� CIRM, Luminy Sept. 28–Oct.2, 2009 Some Results on Integrable Algorithms X ING -B IAO H U ICMSEC, AMSS, Chinese Academy of Sciences P.O.Box 2719, China, 100080 hxb@lsec.cc.ac.cn This is joint work with Yi HE, Hon-Wah TAM and Satoshi Tsujimoto • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Outline � A brief introduction � A general form of sequence transformations and triangular recursion schemes • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Title:Some Results on Integrable Algorithms Keywords: Integrable algorithms • ” ➀ ✾➞ ✙❜✰❛✶❲ ”, in Y. Nakamura’s book: 2006 Fuctionality of Integrable Sys- tems, Kyoritsu Shupan Co., Tokyo, Japan (in Japanese). • Y. Nakamura, ”Why so accurate is an integrable SVC algorithm ?”, ê ♥ ✮ Û ï ➘ ↕ ▲ ➘ ❵ 1473 ë 2006 ❝ 41-61 Questions: • Q1: What is an integrable algorithm? • Q2: Why interesting? • Q3: How to find integrable algorithms? Q1: an algorithm − → a partial difference equation − → integrable equation − → an integrable algorithm • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Q1 − → • What is integrability for an equation? Definition of Integrability: • For a finite Hamiltonian system (2m variables), we say it is completely integrable if it admits m constants of the motion F i , i = 1 , · · · , m , which are independent and in invo- lution under Poisson bracket associated with the Hamiltonian structure and level surface defined by the intersection of surfaces F i = c i is compact and connected. • Infinite-dimensional case: No unified definition working definitions • Lax-integrable: We say an equation is integrable if it can be represented as a compati- bility condition of a pair of linear equations or a commutation relation of a pair of linear operators. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
KdV: u t + 6 uu x + u xxx = 0 ψ xx + uψ = λψ ψ t = u x ψ − (2 u + 4 λ ) ψ x ψ xxt = ψ txx = ⇒ u t + 6 uu x + u xxx = 0 • Symmetries and conservation laws • Bi-Hamiltonian structures • Painlev´ e property • B¨ acklund transformations • N-soliton solutions • C-integrable • · · · · · · • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Examples of integrable algorithms: • ε algorithm: ( ε ( n ) k +1 − ε ( n +1) k − 1 )( ε ( n +1) − ε ( n ) k ) = 1 k ε ( n ) ε ( n ) − 1 = 0 , = S n ( n = 0 , 1 , 2 , · · · ) 0 Discrete potential KdV • ρ algorithm: ( ρ ( n ) k +1 − ρ ( n +1) k − 1 )( ρ ( n +1) − ρ ( n ) k ) = k k ρ ( n ) ρ ( n ) = 0 , = S n ( n = 0 , 1 , 2 , · · · ) 0 1 Continuous limit: cylindrical KdV • η algorithm: 1 − 1 η ( n ) k +1 − η ( n +1) = k − 1 η ( n +1) η ( n ) k k η ( n ) η ( n ) = 0 , = ∆ S n − 1 ( n = 0 , 1 , 2 , · · · ) , S − 1 = 0 0 1 Discrete KdV • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Q2:Why are we interested in integrable algorithms? • They are attractive! Some famous algorithms are integrable ǫ -algorithm, η -algorithm, ρ algorithm, qd algorithm ...... • They have several significant applications For example, qd algorithm: solving matrix eigenvalue problems, algebraic equations, the BCH-Goppa decoding problem and a sorting problem... • They have nice properties and structures and numerical performance ( ε ( n ) k +1 − ε ( n +1) k − 1 )( ε ( n +1) − ε ( n ) k ) = 1 k ε ( n ) ε ( n ) − 1 = 0 , = S n ( n = 0 , 1 , 2 , · · · ) 0 Hankel determinant solution: � � S n S n +1 · · · S n + k � � � � S n +1 S n +2 · · · S n + k +1 � � . . . � . . . � . . . � � � � S n + k S n + k +1 · · · S n +2 k � � ε ( n ) 2 k = � � ∆ 2 S n ∆ 2 S n +1 · · · ∆ 2 S n + k − 1 � � � ∆ 2 S n +1 ∆ 2 S n +2 · · · ∆ 2 S n + k � � � . . . � . . . � . . . � � ∆ 2 S n + k − 1 ∆ 2 S n + k · · · ∆ 2 S n +2 k − 2 � � � � • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
� � ∆ 3 S n ∆ 3 S n +1 · · · ∆ 3 S n + k − 1 � � � ∆ 3 S n +1 ∆ 3 S n +2 · · · ∆ 3 S n + k � � � . . . � . . . � . . . � � � ∆ 3 S n + k − 1 ∆ 3 S n + k · · · ∆ 3 S n +2 k − 2 � � � ε ( n ) 2 k +1 = � � ∆ S n ∆ S n +1 · · · ∆ S n + k � � � � ∆ S n +1 ∆ S n +2 · · · ∆ S n + k +1 � � . . . � . . . � . . . � � � ∆ S n + k ∆ S n + k +1 · · · ∆ S n +2 k � � � • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Comments and Remarks on importance of this subject 1. R. Hirota et al, Mathematics and Computers in Simulation 37(1994)371-383 ”The study of difference scheme of integrable systems is currently the focus of intense activities.” 2. C. Brezinski ➜ Convergence acceleration during the 20th century ➜ Journal of Computa- tional and Applied Mathematics 122 (2000) 1-21 ”...In particular, the connection between convergence acceleration algorithms and contin- uous and discrete integrable systems brings a different and fresh look to both domains and could be of benefit to them....” 3. Moody T. Chu.,Linear algebra algorithms as dynamical systems, Acta Numerica (2008), pp. 1 õ 86 ”...it is truly remarkable that diverse topics, such as soliton theory, integrable systems, continuous fraction, τ -functions, orthogonal polynomials, the sylvester identity, moments, and Hankel determinants, can all play together, interwine, and eventually lead to the fact abstractly, but literally, that the eigenvalues and singular values of a given matrix can be expressed as the limit of some closed-form formulas!” • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
• Q3:How to find integrable algorithms? Integrable discretizations of integrable systems Hirota’s discretization Nonlinear Nonlinear Differential-difference Difference-difference == ⇒ equation equation � � � � � � Dependent Dependent variable variable transformation transformation � � � � � � Bilinear Bilinear == ⇒ Differential-difference Difference-difference equation equation • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
time-discretization of the Lotka-Volterra lattice • Step 1 Lotka-Volterra lattice u n,t = u n ( u n − 1 − u n +1 ) (1) u ( n ) = f ( n +2) f ( n − 1) f ( n ) f ( n +1) , Bilinear form 2 D n + e 2 D n − e 1 3 1 2 D n ] f n · f n = 0 [ D t e (2) e D n a n · b n = a n +1 b n − 1 . D t a · b = a t b − ab t , • Step 2 f m +1 f m n +1 − f m n f m +1 n +1 − δ [ f m n − 1 f m +1 n +2 − f m n f m +1 n +1 ] = 0 (3) n where t = mδ . when δ → 0 , (3) is reduced to Lotka-Volterra lattice (2). • Step 3 n − 1 f m +1 f m Dependent variable transformation u m n = n +1 , n +2 f m,n f m +1 Nonlinear form n = ˆ u m +1 n − 1 − u m +1 u m +1 − u m δ ( u m n u m n +1 ) (4) n n where ˆ δ δ = 1 − δ . • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Stage 1 ➭ Integrable algorithms − → Continuous integrable systems Stage 2 ➭ Fully discrete integrable systems ← → Integrable algorithms Stage 3 ➭ Integrable algorithms − → new continuous integrable systems Stage 4 ➭ Integrable discretizations − → Integrable nemerical algorithms Example 1 The discrete Lotka-Volterra system with variable step-size: singular value compu- tation (M. Iwasaki and Y. Nakamura Inverse Problems 20(2004)553-563) Example 2 The discrete relativistic Toda molecule equation: a Pad´ e approximation algorithm (Y. Minesaki and Y. Nakamura, Numerical Algorithms 27(2001)219-235) Example 3 The discrete mKdV equation: mKdV algorithm to solve a class of algebraic equa- tions (A. Mukaihira and Y. Nakamura Inverse Problems 16(2000)413-424) • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
A general form of sequence transformations and trian- gular recursion schemes Known results: (C. Brezinski, G. Walz, J. Comut. Appl. Math. 34 (1991) 361-383.) • sequence transformations of the form k T ( n ) � α ( n ) = k,i S n + i . (5) k i =0 • a triangular recursion scheme T ( n ) = a ( n ) k T ( n ) k − 1 + b ( n ) k T ( n +1) (6) k k − 1 • E-transformation • E-algorithm: a) Brezinski-Havie E-algorithm b)Ford-Sidi E-algorithm • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Our goals: 1)to generalize C. Brezinski and G. Walz’s results 2) to generalize Brezinski-Havie E-algorithm, Ford-Sidi E-algorithm and so on • a general form of sequence transformations of the form k T ( n ) � α ( n ) = k,i S n + kp + iJ , (7) k i =0 where J > 0 , p > 0 are two integers. • the recursion scheme T ( n ) = a ( n ) k T ( n + p ) + b ( n ) k T ( n + q ) (8) k − 1 k − 1 k where q is an integer and J = q − p . C. Brezinski, M. Redivo Zaglia ✺ Extrapolation Methods: Theory and Practice ✻ 1991 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
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