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On the conditioning of histopolation

Ludovico Bruni Bruno, Stefano Serra-Capizzano

TL;DR

This work analyzes the conditioning of histopolation Vandermonde matrices as the number of segments $d$ grows. It shows that using the standard monomial basis yields exponential conditioning $\kappa_2(V_d)$ across unisolvent segment classes, while a carefully chosen Chebyshev-2 basis on particular segment families (notably class (C2)) yields well-conditioned systems with bounded $\kappa_2$ and linear $\kappa_F$. The authors derive a determinant formula for class (C3), establish unisolvence for the new class (C4), and provide an explicit inversion formula for Chebyshev-segment matrices, highlighting the structural benefits of Chebyshev bases. Extending to generic intervals and connecting to the Fekete problem, the paper delineates when ill-conditioning arises and when conditioning can be controlled, with implications for numerical histopolation and related approximation problems.

Abstract

Histopolation is the approximation procedure that associates a degree $ d-1 $ polynomial $ p_{d-1} \in \mathscr{P}_{d-1} (I) $ with a locally integrable function $ f $ imposing that the integral (or, equivalently, the average) of $p$ coincides with that of $f$ on a collection of $ d $ distinct segments $s_i$. In this work we discuss unisolvence and conditioning of the associated matrices, in an asymptotic linear algebra perspective, i.e., when the matrix-size $d$ tends to infinity. While the unisolvence is a rather sparse topic, the conditioning in the unisolvent setting has a uniform behavior: as for the case of standard Vandermonde matrix-sequences with real nodes, the conditioning is inherently exponential as a function of $d$ when the monomial basis is chosen. In contrast, for an appropriate selection of supports, the Chebyshev basis of second kind has a bounded conditioning. A linear behavior is also observed in the Frobenius norm.

On the conditioning of histopolation

TL;DR

This work analyzes the conditioning of histopolation Vandermonde matrices as the number of segments grows. It shows that using the standard monomial basis yields exponential conditioning across unisolvent segment classes, while a carefully chosen Chebyshev-2 basis on particular segment families (notably class (C2)) yields well-conditioned systems with bounded and linear . The authors derive a determinant formula for class (C3), establish unisolvence for the new class (C4), and provide an explicit inversion formula for Chebyshev-segment matrices, highlighting the structural benefits of Chebyshev bases. Extending to generic intervals and connecting to the Fekete problem, the paper delineates when ill-conditioning arises and when conditioning can be controlled, with implications for numerical histopolation and related approximation problems.

Abstract

Histopolation is the approximation procedure that associates a degree polynomial with a locally integrable function imposing that the integral (or, equivalently, the average) of coincides with that of on a collection of distinct segments . In this work we discuss unisolvence and conditioning of the associated matrices, in an asymptotic linear algebra perspective, i.e., when the matrix-size tends to infinity. While the unisolvence is a rather sparse topic, the conditioning in the unisolvent setting has a uniform behavior: as for the case of standard Vandermonde matrix-sequences with real nodes, the conditioning is inherently exponential as a function of when the monomial basis is chosen. In contrast, for an appropriate selection of supports, the Chebyshev basis of second kind has a bounded conditioning. A linear behavior is also observed in the Frobenius norm.

Paper Structure

This paper contains 13 sections, 13 theorems, 85 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Ill-conditioning: the monomial basis
  3. A motivating example
  4. A formula for the determinant for class (C3)
  5. An application to the Fekete problem
  6. A conditioning result for the reference interval
  7. Segments with uniform length: the class (C4)
  8. Extension to generic intervals
  9. Some remarks on the independence of $c$ on $d$
  10. Well-conditioning: Chebyshev basis and segments
  11. An inversion formula for the Chebyshev segments
  12. A formula for the conditioning of Chebyshev segments
  13. Conditioning with respect to the Frobenius norm

Key Result

Lemma 2.2

Let $s_i = [\alpha, \beta_i]$, with $\alpha < \beta_i < \beta_j$ for $1 \leq i < j \leq d$. The determinant of $V_d$ is positive, and

Figures (6)

  • Figure 1: Exponential conditioning of the Vandermonde matrix in the monomial basis for classes (C3) and (C1), left and right hand panel respectively.
  • Figure 1: Bounded conditioning of the Vandermonde matrix in the Chebyshev basis for the class (C2). The parameter here is $a = \pi / 2$, so the resulting segments also belong to the class (C1).
  • Figure 2: The segments $s_1$, $s_2$ and $s_3$ are translated of the same segment. They are examples of segments in the class (C4).
  • Figure 2: Left: conditioning of the Chebyshev basis for segments in the class (C1). Right: conditioning of the monomial basis for segments in class (C2), compared with segments in class (C1).
  • Figure 3: Exponential asymptotic ill-conditioning for segments in the class (C4).
  • ...and 1 more figures

Theorems & Definitions (35)

  • Example
  • Example
  • Remark 2.1
  • Lemma 2.2
  • Proof 1
  • Proposition 2.3
  • Proof 2
  • Theorem 2.4
  • Proof 3
  • Definition 2.5
  • ...and 25 more