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.

Periodical:

Bulletin of Mathematical Sciences and Applications (Volume 12)

Pages:

1-13

Citation:

R. Purushothaman Nair, "Non-Unit Bidiagonal Matrices for Factorization of Vandermonde Matrices", Bulletin of Mathematical Sciences and Applications, Vol. 12, pp. 1-13, 2015

Online since:

May 2015

Authors:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

BJORCK AND V. PEREYRA, Solution Of Vandermonde Systems Of Equations, Mathematics of Computation, 24, (1970), pp.893-903.

N.J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd edition, SIAM, Philadelphia, (2002).

M. GASCA AND J. M PENA, On Factorization of totally positive matrices , in Total Positivity and Its Applications, Kluwer Academic, Dordrect, The Netherlands, 1996, pp.109-130.

J.H. WILKINSON, Error analysis of floating-point computation, Numerische Math, 2, 1960, pp.319-340.

PLAMEN KOEV, Accurate Eigen Values And SVD's Of Totally Non-Negative Matrices, Siam Journal On Matrix Analysis and Applications, Vol. 27, 2005, No. 1, pp.1-28.

PURUSHOTHAMAN NAIR R, A Simple and Effective Factorization Procedure for Matrices, International Journal of Maths, Game Theory Algebra, 18(2009), No. 2, 145-160.

S.K. SEN AND P.V. SANKAR, Triangular partitioning for matrix inversion, International Journal of Control, Vol. 15, No. 3, 1972, pp.571-575.

VAN DE VEL, Numerical Treatment of a Generalized Vandermonde Systems of Equations, Lin. Alg. & Its Applications, Vo. 17, 1977, pp.149-174.

W. GAUTSCHI, Norm Estimates for Inverses of Vandermonde Matrices, Numer. Math, Vol. 23, 1975, pp.337-47.

W. GAUTSCHI, Optimally Conditioned Vandermonde Matrices, Numer. Math., Vol. 24, 1975, pp.1-12.

Cited By:

This article has no citations.