Accurate Bidiagonal Decomposition and Computations with Generalized Pascal Matrices

TL;DR

The paper tackles reliable, high-relative-accuracy computation for generalized Pascal matrices, which are often ill-conditioned. It derives explicit bidiagonal decompositions for generalized triangular Pascal matrices and lattice-path matrices using Neville elimination, enabling HRA-based algorithms to compute eigenvalues, singular values, inverses, and linear-system solutions. Numerical experiments on lattice-path and generalized Pascal matrices show that HRA-based routines such as $\text{TNEigenValues}$, $\text{TNSingularValues}$, and $\text{TNInverseExpand}$ achieve markedly higher accuracy than standard MATLAB routines at comparable cost. This work extends HRA techniques to a broader class of structured matrices with potential impact in design, probability, and signal processing.

Abstract

This paper provides an accurate method to obtain the bidiagonal factorization of many generalized Pascal matrices, which in turn can be used to compute with high relative accuracy the eigenvalues, singular values and inverses of these matrices. Numerical examples are included.

Figures (7)

• Figure 1: Relative error for the minimal eigenvalues and singular values of $Lp_n(\sqrt{2},\sqrt{3},\sqrt{5})$
• Figure 2: Relative errors for $Lp_n(\sqrt{2},\sqrt{3},\sqrt{5})^{-1}$, $n=5,10,\ldots,50$
• Figure 3: Relative errors for the systems $Lp_n(\sqrt{2},\sqrt{3},\sqrt{5})x=b_n$, $n=5,10,\ldots,50$
• Figure 4: Relative errors for the systems $Lp_n(\sqrt{2},\sqrt{3},\sqrt{5})x=\tilde{b}_n$, $n=5,10,\ldots,50$
• Figure 5: Relative error for the minimal singular values of $P_{n,1}[3/2]$