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Computing the action of a matrix exponential on an interval via the $\star$-product approach

Stefano Pozza, Shazma Zahid

Abstract

We present a new method for computing the action of the matrix exponential on a vector, $e^{At}v$. The proposed approach efficiently evaluates the solution for all $t$ within a prescribed bounded interval by expanding it into an orthogonal polynomial series. This method is derived from a new representation of the matrix exponential in the so-called $\star$-algebra, an algebra of bivariate distributions. The resulting formulation leads to a linear system equivalent to a matrix equation of Stein type, which can be solved by either direct or Krylov subspace methods. Numerical experiments demonstrate the accuracy and efficiency of the proposed approach in comparison with state-of-the-art techniques.

Computing the action of a matrix exponential on an interval via the $\star$-product approach

Abstract

We present a new method for computing the action of the matrix exponential on a vector, . The proposed approach efficiently evaluates the solution for all within a prescribed bounded interval by expanding it into an orthogonal polynomial series. This method is derived from a new representation of the matrix exponential in the so-called -algebra, an algebra of bivariate distributions. The resulting formulation leads to a linear system equivalent to a matrix equation of Stein type, which can be solved by either direct or Krylov subspace methods. Numerical experiments demonstrate the accuracy and efficiency of the proposed approach in comparison with state-of-the-art techniques.
Paper Structure (7 sections, 6 theorems, 76 equations, 1 figure, 4 tables)

This paper contains 7 sections, 6 theorems, 76 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Truncation error
  4. Numerical Methods
  5. Algorithms
  6. Numerical Tests
  7. Conclusions

Key Result

Theorem 3.1

Any matrix exponential can be expanded into a $\star$-algebra Taylor series centered at $c \in \mathbb{C}$, with $|c| < |c-1|$, that is, Moreover, the related truncated series remainder is where $L_k(z)$ is the Laguerre polynomial of $k$-degree.

Figures (1)

  • Figure 1: Comparison of solution norms $\|e^{At}v\|$ for Examples 4 and 8 over their respective time intervals ($[0,4]$ and $[0,0.1]$). The $\star$-method results are shown as curves and expmv_tspan values as discrete points overlaid on that curve.

Theorems & Definitions (12)

  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 2 more