� On the Newton method for the matrix th root Bruno Iannazzo Dipartimento di Matematica Universit` a di Pisa On the Newton method for the matrix th root – p.1/18
✍ ✙ ✔ ✕ ✂ ✖ ✗ ✘ ✚ ✁ � ✛ ✖ ✔ ✕ ✂ ✞ ✏ ✢ ✆ ✁ ✆ ✂ ✄ ☎ ✠ ✁ ✂ ✣ ☛ ✤ ✝ ☞ ✝ ✠ ✜ Problem Given a matrix , with no nonpositive real eigenvalues, integer, find a solution of the equation with ✁✡✠ ✝✟✞ eigenvalues in the sector ✌✎✍ ✑✓✒ S 8 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Existence and uniqueness are guaranteed. We call this solution principal th root of . On the Newton method for the matrix th root – p.2/18
✧ ✛ ✥ ✬ ✰ ✮ ✛ ✂ ✙ ✬ ✙ ✥ ✂ ✫ ✪ ✧ ✩ ✠ ✂ ✛ ✂ ✛ ★ ✂ ✧ ✙ ✥ ✥ ✮ � Available methods Schur decomposition method [Björk & Hammarling ’83, Smith ’03] extremely good accuracy, high cost ✚✎✦ . For taking times square roots requires ✧✭✬✯✮ ✚✎✦ operations! A desirable cost is ✚✎✦ . Newton’s method advantage: low cost, per step, local ✚✎✦ quadratical convergence drawbacks: instability and lack of global convergence. Task Removing these drawbacks On the Newton method for the matrix th root – p.3/18
✂ ✁ ☎ ✞ ✶ ✚ ✠ ✣ ✷ ✸ ✁ ✆ ✆ ✝ ✶ ✂ ✠ ✁ � ✆ ✛ Newton’s Method Applying Newton’s method to the equation ✝✱✞ ☛✳✲ with an initial value which commutes with , one obtains ✆✵✴ ✣✺✹ ✆✵✶ which generalizes the scalar iteration. On the Newton method for the matrix th root – p.4/18
☛ ✁ ☎ ✢ ☛ ✌ ✠ ✛ ✚ ☎ ✾ ✲ ✼ ☎ ✜ ✻ � ✲ Instability of the Newton method Let us consider a well-conditioned problem. Compute the th root of a matrix having eigenvalues ✼✭✽ 5 10 0 10 Res −5 10 −10 10 −15 10 0 5 10 15 20 Iter On the Newton method for the matrix th root – p.5/18
✞ ✶ ✶ ✠ ✆ ✶ ✶ ✆ ❀ ❀ ✷ ❂ ✣ ✠ ❁ ✚ ❀ ✶ ✛ ✸ ❀ ✛ ✚❂ ✚ � ❁ ✆ ✶ ✷ ✣ ✠ ✿ ✆ ✆ ✶ ✛ ✛ ★ ✆ ✠ ✿ ✚ ✥ Nature of the instability The Newton iteration suffers from instability near the solution Def. An iteration is stable in a neighborhood of the solution if the error matrices satisfy where is linear and has bounded powers. Reason: lack of commutativity. On the Newton method for the matrix th root – p.6/18
● ❇ ❊ � ❉ ❈ ❇ ❆ ❄ ❋ ❃ ● ❊ ❈ ❉ ❆ ❉ ❆ ❇ ❈ ❄ ❋ ❃ ❊ Illustration of instability Numerically different iterations for the inverse of a matrix ❃❅❄ ●■❍ ❃❅❄ (unstable) ❃❅❄ ❃❅❄ (stable) ❃❏❄ ❃❏❄ ❃❏❄ ❃❏❄ 2 10 0 10 −2 10 −4 10 −6 10 −8 10 −10 10 −12 10 right left −14 10 interpose −16 10 0 2 4 6 8 10 12 14 16 18 20 Interposition makes iterations stable! It reduces effects of numerical noncommutativity. On the Newton method for the matrix th root – p.7/18
☎ ✚ ✸ ✝ ✛ ✶ ✞ ✂ ✠ ✆ ✣ ✷ ✶ ✆ ✂ � ✁ ✁ Question How to get rid of monolateral multiplication by in the Newton iteration? ✆✵✶ Solution: implicitly found on the known stable square root algorithms. On the Newton method for the matrix th root – p.8/18
✩ ✣ ✣ ✷ ✶ ❑ ✶ ▲ ✸ ✠ ❑ ✣ ✷ ▲ ✆ ✶ ✣ ✹ ✠ ✶ ✸ ✣ ✣ ✹ ✆ ✶ ▲ ✞ ✠ ✷ ✸ ✶ ▲ ✩ ✶ ✣ ✹ ✆ ❑ ✶ ✷ ✆ ✠ ✶ ❑ ✩ ✶ ✣ ✹ ✁ ✹ ✸ ✶ ✆ ✠ ✣ ✷ ✶ ✁ ✣ ✠ ✞ ✣ ✷ ✩ ◆ ◆ ✩ � ✶ ✆ ✣ ✹ ✆ ✁ ✠ ✶ ▲ ✶ ✶ Stable algorithms for the square root ✆▼✶ ✆▼✶ ✆✵✶ ✆✵✶ ✆✵✶ Matrix sign function Graeffe’s iteration [Denman-Beavers] [B. Meini] On the Newton method for the matrix th root – p.9/18
th root – p.10/18 ✸ ✶ ✣ ✹ ❑ ✂ ✶ ✣ ✹ ✆ ✶ ✹ ❑ ✛ ☎ ✞ ✂ ✚ ✠ ✣ ✷ ✝ ★ ❑ ✶ � ✛ ✂ ✙ ✮ ✬ ✚ ✥ ✂ ✣ ✚ ✹ ✆ ✸ ✶ ❑ ✛ ☎ ✞ ✂ ✆ ✶ ✂ ✶ ✣ ✹ ✁ ✠ ✶ ❑ ◆ ✂ ✝ ✝ ✆ ✁ ✸ ✶ ✆ ✠ ✣ ✷ ✶ ✶ ✆ ✹ ✠ ✹ ✛ ☎ ✞ ✂ ✣ ✚ ✣ ❑ ✷ ✶ ✆ ✣ ❖ ◆ ✶ ✸ matrix ops. On the Newton method for the matrix Generalization to The iteration can be implemented with ✣✺✹ ✆✵✶ P◗P❘P P❘P◗P❘❙
✂ � Stability Theorem . The iteration is stable in a neighborhood of the solution. Example . The iteration provides a stable algorithm for computing the matrix th root 5 10 0 10 Res −5 10 −10 10 −15 10 0 5 10 15 20 Iter On the Newton method for the matrix th root – p.11/18
☛ � ✠ ❚ ✛ ✍ ✁ ✚ ✿ Convergence “The problem is to determine the region of the plane, such that [initial point] being taken at pleasure anywhere within one region we arrive ultimately at the point [a solution]” Arthur Cayley , 1879 “J’espére appliquer cette théorie au cas d’une equation cubique, mais les calculs sont beaucoup plus difficiles” Arthur Cayley , 1890 “Donc, en general, la division du plan en régions, qui conduisent chacune à une racine déterminée de , sera un problème impraticable. Voilà la raison de l’échec de la tentative de Cayley” Gaston Julia , 1918 On the Newton method for the matrix th root – p.12/18
☎ ✶ ❳ ❱ ❲ ✸ ✶ ❱ ✛ ✶ ✞ ✂ ✚ ✠ ✣ ✷ ❱ ✝ ✂ P ❖ ❱ ✴ ❯ ✠ ✠ ☎ ❲ ✂ ✁ ✁ � ❨ Choice of the initial value The initial value must Commute with Converge to the principal th root A nice choice is ✆✵✴ The problem of the convergence can be reduced to the scalar iteration ✣✺✹ P❘❙ with eigenvalue of . On the Newton method for the matrix th root – p.13/18
❬ ❇ ❴ ❬ ✝ ✤ ✣ ❲ ✂ ❬ ❭ ❪ ❲ � ✝ ❩ ❇ Question Which is the set of for which the scalar iteration converges to the principal th root ? A set with fractal boundary. in blue color the set of complex numbers for which the sequence converges to the principal ❭❫❪ root in red color the ones that generate sequences converging to secondary roots and so on... On the Newton method for the matrix th root – p.14/18
✄ ✖ ✂ ✜ ☛ ❵ ✍ � ✲ ☎ ❛ ❩ ✍ ❛ ✲ ✑ ✏ ✍ ✌ ✠ ❵ ✝ Main theorem The set Re is such that for any . ❵❝❜ Therefore convergence occurs for any matrix having eigenvalues in . On the Newton method for the matrix th root – p.15/18
✤ ✣ ✩ ✛ ✝ ❡ ✤ ✆ ✠ ❡ ★ ❡ ✂ ❂ ❞ ❂ ★ ✤ ✝ ✠ ✂ ✆ ✕ ✚ ❡ ❞ ❞ � ✁ ✂ ❡ ✠ ❞ ✕ ❂ ❂ ❣ ★ ✝ ❡ ✤ ❢ ✣ ❂ ❞ ❂ ✂ ✂ The algorithm Compute , the principal square root of Normalize: so that the matrix has eigenvalues in the set of convergence By means of the iteration proposed: If is odd compute the th root of and set ✝❤❣ If is even compute the th root of and set Convergence is guaranteed for any matrix having a principal th root! On the Newton method for the matrix th root – p.16/18
� Further results High order rational iterations (König, Halley, Schröder). The behavior and techniques are similar. Some of them have very nice convergence regions (Halley’s method). Scaling to reduce number of steps. It is possible to provide a scaling to reduce the number of steps to a fixed value. Proving this is work in progress. On the Newton method for the matrix th root – p.17/18
✂ ✙ ✂ � ✛ ✥ Conclusions We have presented stable iterations for the th root ✧✭✬✯✮ Their cost is ✚✎✦ Convergence ensured whenever a solution exists Further developments Proving convergence for the Halley’s method and designing a rigorous scaling procedure On the Newton method for the matrix th root – p.18/18
Recommend
More recommend
