Module structured 

Structured matrix-vector products — the unifying object: semiseparable, Toeplitz, Cauchy, and Vandermonde matvecs.

Functions

cauchy_matvec

Cauchy kernel evaluation — original S4’s (DPLR) kernel-generation step. Given complex evaluation points nodes (z_k, typically roots of unity), complex poles (lambda_i, the diagonalized state matrix’s eigenvalues), and complex weights (w_i, the per-pole residues, e.g. C_i * B_i), computes for each node: out[k] = sum_i weights[i] / (nodes[k] - poles[i])

semiseparable_matvec

Semiseparable matrix-vector product, diagonal/selective parameterization (Mamba-2 / SSD style). Each row i computes: y_i = C_i . h_i, where h_i = a_i * h_{i-1} + B_i * x_i — i.e. the matvec IS the SSM recurrence; this function just exposes it as “apply this implicit matrix to a vector” for callers that think in those terms (e.g. an attention-equivalent API).

toeplitz_matvec

Causal Toeplitz matrix-vector product — the convolutional view of an SSM. Implicitly applies the lower-triangular matrix M[i][j] = kernel[i - j] for j <= i, 0 otherwise to x, i.e. computes the causal convolution: y[i] = sum_{j=0}^{i} kernel[i - j] * x[j]

vandermonde_matvec

Vandermonde kernel evaluation — S4D’s (diagonal-only) direct kernel-generation step. Given complex poles (lambda_i, the diagonalized state matrix’s eigenvalues) and complex weights (w_i, the per-pole residues, e.g. C_i * B_i), computes the SSM’s convolution kernel directly: kernel[t] = sum_i weights[i] * poles[i]^t, for t = 0..length