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Scaling-and-squaring method for computing the inverses of matrix $\varphi$-functions

Lidia Aceto, Luca Gemignani

TL;DR

An adaptation of the standard scaling-and-squaring technique for computing $\psi_\ell(A)$ based on the Newton-Schulz iteration for matrix inversion is proposed, based on the Newton-Schulz iteration for matrix inversion.

Abstract

This paper aims to develop efficient numerical methods for computing the inverse of matrix $\varphi$-functions, $ψ_\ell(A) := (\varphi_\ell(A))^{-1}$, for $\ell =1,2,\ldots,$ when $A$ is a large and sparse matrix with eigenvalues in the open left half-plane. While $\varphi$-functions play a crucial role in the analysis and implementation of exponential integrators, their inverses arise in solving certain direct and inverse differential problems with non-local boundary conditions. We propose an adaptation of the standard scaling-and-squaring technique for computing $ψ_\ell(A)$, based on the Newton-Schulz iteration for matrix inversion. The convergence of this method is analyzed both theoretically and numerically. In addition, we derive and analyze Padé approximants for approximating $ψ_1(A/2^s)$, where $s$ is a suitably chosen integer, necessary at the root of the squaring process. Numerical experiments demonstrate the effectiveness of the proposed approach.

Scaling-and-squaring method for computing the inverses of matrix $\varphi$-functions

TL;DR

An adaptation of the standard scaling-and-squaring technique for computing based on the Newton-Schulz iteration for matrix inversion is proposed, based on the Newton-Schulz iteration for matrix inversion.

Abstract

This paper aims to develop efficient numerical methods for computing the inverse of matrix -functions, , for when is a large and sparse matrix with eigenvalues in the open left half-plane. While -functions play a crucial role in the analysis and implementation of exponential integrators, their inverses arise in solving certain direct and inverse differential problems with non-local boundary conditions. We propose an adaptation of the standard scaling-and-squaring technique for computing , based on the Newton-Schulz iteration for matrix inversion. The convergence of this method is analyzed both theoretically and numerically. In addition, we derive and analyze Padé approximants for approximating , where is a suitably chosen integer, necessary at the root of the squaring process. Numerical experiments demonstrate the effectiveness of the proposed approach.
Paper Structure (7 sections, 5 theorems, 57 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 7 sections, 5 theorems, 57 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Padé approximation of Lg
  3. Exploring the case Lg1
  4. The scaling-and-squaring method
  5. Convergence analysis
  6. Numerical Results
  7. Conclusions

Key Result

Theorem 2.1

Let $r_{\mu,\nu}(z)= p_{\mu}(z)/q_{\nu}(z)$ be the $[\mu/\nu]$-Padé approximant for a function $f(z)$ with the formal power series $f(z) = \sum_{j=0}^{+ \infty} c_jz^j,$$c_j \in \mathbb{C},$$c_0 \neq 0.$ Then, the $[\nu/\mu]$-Padé approximant for $1/f(z)$ is given by

Figures (5)

  • Figure 1: Plots of $h_\ell(z)$ for $\ell= 2,3, 4, 16,$ with $z\geq 0$.
  • Figure 2: Convergence of the Newton-Schulz algorithm applied to $\varphi_2(\mathcal{A}_1/2^{19})$ with starting point $\psi_1(\mathcal{A}_1/2^{19})$. The plots illustrate the convergence of $err_2$ of Algorithm 1 for the matrix $\mathcal{A}_1$ with $n=1024$.
  • Figure 3: Convergence of the Newton-Schulz algorithm called by the UpdateMatrix procedure to perform the squaring steps for matrix $\psi$-functions. The plots illustrate the convergence of $err_2$ of Algorithm 1 in each squaring step for the matrix $\mathcal{A}_1$ with $n=1024$.
  • Figure 4: Illustration of the spectrum of $\mathcal{A}_2$ for $n=10000$.
  • Figure 5: Convergence of the Newton-Schulz algorithm called by the UpdateMatrix procedure to perform the squaring steps for matrix $\psi$-functions. The plots illustrate the convergence of $err_2$ of Algorithm 1 in each squaring step for the matrix $\mathcal{A}_2$ with $n=10000$.

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof