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I have a linear system to solve, set up as:

$\bf{Ax}=\bf{b}$

with a non-squared matrix A,

$ \bf{A}= \begin{bmatrix} 1 & A_{1} & A_{1}^2 & \cdots & A_{1}^n \\ 1 & A_{2} & A_{2}^2 & \cdots & A_{2}^n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & A_{m} & A_{m}^2 & \cdots & A_{m}^n \end{bmatrix} $,

where $m$ and $n$ satisfy $m<n$, and

$A_{j} = \exp((i-1)a_j) $.

This $a_j$ satisfy $a_j >0$, and $i$ is an imaginary unit. This matrix is not a pure Vandermonde matrix because it’s not a square matrix, but it has similar formula.

I applied SVD and Tikhonov regularization to solve this system, but its solution is far from the true value. What method can I use to solve this? Is there any analytical solution for this system?

Thanks.

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  • $\begingroup$ you have fewer equations than unknown, so $x$ is not uniquely determined --- what do you mean by "solve" ? $\endgroup$
    – Carlo Beenakker
    Commented May 27, 2017 at 21:00
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    $\begingroup$ The condition number of these matrices is usually spectacularly bad. if $m < n$ then you must apply additional constraints to get a unique solution. A lot of times people search for the solution that has the smallest 1 norm ($L_1$ optmization). My favorite is the Fast Linearized Bregman Iteration found at math.ucla.edu/applied/cam . You will need to modify the shrinkage operator to handle the complex case (shrink into the unit circle) $\endgroup$
    – eigenjohnson
    Commented Nov 8, 2017 at 1:58

2 Answers 2

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Read the ancient paper entitled "Inverse of the Vandermonde Matrix with Applicataions", by L. Richard Turner, and be enlightened.

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Vandermonde matrices are famously unstable to solve numerically. Here's some of the latest research on inverting Vandermonde matrixes: https://arxiv.org/abs/1909.08155 -- Mahdi S. Hosseini and Alfred Chen and Konstantinos N. Plataniotis (2019) "On the Closed Form Expression of Elementary Symmetric Polynomials and the Inverse of Vandermonde Matrix" -- it's about square matrices, but it will at least give you a stable algorithm.

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