I want to calculate the pseudoinverse $A^+$ of a matrix $A$ whose columns are exponential decays:
$$ \begin{pmatrix} e^{-\alpha_{0}t_{0}} & e^{-\alpha_{1}t_{0}} & e^{-\alpha_{2}t_{0}} \\ e^{-\alpha_{0}t_{1}} & e^{-\alpha_{1}t_{1}} & e^{-\alpha_{2}t_{1}} \\ e^{-\alpha_{0}t_{2}} & e^{-\alpha_{1}t_{2}} & e^{-\alpha_{2}t_{2}} \\ \vdots & \vdots &\vdots \\ e^{-\alpha_{0}t_{n}} & e^{-\alpha_{1}t_{n}} & e^{-\alpha_{2}t_{n}} \\ \end{pmatrix} $$
where $ t = 0,1,2,...n $.
$\alpha_{0}, \alpha_{1}, \alpha_{2}$ are positive numbers. One of these alphas could be zero, meaning a column would be all ones.
Using exponential sum formulas I can calculate $A^TA$, but I wonder if there is a method to compute $A^+$ directly