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Quasiseparable LU decay bounds for inverses of banded matrices

Paola Boito, Yuli Eidelman

TL;DR

This paper derives new exponential decay bounds for the inverse of banded matrices by exploiting a quasiseparable Green-matrix representation under a simple diagonal-dominance condition. The main LU-based result shows that if a lower-banded matrix $A$ of order $r$ satisfies a strong diagonal dominance with parameter $0<\mu<1$, then the Green-block form of $A^{-1}$ decays as $|(A^{-1})'(i,j)| \le M\gamma^{i-j-r}$ with $\gamma=\mu^{1/r}$ and a computable $M$, yielding a scalar bound $|(A^{-1})(i,j)| \le M\gamma^{i-j}$ for $i\ge j$. A QR-based counterpart provides bounds under a related hypothesis but is typically less sharp; numerical experiments show the LU-based bounds are particularly effective for nonsymmetric or indefinite matrices and for larger bandwidths, often outperforming classical spectral-function bounds like DMS84. The results advance practical, spectrally agnostic decay estimates and open avenues for extending the quasiseparable framework to broader matrix functions $f(A)$.

Abstract

We develop new, easily computable exponential decay bounds for inverses of banded matrices, based on the quasiseparable representation of Green matrices. The bounds rely on a diagonal dominance hypothesis and do not require explicit spectral information. Numerical experiments and comparisons show that these new bounds can be advantageous especially for nonsymmetric or symmetric indefinite matrices.

Quasiseparable LU decay bounds for inverses of banded matrices

TL;DR

This paper derives new exponential decay bounds for the inverse of banded matrices by exploiting a quasiseparable Green-matrix representation under a simple diagonal-dominance condition. The main LU-based result shows that if a lower-banded matrix of order satisfies a strong diagonal dominance with parameter , then the Green-block form of decays as with and a computable , yielding a scalar bound for . A QR-based counterpart provides bounds under a related hypothesis but is typically less sharp; numerical experiments show the LU-based bounds are particularly effective for nonsymmetric or indefinite matrices and for larger bandwidths, often outperforming classical spectral-function bounds like DMS84. The results advance practical, spectrally agnostic decay estimates and open avenues for extending the quasiseparable framework to broader matrix functions .

Abstract

We develop new, easily computable exponential decay bounds for inverses of banded matrices, based on the quasiseparable representation of Green matrices. The bounds rely on a diagonal dominance hypothesis and do not require explicit spectral information. Numerical experiments and comparisons show that these new bounds can be advantageous especially for nonsymmetric or symmetric indefinite matrices.

Paper Structure

This paper contains 7 sections, 11 theorems, 114 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. LU-based decay bounds: the main result
  4. Proof of LU-based decay bounds
  5. QR-based decay bounds
  6. Numerical experiments
  7. Conclusions

Key Result

Theorem 2.3

Let $A=\{A(i,j)\}_{i,j=1}^N$ be a strongly regular lower band matrix of order $r$. Then $A$ admits the factorization where $L$ is a unit lower triangular matrix and $R$ is an upper triangular matrix. The inverse $L^{-1}$ of the lower triangular factor $L$ may be represented as the product with and $(r+1)\times(r+1)$ lower triangular matrices $L_k,\;k=1,\dots,N-r$ and $r\times r$ lower triangula

Figures (5)

  • Figure 1: Plots for Example 1: behavior of the absolute value of the first column of $A^{-1}$, in logarithmic scale. Row index is on the $x$ axis, entry values and bounds on the $y$ axis. DMS stands for the bounds from DMS84, LU bounds are from Theorem \ref{['thm:bound']}.
  • Figure 2: LU bounds from Theorem \ref{['thm:bound']} for Example 2. As usual, bounds are compared to the behavior of the first column of $A^{-1}$. Row index is on the horizontal axis. DMS stands for the bounds from DMS84.
  • Figure 3: Left plot: decay bounds for Example 3. As usual, bounds are compared to the behavior of the first column of $A^{-1}$; row index is on the horizontal axis. DMS stands for the bounds from DMS84. Right plot: eigenvalues of matrix $A$.
  • Figure 4: First plot: LU bounds for Example 4a. Second plot: LU bounds for Example 4b. As usual, bounds are compared to the behavior of the first column of $A^{-1}$. Row index is on the horizontal axis. The two bottom plots show eigenvalue distributions in the complex plane for the two examples.
  • Figure 5: Comparison between LU- and QR-based bounds from Theorems \ref{['thm:bound']} and \ref{['TTRQR']} (Example 5). As usual, bounds are compared to the behavior of the first column of $A^{-1}$. Row index is on the horizontal axis. In the left plot, LU-based bounds are very close to exact values.

Theorems & Definitions (22)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • ...and 12 more