# Numerically solving for pseudo inverse of non-squared Vandermonde matrix

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.

• you have fewer equations than unknown, so $x$ is not uniquely determined --- what do you mean by "solve" ? May 27, 2017 at 21:00
• 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) Nov 8, 2017 at 1:58