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Algorithm 1 : An Iterative Rational Krylov Algorithm (IRKA) for MIMO Systems

Input: Matrices A, B, and C; desired reduced system size r;

shift convergence tolerance ε_s

Output: Reduced order matrices A_r, B_r, C_r

1: Initialize A_r, B_r, C_r to be random matrices of appropriate reduced dimensions

2: Compute shifts σ_i «--Aj(A_r) for i = 1 : r

3: Construct V_r and W_r so that

Ran(K) = span{(σ√- A) ¹Bb₁,... ,(σ_rI — A) ¹Bb_r}
Ran(IK_r) = span{(σ₁/ — A^τ)~¹C^τc₁,..., (σ_rI — A^τ)~¹C^τc_r},

where

A_rx_i = x_iλ_l, yy A_r--= Xyy? and y[x_i = 1,
b₁ y₁B_r, C₁ C_rXi-
4: while (relative change in σ₂ > ε_s) do

5: Set A_r = (W_r^τV_r)~¹W_r^τAV_r, B_r = (W_r^τV_r)~¹W_r^τB, and C_r = CV_r

6.∙ σ_i <--A_z(A_r) for i = 1,..., r

7: Update V_r and W_r so that

Ran(U_r) = span{(σι∕- A)~¹Bb₁,... ,{σ_rI - A)~¹Bb_r}
Ran(W_r) = span{(σj/ — A^τ)~¹C^τc₁,..., (σ_rI — A^τ)~¹C^τc_r},

where

A_rx_i = x_i∖_i, yf A_r = ∖_iyl and yξx_i = 1,
b₁ yiB_r, c_l C_rXi

8: A_r = (W_r^τV_r)-¹W_r^τAV_r, B_r = (W_r^τV_r)-¹W_r^τB, and C_r - CV_r

9: end while

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