相关论文: The largest 5th pivot may be the root of a 61st de…

This paper combines modern numerical computation with theoretical results to improve our understanding of the growth factor problem for Gaussian elimination. On the computational side we obtain lower bounds for the maximum growth for…

The growth factor in Gaussian elimination measures how large the entries of an LU factorization can be relative to the entries of the original matrix. It is a key parameter in error estimates, and one of the most fundamental topics in…

Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth factor is large. In the study of numerical linear algebra,…

The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although in specific cases the loss of precision in GEPP due to roundoff errors can be very significant, empirical…

Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible…

We show that the problem of finding the measure supported on a compact subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to K is a minimum, is…

We consider the problem of finding a local optimum for Max-Cut with FLIP-neighborhood, in which exactly one node changes the partition. Schaeffer and Yannakakis (SICOMP, 1991) showed PLS-completeness of this problem on graphs with unbounded…

We describe a factor-revealing convex optimization problem for the integrality gap of the maximum-cut semidefinite programming relaxation: for each $n \geq 2$ we present a convex optimization problem whose optimal value is the largest…

We study the problem of approximating the largest root of a real-rooted polynomial of degree $n$ using its top $k$ coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in $k$ that…

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain $5$ elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that \[r_5(N)\ll \frac{N}{\exp((\log\log N)^{c})}.\] Our work…

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input…

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which…

The set of 2-dimensional packing problems builds an important class of optimization problems and Strip Packing together with 2-dimensional Bin Packing and 2-dimensional Knapsack is one of the most famous of these problems. Given a set of…

In this paper we investigate the following related problems: (A) the separation of $p$-adic roots of integer polynomials of a fixed degree and bounded height; and (B) counting integer polynomials of a fixed degree and bounded height with…

The growth problem in Gaussian elimination (GE) remains a foundational question in numerical analysis and numerical linear algebra. Wilkinson resolved the growth problem in GE with partial pivoting (GEPP) in his initial analysis from the…

The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite and $\mathrm{i}=\sqrt{-1}$ is the imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth factor in…

We consider the problem of Gaussian multiplier bootstrap procedures for the $k$th largest statistics and functions of the top $k$ order statistics, which are commonly encountered in high-dimensional statistical inference. Such a problem has…

Given polynomials $g$ and $f_1,\dots,f_p$, all in $\Bbbk[x_1,\dots,x_n]$ for some field $\Bbbk$, we consider the problem of computing the critical points of the restriction of $g$ to the variety defined by $f_1=\cdots=f_p=0$. These are…

The slope of the best fit line from minimizing the sum of the squared oblique errors is the root of a polynomial of degree four. This geometric view of measurement errors is used to give insight into the performance of various slope…

Inspired by Armin Straub's conjecture (arXiv:1601.07161) about the number and maximal size of (2n+1, 2n+3)-core partitions with distinct parts, we develop relatively efficient, symbolic-computational algorithms, based on non-linear…