QR-algorithms for eigenvalue computation of structured matrices. The case of low-rank corrections of unitary matrices L. Gemignani University of Pisa gemignan@dm.unipi.it http://www.dm.unipi.it/˜gemignan Cortona, 2004 – p.1/19
✜ ☎ ✝ ✕ ✓ � ✔ � ✝ ☞ ✑ ✂ ✔ ✁ ✕ ✓ � � ✖ ✔ � ✝ ✚ ☞ � ✡ � ✣ ☎ ✆ ✝ ✢ ✡ � ✞ ✂ ✞ ☛ ☞ ✆ ✌ ✏ ✑ ✒ ✒ ✒ ✑ ✓ The Companion Matrix . ... . ✍✎✍✎✍✎✍ ✛✎✛✎✛✎✛ . �✗✖ ✁✄✂ . ... . . ✞✠✟ ✝✙✘ Eigenvalues of == Zeros of is upper Hessenberg MATLAB uses the shifted eigenvalue algorithm for the computation of polynomial zeros MATLAB only exploits the Hessenberg structure Cortona, 2004 – p.2/19
✣ ✡ ✤ ✧ ✝ ☛ ☞ ✢ ✣ ☎ ☞ ✮ ✬ ✫ ☞ ✖ ✬ ✬ ☎ ✮ ☎ ✯ ✢ ✦ ✪ ✤ ✢ ✡ ✆ ☞ ✓ ✦ ✤ ✧ ✝ ✆ ✤ ✢ ✣ ✤ ★ ✖ ✩ ✆ ✣ ✤ The (Shifted) QR Algorithm ☞✥✤ ☞✥✤ Classical Theory: The Hessenberg structure is invariant under ( Hessenberg Hessenberg) flops for each step of applied to a ✁✄✭ Hessenberg matrix The MATLAB implementation generally requires ✁✄✭ flops and storage for computing all the zeros ✁✄✭ Cortona, 2004 – p.3/19
✢ ✣ ✬ ☎ ✬ ☎ The Research Problem To design a eigenvalue algorithm for companion matrices which requires flops and storage per iteration ✁✄✭ ✁✄✭ Recent Contributions: 1. [ Bini, Daddi, G. ]. (To appear in ETNA) FAST but UNSTABLE 2. [ Chandrasekaran, Gu ]. (Talk at the Workshop in Banff, November 2003) www.pims.math.ca/birs/workshops/2003/03w5008 FAST but (probably) UNSTABLE The main goal: To achieve both efficiency and stability Cortona, 2004 – p.4/19
✕ ✳ ✚ ✑ ☞ ✑ ✳ ✝ ✆ ✑ ✴ ✑ ✲ ✝ ✜ ✴ ✕ ✕ ✑ ✒ ✒ ✒ ✑ ✏ ☞ ✌ ✆ ✲ The Novel Approach . ... . ✍✎✍✎✍✎✍ ✛✎✛✎✛✎✛ . ✪✱✰ ✪✱✰ . ... . . is a very special unitary Hessenberg matrix Exploit the representation of as a unitary Hessenberg plus a rank-one matrix Unitary Hessenberg matrices belong to the class of rank-structured matrices Cortona, 2004 – p.5/19
❊ ❅ ❈ ❈ ❈ ❈ ✿ ❀ ❁ ❆ ❅ ❉ ❉ ❉ ❅ ❁ ❈ ❉ ❉ ❅ ✿ ❀ ❉ ❇ ❈ ❈ ❈ ❈ ❈ ❈ ✿ ❈ ❊ ❆ ✆ ✵ ❏ ✵ ✆ ✢ ✣ ❊ ✢ ✣ ❁ ✣ ❅ ❁ ✣ ✧ ❅ ✝ ❀ ✵ ✆ ✢ ✿ ✹ ❅ ✿ ❆ ❀ ✿ ❇ ❆ ❀ Unitary Hessenberg Matrices 1. Let be unitary Hessenberg and its factorization, where has nonnegative diagonal entries 2. We have and, therefore, ❆❋❊✗● ❀❂❁❄❃ ❀❂❁❄❃ ❁❄❅ ❁❄❇ ❁❄❃ ✺✻✺✼✺✽✺✻✺✼✺✽✺✻✺✼✺✽✾ ❑✻❑✼❑✽❑✻❑✼❑✽❑✻❑✼❑✽▲ ❀✱❆ ❆■❊✗● ❁❍❅ ❁❍❇ ❁❍❅ . ... . ❀✱❆ ✶✸✷ . . ... ... . . ❊✗● ❊✗● [ Gill, Golub, Murray, Saunders , 1974] Cortona, 2004 – p.6/19
❘ ❙ ✆ ◆ ❖ P ✖ ◗ ▼ ◗ ✵ ✖ ✁ ✁ ✵ ✕ ✵ ✩ ❚ ☛ ❚ ✪ ✕ ✩ ✭ ❯ ☎ ▼ ✁ ✵ ☎ ✁ ✕ ✕ ✁ ✵ ☎ ✆ ◆ ❖ P ✖ ◗ ❘ ◗ ✵ ✖ ✁ ❱ ✵ ❙ ❚ ✪ ✕ ✩ ✭ ☛ ✕ ✩ ❚ ❯ ☎ ❱ The Rank Structure rank ✝✙✘ rank ✝✙✘ ☎❳❲ and ☎❳❲ for unitary Hessenberg Depending on some small differences in the definition and in the representation, we may have: 1. Quasiseparable matrices [ Eidelman, Gohberg ] 2. Matrices with low Hankel rank [ Dewilde, Van der Veen ] 3. Weakly semiseparable matrices [ Tyrtyshnikov ] 4. Sequentially/Hierarchically semiseparable matrices [ Chandrasekaran, Gu ] Cortona, 2004 – p.7/19
✤ ☞ ✇ ❡ ❵ ❴ ✪ ✤ ✴ ✆ s ✤ ❡ t ❫ ❜ ❛ ❵ ✞ ❛ ❵ s ❵ ❜ ✴ ♠ r t ❘ ✆ q ❥ ♠ ♣ ❣ ♦ ❫ ✓ ❦ ❥ ❖ ✐ ❤ ❢❣ ✤ ✴ ☛ ✡ ✆ ✤ ✆ ✕ ❲ ☎ ✓ ✁ ▼ ❚ ✡ ☞ ☞ ❱ ⑧ ⑨ ✣ ✢ ✴ ✤ ❩ ✤ ☞ ❨ ⑦ ✁ ✡ ✤ ☞ ✤ ✤ ❫ ✢ ✖ ✘ ✤ ☞ ✢ ❥ ✆ ✤ ☞ t ❪ ❘ ☛ ❭ ❲ ☎ s Companion Matrices Under QR Theorem 1 Let be the sequence of matrices generated by the shifted algorithm applied to the companion matrix . Then and for ☞❬✤ ☞❬✤ all Proof: From ❜❞❝ and , one ✪✱❴ gets ❜❞❝ ❧♥♠♦ Since is Hessenberg, it also holds ☞✥✤ ④⑥⑤ ①③② ✞✈✉ To conclude: Characterize the rank structure of such a matrix Cortona, 2004 – p.8/19
✭ ❥ ✮ ④ ❷ ❶ ❘ ❹ ⑤ ⑨ ✓ ⑦ ⑨ ✭ ❲ ❚ ❲ ⑨ ✪ ✮ ❷ ❶ ❘ ☛ ❲ ✕ ✞ ❘ ❹ ❻ ❻ ❻ ✖ ★ ✞ ❹ ✳ ❚ ⑩ ✆ ❘ ❸ r ❘ ✕ ✓ ✭ ❲ ❸ ✕ ✖ ✁ ❱ ✬ ☎ ⑨ ✓ ☛ ❫ ❝ ❴ ☎ ✴ ✆ ☎ ⑨ ✓ ☛ ✴ ✁ ✴ ✢ ✴ ✁ ✢ ✓ ❚ ✭ ❲ ✕ ✓ ✭ ❲ ❚ ❲ ✕ ✮ ⑨ ❷ ❶ ❘ ⑩ ❲ ✕ ✓ ✭ ✭ ✴ ✘ Unitary Rank-Structured Matrices Theorem 2 Let be a unitary matrix such that tril tril . Then, . In particular, ☎❳❲ there exist vectors , , vectors , and lower triangular matrices , , such that ✮❂❺ ✞✈✉ Proof: Compute the QR factorization of . The unitary factor is the product of ✁✄✭ rotations. The rank structure of coincides with the rank structure of Cortona, 2004 – p.9/19
❵ ❥ ✤ t ❘ ☛ ④ ⑤ ⑦ ⑤ ✪ ✕ ❲ ❚ ❲ ✭ s ✞ � ❸ s ✤ t ❘ ✘ ✖ ❼ t ❡ ❜ ❾ ✪ ✇ s ✤ ➀ s ❻ ✖ � s ✤ t ✞ ★ ✉ ✭ ✞ ✆ ➁ s ✤ t ➀ ❲ ✤ s t ✞ ✉ ✞ ✆ � ✤ ⑦ t ✞ ☛ ④ ❥ ✕ ❲ ❹ ❻ ☛ ✕ ❶ ❷ ✮ ✭ ✭ ✓ ❲ ❜ ❸ ❵ ❡ ❜ ❾ ❶ ❾ ❡ ✮ ☞ ✭ ✕ ✕ ➂ ☞ ✆ ✡ ❵ ➀ ✕ ✭ ✓ ✕ ✓ ⑩ ❷ ⑦ ❻ ❿ ❘ ✆ ⑩ ❵ ❡ ❜ ✳ ✞ ❹ s ✤ t ✞ ★ ✖ ✉ t ✭ t ✓ ⑨ ❲ ❹ s ✤ ❘ ✤ ❶ ❷ ✕ ✮ ❥ � s ✞ A Representation for Theorem 3 For each matrix generated by shifted QR ☞❽✤ algorithm applied to the companion matrix , there exist vectors , vectors and lower triangular matrices such that ✮❂❺ ①③② ④⑥⑤ Each iterate can be represented by means of ☞✥✤ parameters. Cortona, 2004 – p.10/19
Cortona, 2004 – p.11/19 ☛ ❡ ➏ ➐ ❜ ☛ ❴ ❵ ❡ ➏ ➐ ❜ ❝ ➄ ❵ ❡ ➏ ➐ ❜ ☎ ★ ✖ ➃ ✁ ❨ ❵ ☛ ❵ ❹ ❨ ❸ ❵ ❡ ➏ ➐ ❜ ❿ ❩ ☛ ❨ s ❜ ✤ ★ ✖ t ✞ ❩ ☛ ✰ ❵ ❡ ➏ ➐ ⑩ ❡ ❿ ❵ ➐ ❜ ☛ ➄ ❵ ❡ ➏ ➐ ❜ ☛ ❴ ❡ ❡ ➏ ➐ ❜ ☛ ❝ ❵ ❡ ➏ ➐ ❜ ☎ ➏ ❵ ➏ ❜ ➐ ❜ ❿ ❩ ☛ ❨ ❸ ❵ ❡ ➏ ➐ ❿ ✰ ❩ ☛ ❨ ❹ s ✤ ★ ✖ t ✞ ❩ ☛ ☛ ❩ ❜ ❵ ☛ ➄ ❵ ❡ ☛ ❴ ❵ ❡ ❜ ☛ ❝ ❡ ❡ ❜ ☎ ✆ ☞ ✤ ✁ ❨ ⑩ ❵ ❡ ❜ ❿ ❜ ❵ ➐ ❵ ➃ ✁ ❨ ⑩ ❵ ❡ ❜ ❿ ❩ ☛ ❨ ❸ ❡ ✰ ❜ ❿ ❩ ☛ ❨ ❹ s ✤ t ✞ ❩ ☛ ❩ ☛ ❨ ➇ ❵ ❡ ❜ ☛ ❝ ❵ ❡ ❜ ☛ ➅ ➆ ➈ ❸ ➉ ➊ ✦ ✤ ☎ ✁ ❨ ⑩ ❵ ❡ ➏ ❴ ☛ ❜ ✤ ❵ ❡ ❜ ❿ ❩ ☛ ❨ ❡ s ❹ t ❡ ❵ ➄ ☛ ✞ ❜ ❵ ✰ ☛ ❩ The Structured QR Iteration ➋➌➎➍ The structured QR iteration takes in input is equal to and returns in output ☞✥✤ such that Let
Cortona, 2004 – p.12/19 ✤ ✤ ✵ ✤ ✢ ✤ ☞ ✤ ★ ✖ ✩ ✆ ✣ ✤ ✢ ✪ ✢ ✆ ❡ ❵ ❴ ✓ ✖ ★ ✖ ✦ ★ ✤ ✵ ✝ ✧ ✤ ❫ ✆ ➐ ✓ ❡ ❜ ☛ ❵ ❡ ❜ ☎ ✆ ✵ ✤ ✢ ✤ ☞ ✤ ✦ ✖ ✵ ★ ✤ ✵ ✖ ★ ✤ ✤ ✤ ✣ ✤ ✢ ✆ ✝ ✧ ➏ ❵ ❴ ➐ ✞ ❩ ☛ ✰ ❵ ❡ ➏ ➐ ❜ ☛ ➄ ❵ ❡ ➏ ❜ ✖ ❝ ☎ ❜ ➐ ➏ ❡ ❵ ☛ ☛ ❜ ➐ ➏ ❡ ❵ ❴ t ★ ❡ ✖ ➏ ❡ ❵ ⑩ ❨ ✁ ★ ❜ ✤ ✵ ❫ ❜ ➐ ➏ ➐ ❿ ✤ ❜ s ❹ ❨ ☛ ❩ ❿ ➐ ❩ ➏ ❡ ❵ ❸ ❨ ☛ ❵ ❝ ☛ ➅ ✦ ➊ ➉ ➈ ➇ ➆ ☛ ☎ ❜ ❡ ❵ ❝ ☛ ❜ ✤ ➃ ❵ ❨ ❩ ❿ ❜ ❡ ❵ ❸ ☛ ✁ ❩ ❿ ❜ ❡ ❵ ⑩ ❨ ❡ ❴ ❨ ❩ ❜ ❡ ❵ ❸ ❨ ☛ ❿ ❩ ❜ ❡ ❵ ⑩ ❨ ✁ ❿ ☛ ☛ ❵ ❡ ❵ ➄ ☛ ❜ ❡ ✰ ❨ ☛ ❩ ✞ t ✤ s ❹ t ☛ ❹ ☛ ❿ ❜ ❡ ❵ ❸ ❨ ❩ ☛ ❿ ❜ ❡ ❵ ⑩ s ❩ ❨ ➑ ❵ ✤ s ➄ ☛ ❜ ❡ ✰ ❹ ☛ ❩ ✞ t ✤ s ✁ ❨ ✤ ☛ ✤ t ✞ ❩ ☛ ✰ ❵ ❡ ❜ ☛ ➄ ☞ ❡ ❜ ❵ ❴ ❵ ❡ ❜ ✆ ☎ ❜ ☛ ❝ ❡ ❵ A Fast and Stable Structured Iteration 3. (Compression Phase) Compute a QR factorization of ➋➌➎➍ ❜❞➒ ☞✥✤ : 2. (Expansion Phase) Compute , such that 1. Compute OUTPUT: INPUT: (a) (b)
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