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Flexible rational approximation and its application for matrix functions

Nir Sharon, Vinesha Peiris, Nadia Sukhorukova, Julien Ugon

TL;DR

The paper tackles evaluating real-valued matrix functions f(A) by minimax rational approximation and develops a flexible optimization framework that accommodates generalized rational forms with linear constraints. The optimization targets min_{α,β} max_{x ∈ [a,b]} |f(x) - (α^T G(x)) / (β^T H(x))| with the constraint β^T H(x) > 0, and uses a bisection-based convex feasibility approach. A key theoretical result shows that if 0<ℓ ≤ |q(x)| ≤ u on the domain and A is real normal with eigenvalues in Ω, then cond(q(A)) ≤ u/ℓ, enabling conditioning control. The paper demonstrates spectrum filtering and projection to the positive semidefinite cone as practical matrix-function tasks, and provides numerical evidence of accuracy and speed gains while offering reproducible open-source code. These contributions yield a practical tool for efficient matrix function evaluation in large-scale settings and support extensions to constrained rational approximations.

Abstract

This paper proposes a unique optimization approach for estimating the minimax rational approximation and its application for evaluating matrix functions. Our method enables the extension to generalized rational approximations and has the flexibility of adding constraints. In particular, the latter allows us to control specific properties preferred in matrix function evaluation. For example, in the case of a normal matrix, we can guarantee a bound over the condition number of the matrix, which one needs to invert for evaluating the rational matrix function. We demonstrate the efficiency of our approach for several applications of matrix functions based on direct spectrum filtering.

Flexible rational approximation and its application for matrix functions

TL;DR

The paper tackles evaluating real-valued matrix functions f(A) by minimax rational approximation and develops a flexible optimization framework that accommodates generalized rational forms with linear constraints. The optimization targets min_{α,β} max_{x ∈ [a,b]} |f(x) - (α^T G(x)) / (β^T H(x))| with the constraint β^T H(x) > 0, and uses a bisection-based convex feasibility approach. A key theoretical result shows that if 0<ℓ ≤ |q(x)| ≤ u on the domain and A is real normal with eigenvalues in Ω, then cond(q(A)) ≤ u/ℓ, enabling conditioning control. The paper demonstrates spectrum filtering and projection to the positive semidefinite cone as practical matrix-function tasks, and provides numerical evidence of accuracy and speed gains while offering reproducible open-source code. These contributions yield a practical tool for efficient matrix function evaluation in large-scale settings and support extensions to constrained rational approximations.

Abstract

This paper proposes a unique optimization approach for estimating the minimax rational approximation and its application for evaluating matrix functions. Our method enables the extension to generalized rational approximations and has the flexibility of adding constraints. In particular, the latter allows us to control specific properties preferred in matrix function evaluation. For example, in the case of a normal matrix, we can guarantee a bound over the condition number of the matrix, which one needs to invert for evaluating the rational matrix function. We demonstrate the efficiency of our approach for several applications of matrix functions based on direct spectrum filtering.

Paper Structure

This paper contains 17 sections, 1 theorem, 32 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation and related background
  3. Evaluating matrix functions
  4. Uniform rational approximation
  5. Optimization based on quasiconvexity
  6. Flexible uniform rational approximation via optimization
  7. Optimizing a rational approximation
  8. Quasiconvexity and the optimization setup
  9. Bisection method and convex feasibility problem
  10. Denominator Bounds
  11. Approximating matrix functions
  12. Spectrum filtering of a matrix
  13. Applying filtered matrix to a vector
  14. From symmetric matrices to positive semidefinite matrices
  15. Conclusions and future research directions
  16. ...and 2 more sections

Key Result

Lemma 1

Let $r=\dfrac{p}{q} \in { \text{R}_{ {m} , {n} } }$ and assume that Furthermore, assume $A$ is a real normal matrix with all eigenvalues in $\Omega$. Then,

Figures (8)

  • Figure 1: Approximating matrix function via scalar rational approximation.
  • Figure 2: The filter function $F$ of \ref{['eqn:filter_func']}. The effective zone of eigenvalue preservation, $[0.25 , 0.55]$, is marked, together with the corresponding ideal filter function
  • Figure 3: The approximation of the filter function $F$ of \ref{['eqn:filter_func']}
  • Figure 4: Algebraic multiplicity of the eigenvalues of the matrix, before and after applying the filter $F$ of \ref{['eqn:filter_func']}.
  • Figure 5: A runtime comparison of evaluating $f(A)v$. The results imply that evaluating the rational approximation is faster than extracting the spectral decomposition, and the difference becomes more significant as the matrix grows
  • ...and 3 more figures

Theorems & Definitions (3)

  • Remark 1
  • Lemma 1
  • proof