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Multicentric representation of piecewise constant holomorphic functions and Hermite interpolation

Olavi Nevanlinna, Tiina Vesanen

TL;DR

The paper analyzes multicentric calculus for piecewise holomorphic, especially piecewise-constant, functions, showing that truncating the power-series coefficients yields Hermite-like polynomials while preserving numerical stability as the degree grows. It develops several computational routes (Cauchy integrals, residues, recursions, and polyproducts) to form multicentric representations and links them to Hermite interpolation via basis polynomials $oldsymbol{ abla}_{k,j,n}$. Numerical experiments compare different implementations, revealing that multicentric formulations maintain bounded errors with increasing $n$, whereas conventional Hermite interpolation exhibits exponential error growth; this resilience persists even when eigenvalues are imprecisely known. The results support using multicentric representations in holomorphic functional calculus, particularly for large-scale problems where spectrum information is approximate, and highlight the potential for exact arithmetic in coefficient computation to further reduce numerical noise.

Abstract

In multicentric representation of piecewise holomorphic functions one combines Lagrange interpolation at roots of a polynomial $p$ with convergent power series of $p$ as the "coefficients" multiplying the Lagrange basis polynomials. When these power series are truncated one obtains Hermite interpolation polynomials. In this paper we first review different approaches to obtain multicentric representations with emphasis in piecewise constant holomorphic functions. When the polynomial is of degree $d$ and all power series are truncated after $n^{th}$ power, we formally arrive into a Hermite interpolation polynomial of degree $d(n+1)-1$. The natural way to represent Hermite interpolation is to have for each interpolation condition a basis polynomial which in this case leads to $d(n+1)$ basis polynomials. We then consider the numerical accumulation of errors in the different ways to represent and evaluate the Hermite interpolation. In the multicentric representation due to the convergence of the power series, numerical errors stay bounded as $n$ grows. When we assume that the piecewise constant holomorphic function takes the value $1$ in one of the components and vanishes in the other so that the Hermite interpolation agrees with just one basis polynomial, even then the truncated multicentric representation is favorable. In the general case one would take a linear combination of all $d(n+1)$ basis polynomials.

Multicentric representation of piecewise constant holomorphic functions and Hermite interpolation

TL;DR

The paper analyzes multicentric calculus for piecewise holomorphic, especially piecewise-constant, functions, showing that truncating the power-series coefficients yields Hermite-like polynomials while preserving numerical stability as the degree grows. It develops several computational routes (Cauchy integrals, residues, recursions, and polyproducts) to form multicentric representations and links them to Hermite interpolation via basis polynomials . Numerical experiments compare different implementations, revealing that multicentric formulations maintain bounded errors with increasing , whereas conventional Hermite interpolation exhibits exponential error growth; this resilience persists even when eigenvalues are imprecisely known. The results support using multicentric representations in holomorphic functional calculus, particularly for large-scale problems where spectrum information is approximate, and highlight the potential for exact arithmetic in coefficient computation to further reduce numerical noise.

Abstract

In multicentric representation of piecewise holomorphic functions one combines Lagrange interpolation at roots of a polynomial with convergent power series of as the "coefficients" multiplying the Lagrange basis polynomials. When these power series are truncated one obtains Hermite interpolation polynomials. In this paper we first review different approaches to obtain multicentric representations with emphasis in piecewise constant holomorphic functions. When the polynomial is of degree and all power series are truncated after power, we formally arrive into a Hermite interpolation polynomial of degree . The natural way to represent Hermite interpolation is to have for each interpolation condition a basis polynomial which in this case leads to basis polynomials. We then consider the numerical accumulation of errors in the different ways to represent and evaluate the Hermite interpolation. In the multicentric representation due to the convergence of the power series, numerical errors stay bounded as grows. When we assume that the piecewise constant holomorphic function takes the value in one of the components and vanishes in the other so that the Hermite interpolation agrees with just one basis polynomial, even then the truncated multicentric representation is favorable. In the general case one would take a linear combination of all basis polynomials.

Paper Structure

This paper contains 16 sections, 4 theorems, 76 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic formulas for multicentric representation
  3. Cauchy integrals
  4. Using residue calculus
  5. Recursion by differentiating
  6. Solving $g\circledcirc g=g$
  7. Formulas for $z^k$
  8. Errors due to inexact evaluation of the variables
  9. Formulas for related Hermite interpolation
  10. Numerical experiments
  11. Testpoints
  12. Results
  13. Level curves $\gamma_\rho$
  14. Evaluation error $\nu$
  15. Rounding error $\mu$
  16. ...and 1 more sections

Key Result

Proposition 1.1

Assume that all roots $\lambda_k$ of $p$ have rational real and imaginary parts. If $\varphi$ is holomorphic at the roots and all values $\varphi^{(\nu)}(\lambda_k)$ have also rational real and imaginary parts then the series expansions of $f_k$ have coefficients with rational real and imaginary par

Figures (7)

  • Figure 1: (Example 1) Level curves of polynomial $p(z)=z(1-z^2)$ with roots $\lambda_1=0$, $\lambda_2=-1$, and $\lambda_3=+1$. The roots are marked with 'x'. The critical points $c$, from $p'(c)=0$, are $c=\pm 1/\sqrt{3} \approx \pm 0.577$. At corresponding level $|p(c)| = 2/(3\sqrt{3}) =:\rho_{crit} \approx 0.3849$ the three components join together. The piecewise constant function $\chi$ we approximate with Hermite polynomials is 1 at $\lambda_1=0$ and 0 at the other two roots. The component enclosing $\lambda_1=0$ is colored. Four sets of smaller level curves $\gamma_\rho$ where $\rho$ is $t \times \rho_{crit}$ and $t$ is either $0.30$, $0.60$, $0.90$ or $0.99$ are drawn in addition to the lemniscate $|p(z)|=\rho_{crit}$. Example of the testpoints used in our numerical calculations are marked with '+' on the lemniscate $|p(z)|=0.6\times\rho_{crit}$.
  • Figure 2: (Example 2) Level curves of polynomial $p(z)=z(z^2+4z+5$) with roots $\lambda_1=0$, $\lambda_2=-2+i$, and $\lambda_3=-2-i$. The roots are marked with 'x'. There are two critical points $c$, $p'(c)=0$, $c_1=-1$ and $c_2=-5/3$. At corresponding level $|p(c_1)| = 2 =:\rho_{crit}$ the two remaining components join together. The component including the root $\lambda_1=0$ is colored. Four sets of smaller level curves $\gamma_\rho$ where $\rho$ is $t \times \rho_{crit}$ and $t$ is either $0.30$, $0.60$, $0.90$ or $0.99$ are drawn in addition to the lemniscate $|p(z)|=\rho_{crit}$. Example of the testpoints used in our numerical calculations are marked with '+' on the lemniscate $|p(z)|=0.6\times\rho_{crit}$.
  • Figure 3: (Example 1 in the left column and Example2 on the right hand side) Effect of the size of $\rho$. The errors $\nu$ and $\mu$ are zero. The smallest level $\rho=0.3\times\rho_{crit}$ is in the first row and the levels grow with each line according the Table \ref{['rhootaulukko']}.
  • Figure 4: (Example 1 on the left and and 2 in the right hand side column) Effect of evaluation error $\nu$ when $\rho=0.02\times \rho_{crit}$ and rounding error $\mu=0$. The tested values for $\nu$ are $0$, $10^{-16}$, $10^{-8}$, and $10^{-4}$ (from top to bottom).
  • Figure 5: (Example 1 on the left and 2 on the right) The effect of rounding error $\mu$ when $\nu=0$ and $\rho=0.02\times \rho_{crit}$. Values used for $\mu$ are $0$, $10^{-16}$, $10^{-8}$, and $10^{-4}$ (from top to bottom). The error values for Special (red contiguous line) on the bottom right are $6.54\cdot10^{-4}$ at $n=4$ and $2.9\cdot10^{-3}$ at $n=24$.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 3.1
  • ...and 3 more