Fast algorithms for block Vandermonde systems

Develop a fast numerical algorithm that exploits the block Vandermonde structure of the KEG-D coefficient matrix to achieve computational complexity comparable to that of specialized algorithms for ordinary Vandermonde systems.

Background

The paper derives an explicit determinant formula and an interpolation-based inverse for the block Vandermonde matrix arising in the KEG-D system. However, the determinant formula contains (2qq)\binom{2q}{q} terms and is therefore impractical for large qq. The authors note that Gaussian elimination requires O(q3)O(q^3) operations, whereas specialized algorithms for ordinary Vandermonde systems can be faster. The unresolved question is whether the additional block structure can likewise be exploited algorithmically.

References

Whether the particular block Vandermonde structure of eq:bv can be exploited to obtain a comparable fast algorithm is an interesting numerical question, which we do not pursue here.

eq:bv:

Λ=(α1α1x1⋯α1x1q−1β1β1x1⋯β1x1q−1α2α2x2⋯α2x2q−1β2β2x2⋯β2x2q−1⋮⋮⋱⋮⋮⋮⋱⋮α2qα2qx2q⋯α2qx2qq−1β2qβ2qx2q⋯β2qx2qq−1).\Lambda = \begin{pmatrix} \alpha_1 & \alpha_1x_1 & \cdots & \alpha_1x_1^{q-1} & \beta_1 & \beta_1x_1 & \cdots & \beta_1x_1^{q-1} \\ \alpha_2 & \alpha_2x_2 & \cdots & \alpha_2x_2^{q-1} & \beta_2 & \beta_2x_2 & \cdots & \beta_2x_2^{q-1} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ \alpha_{2q} & \alpha_{2q}x_{2q} & \cdots & \alpha_{2q}x_{2q}^{q-1} & \beta_{2q} & \beta_{2q}x_{2q} & \cdots & \beta_{2q}x_{2q}^{q-1} \end{pmatrix}.

— Asymptotic behavior of Eckhoff's method for convergence acceleration of Dirac eigenfunction expansions  (2609.00895 - Barkhudaryan et al., 1 Sep 2026) in Remark following Theorem 2.3, Section 2