Irka

IRKA, or Iterative Rational Krylov Algorithm, is an algorithm for model order reduction of linear time-invariant (LTI) systems. It aims to produce a reduced-order model that closely approximates the original system in the H2 norm by enforcing interpolatory tangential conditions and iteratively adjusting interpolation points (shifts) and tangential directions.

For a full-order system with transfer function H(s) = C(sI − A)⁻¹B + D, IRKA starts with initial shifts and right/left tangential directions. It builds a rational Krylov subspace using A, B, C and the current shifts, then computes projection matrices V and W that define a Petrov-Galerkin projection. The reduced system is obtained as A_r = WᵀAV, B_r = WᵀB, C_r = CV, D_r = D. The reduced transfer function H_r(s) = C_r(sI − A_r)⁻¹B_r + D_r is formed, and the poles of H_r(s) (the eigenvalues of A_r) are used to update the shifts to σ_i := −λ_i(A_r). The tangential directions are updated from the current B and C so that the next iteration uses new interpolation directions. The process repeats until the shifts (and often the reduced model) converge.

IRKA is known for producing high-fidelity reduced models that are locally optimal in the H2 sense for many systems, scalable to large sparse matrices, and applicable to both single-input/single-output and multi-input/multi-output setups. Extensions exist for discrete-time systems, parametric or interval model reduction, and data-driven settings where system matrices are not fully available. As with many iterative optimization methods, convergence is local and depends on initialization.