singular value of the matrix M + N n, generalizing an earlier result of Spielman and Teng for the case when x is gaussian and involves the norm ∥M∥.

Inverse Littlewood-Offord theorems and the condition number of random discrete matrices

Consider a random sum r)\V\ + • • • + r]nvn, where 771, . . . , rin are independently and identically distributed (i.i.d.) random signs and vi, . . . , vn are integers. The Littlewood-Offord problem

A numerical algorithm for zero counting. II: Distance to ill-posedness and smoothed analysis

A Condition Number Theorem is shown that this condition number of zero counting for real polynomial systems equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those having multiple real zeros).