matrixcalc (version 1.0-6)

vandermonde.matrix: Vandermonde matrix

Description

This function returns an m by n matrix of the powers of the alpha vector

Usage

vandermonde.matrix(alpha, n)

Value

A matrix.

Arguments

alpha

A numerical vector of values

n

The column dimension of the Vandermonde matrix

Author

Frederick Novomestky

Details

In linear algebra, a Vandermonde matrix is an $m\times n$ matrix with terms of a geometric progression of an $m\times 1$ parameter vector $\alpha ={\left[\begin{array}{cccc}{\alpha }_1& {\alpha }_2& \cdots & {\alpha }_m\end{array}\right]}^{\prime }$

such that $V\left(\alpha \right)=\left[\begin{array}{ccccc}1& {\alpha }_1& {\alpha }_1^2& \cdots & {\alpha }_1^{n-1}\\ 1& {\alpha }_2& {\alpha }_2^2& \cdots & {\alpha }_2^{n-1}\\ 1& {\alpha }_3& {\alpha }_3^2& \cdots & {\alpha }_3^{n-1}\\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 1& {\alpha }_m& {\alpha }_m^2& \cdots & {\alpha }_m^{n-1}\end{array}\right]$.

References

Horn, R. A. and C. R. Johnson (1991). Topics in matrix analysis, Cambridge University Press.

Examples

alpha <- c( .1, .2, .3, .4 )
V <- vandermonde.matrix( alpha, 4 )
print( V )