Bidiagonal Decompositions of Vandermonde-Type Matrices of Arbitrary Rank

TL;DR

This paper introduces a method for deriving explicit bidiagonal decompositions of Vandermonde-type matrices, including those of arbitrary rank, enabling efficient and accurate computations for various matrix operations.

Contribution

It generalizes existing bidiagonal decomposition expressions to matrices of any rank, including singular and nonnegative matrices, improving computational efficiency and accuracy.

Findings

Decomposition formulas derived for arbitrary rank matrices

Efficient, high-accuracy computations for eigenvalues and related operations

Applicable to totally nonnegative Vandermonde-type matrices

Abstract

We present a method to derive new explicit expressions for bidiagonal decompositions of Vandermonde and related matrices such as the (q-, h-) Bernstein-Vandermonde ones, among others. These results generalize the existing expressions for nonsingular matrices to matrices of arbitrary rank. For totally nonnegative matrices of the above classes, the new decompositions can be computed efficiently and to high relative accuracy componentwise in floating point arithmetic. In turn, matrix computations (e.g., eigenvalue computation) can also be performed efficiently and to high relative accuracy.