Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Advances on the recovery of (perturbed) Cauchy matrices

Paola Boito, Dario Fasino, Beatrice Meini

Abstract

Given a (possibly approximate) Cauchy matrix, how can we efficiently compute its generators? Expanding on previous work by Liesen and Luce [Linear Algebra Appl. 493 (2016) 261--280], we present a general family of algorithms for Cauchy parameter recovery, together with new error estimates. We also introduce a displacement-based approximation, which leads to a new algorithm for Cauchy parameter recovery. Numerical experiments show that the algorithm based on the displacement approximation is generally more accurate than the other algorithms.

Advances on the recovery of (perturbed) Cauchy matrices

Abstract

Given a (possibly approximate) Cauchy matrix, how can we efficiently compute its generators? Expanding on previous work by Liesen and Luce [Linear Algebra Appl. 493 (2016) 261--280], we present a general family of algorithms for Cauchy parameter recovery, together with new error estimates. We also introduce a displacement-based approximation, which leads to a new algorithm for Cauchy parameter recovery. Numerical experiments show that the algorithm based on the displacement approximation is generally more accurate than the other algorithms.
Paper Structure (9 sections, 18 theorems, 92 equations, 6 figures, 4 algorithms)

This paper contains 9 sections, 18 theorems, 92 equations, 6 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries on Cauchy matrices and the recovery of Cauchy points
  4. A general family of algorithms for Cauchy parameter recovery
  5. A deeper analysis of Algorithm 1 using CUR approximation theory
  6. A displacement-based recovery method
  7. Minimizing the entrywise relative error
  8. Numerical experiments
  9. A different parametrization of ${\mathcal{D}}$

Key Result

Lemma 1.1

If $A,B\in{\mathbb R}^{n\times n}$ are two matrices such that $|A_{ij} - B_{ij}|/|A_{ij}| \leq \alpha$ for every $i,j=1,\ldots,n$ for some $\alpha \geq 0$, then $\|A - B\|_\star/\|A\|_\star \leq \alpha$.

Figures (6)

  • Figure 1: Relative normwise errors for Example \ref{['ex:ex1']}. Here $C$ is the original Cauchy matrix and $C_i$, $i=1,2,3,4$ are the Cauchy matrices recovered from the corresponding algorithms from $A$. The parameter $\delta$ measures the magnitude of entrywise relative perturbations applied to $C$. The dashed line represents $\beta_{\max}(A)$. Matrix size is $100\times 100$.
  • Figure 2: Relative normwise errors for Example \ref{['ex:ex1']}. Here $C$ is the original Cauchy matrix and $C_i$, $i=1,2,3,4$ are the Cauchy matrices recovered from the corresponding algorithms from the perturbed matrix $A$ with $\delta=10^{-5}$. Matrix size increases from $100$ to $2000$ by steps of $100$.
  • Figure 3: Relative normwise errors for Example \ref{['ex:ex2']}. Here $C_i$, $i=1,2,3,4$ are the Cauchy matrices recovered by the corresponding algorithms. The dashed lines represent the upper bound from \ref{['eq:thm3.5']}. Matrix size is $100\times 100$.
  • Figure 4: Timings and linear fits for Algorithm \ref{['alg:LL2']} and \ref{['alg:last']}, in log-log scale. The slope of the linear fits gives the exponent in the power law $cn^\alpha$.
  • Figure 5: Relative normwise errors for Example \ref{['ex:ex4']}. Here $C_i$, $i=1,2,3,4$ are the Cauchy matrices recovered by the corresponding algorithms. The matrix $A$ is subject to an unbalanced perturbation with size $\delta=10^{-5}$. Matrix size increases from $100$ to $2000$ by steps of $100$.
  • ...and 1 more figures

Theorems & Definitions (37)

  • Lemma 1.1
  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 27 more