Non-Unit Bidiagonal Matrices for Factorization of Vandermonde Matrices

R Purushothaman Nair

doi:10.18052/www.scipress.com/bmsa.12.1

R Purushothaman Nair

Open Access

https://doi.org/10.18052/www.scipress.com/bmsa.12.1

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Abstract

A non-unit bidiagonal matrix and its inverse with simple structures are introduced. These matrices can be constructed easily using the entries of a given non-zero vector without any computations among the entries. The matrix transforms the given vector to a column of the identity matrix. The given vector can be computed back without any round off error using the inverse matrix. Since a Vandermonde matrix can also be constructed using given n quantities, it is established in this paper that Vandermonde matrices can be factorized in a simple way by applying these bidiagonal matrices. Also it is demonstrated that factors of Vandermonde and inverse Vandermonde matrices can be conveniently presented using the matrices introduced here.

Highlights

Bjorck and Pereyra in 1970 used in their classical work [1] unit bidiagonal matrices with constant off-diagonal entries and diagonal matrices for the LU representation of the inverse of Vandermonde matrices
Since a Vandermonde matrix can be constructed using given n quantities, it is established in this paper that Vandermonde matrices can be factorized in a simple way by applying these bidiagonal matrices
It is demonstrated that factors of Vandermonde and inverse Vandermonde matrices can be conveniently presented using the matrices introduced here

Summary

Introduction

Bjorck and Pereyra in 1970 used in their classical work [1] unit bidiagonal matrices with constant off-diagonal entries and diagonal matrices for the LU representation of the inverse of Vandermonde matrices. Equations (2.6) and (2.7) are the elementary bidiagonal decomposition (EBD) of the matrices used in the factorization technique This is an interesting lower triangular matrix and this construction (2.8) is possible only when the entries of x are distinct and non-zero. This matrix corresponds to all strictly monotonic decreasing and increasing sequences in the interval (0,1) and correspondence among such sequence matrices are realized by similarity transformation using appropriate diagonal matrices.Recall the remarks by Higham [2] that entries of the inverse Vandermonde lower triangular components should be distinct and in ascending order This is a pointer to the association of the eigen vectors of matrices (2.9) to such special matrices which are basically generated out of given n distinct quantities. STEP-1 Observe that factors T1 and T1-1 of order n are the basic matrices which generate the matrices B(x) and its inverse from the corresponding diagonal matrices as described in (2.5)

STEP-2

Conclusions