Multivariate Barycentric Rational Interpolation of Matrix-Valued Data with a Shared Scalar Denominator

We study a multivariate barycentric rational interpolant for scattered matrix-valued data. The interpolant is defined as a weighted sum of kernel-sample products normalized by a weighted sum of kernels, where the kernels possess isolated singularities at the nodes. This formulation enforces a shared scalar denominator across all matrix entries and bilinear forms, creating a coupling that allows for shared pole candidates (resonances) while ensuring exact cancellation in ratios of bilinear forms. We prove the interpolation property using removable-singularity limits, derive a common polynomial-denominator representation for rational kernels, and provide stable evaluation algorithms that detect spurious poles (Froissart doublets). We clarify the relationship to classical inverse-distance (Shepard) interpolation, showing that our method extends the pole-free Shepard regime to an indefinite-weight regime capable of modeling resonant dynamics.