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Computation of quasiseparable representations of Green matrices

Paola Boito, Yuli Eidelman

TL;DR

This paper addresses the problem of efficiently inverting banded matrices by exploiting the Asplund theorem, which links the inverse of a lower band matrix to a lower Green matrix with a rank-structured, quasiseparable representation. It develops two complementary, linear-time inversion schemes based on QR factorization and LU factorization, respectively, to compute the quasiseparable generators of $A^{-1}$ and to handle both one-sided and two-sided band structures. The key contributions are explicit constructions of the Green generators through transform-factorizations and recursions, along with unitary and triangular factorization strategies that preserve the rank-structured form during inversion. Numerical experiments confirm linear-time behavior and stability of the proposed methods, and the work sets the stage for decay-bound analyses and efficient subsequent structured computations in large-scale problems.

Abstract

The well-known Asplund theorem states that the inverse of a (possibly one-sided) band matrix $A$ is a Green matrix. In accordance with quasiseparable theory, such a matrix admits a quasiseparable representation in its rank-structured part. Based on this idea, we derive algorithms that compute a quasiseparable representation of $A^{-1}$ with linear complexity. Many inversion algorithms for band matrices exist in the literature. However, algorithms based on a computation of the rank structure performed theoretically via the Asplund theorem appear for the first time in this paper. Numerical experiments confirm complexity estimates and offer insight into stability properties.

Computation of quasiseparable representations of Green matrices

TL;DR

This paper addresses the problem of efficiently inverting banded matrices by exploiting the Asplund theorem, which links the inverse of a lower band matrix to a lower Green matrix with a rank-structured, quasiseparable representation. It develops two complementary, linear-time inversion schemes based on QR factorization and LU factorization, respectively, to compute the quasiseparable generators of and to handle both one-sided and two-sided band structures. The key contributions are explicit constructions of the Green generators through transform-factorizations and recursions, along with unitary and triangular factorization strategies that preserve the rank-structured form during inversion. Numerical experiments confirm linear-time behavior and stability of the proposed methods, and the work sets the stage for decay-bound analyses and efficient subsequent structured computations in large-scale problems.

Abstract

The well-known Asplund theorem states that the inverse of a (possibly one-sided) band matrix is a Green matrix. In accordance with quasiseparable theory, such a matrix admits a quasiseparable representation in its rank-structured part. Based on this idea, we derive algorithms that compute a quasiseparable representation of with linear complexity. Many inversion algorithms for band matrices exist in the literature. However, algorithms based on a computation of the rank structure performed theoretically via the Asplund theorem appear for the first time in this paper. Numerical experiments confirm complexity estimates and offer insight into stability properties.
Paper Structure (11 sections, 15 theorems, 135 equations, 8 figures)

This paper contains 11 sections, 15 theorems, 135 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Band and Green matrices, the block partitions
  3. The inversion of lower band matrices
  4. The transform and the factorization
  5. The multiplication by triangular matrix
  6. The QR algorithm
  7. Inverses of lower band matrices
  8. Two-sided band matrices
  9. The LU algorithm
  10. Numerical tests
  11. Conclusions and future work

Key Result

Theorem 2.1

An invertible matrix $A$ is a lower band matrix of order $r$ if and only if its inverse $B=A^{-1}$ is a lower Green matrix of order $r$.

Figures (8)

  • Figure 1: Logarithmic plot of running times for Example 1 (red dots), i.e., QR-based inversion of two-sided banded matrices. Matrix size ranges from $250$ to $2000$. The slope of the linear fit is $0.9373$, therefore consistent with theoretical complexity analysis.
  • Figure 2: Relative forward errors for Example 1 (red dots), i.e., QR-based inversion of two-sided banded matrices. Matrix size ranges from $250$ to $2000$. Experimental accuracy is consistent with theoretical estimates (black stars) given by the machine epsilon times the 2-norm condition number, and does not appear to deteriorate with increasing size.
  • Figure 3: Logarithmic plot of running times for Example 2 (blue dots), i.e., LU-based inversion of two-sided banded matrices. Matrix size ranges from $500$ to $2500$. The slope of the linear fit is $1.016$, therefore consistent with theoretical complexity analysis.
  • Figure 4: Relative forward errors for Example 2, i.e., LU-based inversion of two-sided banded matrices. Matrix size ranges from $500$ to $2500$.
  • Figure 5: Logarithmic plot of running times for Example 3 (two-sided banded case). Matrix size ranges from $600$ to $2500$. Linear fits (not shown here) have a slope of about $1.04$ for both quasiseparable algorithms, $2.40$ for sparseinv, $2.75$ for inv and $2.78$ for classical QR.
  • ...and 3 more figures

Theorems & Definitions (16)

  • Theorem 2.1: The Asplund theorem
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 5.1
  • Corollary 5.2
  • ...and 6 more