invvander

inverse or pseudoinverse of Vandermonde matrix

Fast Analytical-Form Inverse of Vandermonde Matrix

1. Introduction

`invvander` inverses an m-by-n Vandermonde matrix:

Its syntax is similar to the MATLAB built-in function `vander`.

When V is a square matrix, the `invvander` computes the analytic-form inverse of any square Vandermonde matrix based on [1] and the computational complexity is 5.5n^2 floating point operations (flops) [1]. `invvander` introduces significantly less rounding errors because it avoids numerical matrix inversion (Vandemonde matrices are usually ill-conditioned).

When V isn't a square matrix, the algorithm of the calculating the pseudoinverse of a rectangular Vandermonde matrix is standard. It implemented based on the QR decomposition, followed by a forward and a back substitutions.

2. Syntax and Function Description

V = invvander(v) returns the inverse of a square Vandermonde Matrix, i.e., m = n for the above matrix V. v has to be a row vector and v = [x1, x2, ..., xn].

V = invvander(v, m) returns the pseudoinverse of an m-by-n rectangular Vandermonde Matrix. v has to be a row vector and v = [x1, x2, ..., xn] while m has to be a scalar and positive integer of the above matrix V. If m equals the number of v, then B is the inversed square Vandermonder matrix.

3. Examples

3.1 Example 1: inverse of an n-by-n square Vandermonde matrix:

v = 1:.5:6;

B = invvander(v);

3.2 Example 2: pseudoinverse of an m-by-n rectangular Vandermonde matrix:

v = 1:.5:6;

B = invvander(v, 20);

4. References

1. Gohberg, Israel, and Vadim Olshevsky. "The fast generalized Parker–Traub algorithm for inversion of Vandermonde and related matrices." Journal of Complexity 13.2 (1997): 208-234.

2. F. Parker, Inverses of Vandermonde matrices, Amer. Math. Monthly 71,410-411, (1964).