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5858
5959## Complexity Analysis
6060
61- | Case | Time | Space | Notes |
62- | ------------------------| --------------------| -----------------------------| ---------------------------------------------|
63- | $N{=}M{=}2$ | $O(64)$ flops | $O(NM)^2 = 16$ floats for K | 4×4 linear system |
64- | $N{=}M{=}3$ | $O(729)$ flops | 81 floats for K | 9×9 linear system |
65- | General $N{=}M$ | $O(N^6)$ flops | $O(N^4)$ words | Gaussian elimination on the $N^2 \times N^2$ Kronecker system |
66- | Alternative (Bartels–Stewart) | $O(N^3)$ | $O(N^2)$ | Requires Schur decomposition; preferred for $N \geq 6$ |
61+ | Case | Time | Space | Notes |
62+ | -------------------------------| ---------------- | -----------------------------| ------------------ ---------------------------------------------|
63+ | $N{=}M{=}2$ | $O(64)$ flops | $O(NM)^2 = 16$ floats for K | 4×4 linear system |
64+ | $N{=}M{=}3$ | $O(729)$ flops | 81 floats for K | 9×9 linear system |
65+ | General $N{=}M$ | $O(N^6)$ flops | $O(N^4)$ words | Gaussian elimination on the $N^2 \times N^2$ Kronecker system |
66+ | Alternative (Bartels–Stewart) | $O(N^3)$ | $O(N^2)$ | Requires Schur decomposition; preferred for $N \geq 6$ |
6767
6868For the embedded sizes ($N \leq 5$) targeted by this library the Kronecker approach is negligibly
6969more expensive than Bartels–Stewart and avoids the full Schur machinery.
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