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When Lanczos Iterations Generate Symmetric Quadrature Nodes?

Wenhao Li, Zongyuan Han, Shengxin Zhu

TL;DR

The paper examines when Lanczos-based Gaussian quadrature for estimating matrix quadratic forms yields symmetric quadrature nodes, challenging the long-standing assumption of inherent symmetry. It develops a sufficient condition based on symmetric eigenvalue distributions and the discrete measure $\mu(t)$, together with a constructive method for choosing starting vectors in Jordan-Wielandt matrices to enforce symmetric Ritz values, thereby achieving symmetric quadrature rules. The main theoretical payoff is that, for a wide class of Jordan-Wielandt matrices, one can guarantee symmetry (and equal weights for symmetric nodes) by ensuring the initial vector induces an absolute-palindrome measure $\boldsymbol{\mu}$, with practical impact in estimating network measures like the Estrada index. Numerical experiments on synthetic and real network data validate the theory, demonstrate variance reduction from symmetry-driven starts, and quantify iteration savings relative to ill-conditioned problems. Overall, the work clarifies conditions under which Lanczos quadrature remains efficient and symmetric in realistic applications, while acknowledging that symmetry for general matrices remains problem-specific and initial-vector dependent.

Abstract

The Golub-Welsch algorithm [ Math. Comp., 23: 221-230 (1969)] has long been assumed symmetric for estimating quadratic forms. Recent research indicates that asymmetric quadrature nodes may be more often and the existence of a practical symmetric quadrature for estimating matrix quadratic form is even doubtful.This paper derives a sufficient condition for symmetric quadrature nodes for estimating quadratic forms involving the Jordan-Wielandt matrices which frequently arise from many applications. The condition is closely related to how to construct an initial vector for the underlying Lanczos process. Applications of such constructive results are demonstrated by estimating the Estrada index in complex network analysis.

When Lanczos Iterations Generate Symmetric Quadrature Nodes?

TL;DR

The paper examines when Lanczos-based Gaussian quadrature for estimating matrix quadratic forms yields symmetric quadrature nodes, challenging the long-standing assumption of inherent symmetry. It develops a sufficient condition based on symmetric eigenvalue distributions and the discrete measure , together with a constructive method for choosing starting vectors in Jordan-Wielandt matrices to enforce symmetric Ritz values, thereby achieving symmetric quadrature rules. The main theoretical payoff is that, for a wide class of Jordan-Wielandt matrices, one can guarantee symmetry (and equal weights for symmetric nodes) by ensuring the initial vector induces an absolute-palindrome measure , with practical impact in estimating network measures like the Estrada index. Numerical experiments on synthetic and real network data validate the theory, demonstrate variance reduction from symmetry-driven starts, and quantify iteration savings relative to ill-conditioned problems. Overall, the work clarifies conditions under which Lanczos quadrature remains efficient and symmetric in realistic applications, while acknowledging that symmetry for general matrices remains problem-specific and initial-vector dependent.

Abstract

The Golub-Welsch algorithm [ Math. Comp., 23: 221-230 (1969)] has long been assumed symmetric for estimating quadratic forms. Recent research indicates that asymmetric quadrature nodes may be more often and the existence of a practical symmetric quadrature for estimating matrix quadratic form is even doubtful.This paper derives a sufficient condition for symmetric quadrature nodes for estimating quadratic forms involving the Jordan-Wielandt matrices which frequently arise from many applications. The condition is closely related to how to construct an initial vector for the underlying Lanczos process. Applications of such constructive results are demonstrated by estimating the Estrada index in complex network analysis.
Paper Structure (18 sections, 10 theorems, 72 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 18 sections, 10 theorems, 72 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Lanczos quadrature method
  3. Symmetric Ritz values in the Lanczos iterations
  4. Symmetric matrices with symmetric eigenvalues
  5. Central symmetric discrete measure functions
  6. A sufficient condition of symmetric Ritz values in the Lanczos iterations
  7. Construction of initial vectors for Jordan-Wielandt matrices
  8. Case 1: $n_1 = n_2$
  9. Case 2: $n_1 > n_2$
  10. A simple example
  11. Computation complexity for symmetric and asymmetric quadrature nodes
  12. Numerical experiments
  13. Test on Theorem \ref{['Thm:main']}
  14. Test on Theorem \ref{['prop:1']} in applications of complex network
  15. Test on synthetic matrix
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $T_m \in \mathbb{R}^{m\times m}$ be the symmetric tridiagonal matrix generated in the $m$-step Lanczos method with symmetric matrix $A \in \mathbb{R}^{n \times n}$ and initial vector $\boldsymbol{u}$. Then under the assumption that the values are non-decreasing by index, the eigenvalues of $A$,

Figures (4)

  • Figure 1: Plots of discrete measure functions \ref{['eq:measure_f']} and the locations of Ritz values in four cases.
  • Figure 2: 100 trials of stochastic Lanczos quadrature estimators of $\mathrm{tr}(e^{\beta A})$ with $\beta = 1$, $m=100$, synthetic bipartite matrix $A$ and different initial vectors. $\boldsymbol{v}^1_u$ and $\boldsymbol{v}_d^1$ are half-zero-half-Rademacher vectors with either upper or lower part of entries taking $\pm 1$, while all elements of $\boldsymbol{v}^1_r$ are Rademacher distributed.
  • Figure 3: 100 trials of stochastic Lanczos quadrature estimators of $\mathrm{tr}(e^{\beta A})$ with $\beta = 0.5/\lambda_{\max}$, $m=100$, bipartite matrix based on email-Eu-core-temporal data set PBL17DH11 and different initial vectors. $\boldsymbol{v}^1_u$ and $\boldsymbol{v}_d^1$ are half-zero-half-Rademacher vectors with either upper or lower part of entries taking $\pm 1$, while all elements of $\boldsymbol{v}_r^1$ are Rademacher distributed.
  • Figure 4: Lower, upper bounds, the average of the two bounds, and the exact values of the absolute difference $m^*$ between the Lanczos iterations required for equivalent accuracy in approximating quadratic forms \ref{['eq:GQ']} with asymmetric and symmetric quadrature nodes, where $\kappa = 10,50,100,500,\ldots,5\times 10^4,10^5$.

Theorems & Definitions (20)

  • Theorem 3.1
  • proof
  • Theorem 3.3
  • Definition 3.5
  • Theorem 3.7
  • proof
  • Remark 3.8
  • Lemma 3.9
  • proof
  • Theorem 3.10
  • ...and 10 more