1 / 29 Complexity of the Creative Telescoping for Bivariate Rational Functions Shaoshi Chen (joint work with Alin Bostan, Fr´ ed´ eric Chyzak & Ziming Li) Algorithms Project-Team, INRIA (France) KLMM, Chinese Academy of Sciences (China) February 24, 2010 Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
2 / 29 Symbolic integration From differential algebra to creative telescoping → Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
3 / 29 Outline Introduction Minimal telescopers Hermite reduction approach Almkvist and Zeilberger’s approach Non-minimal telescopers Implementation and Application Conclusion Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Introduction 4 / 29 Introduction Minimal telescopers Hermite reduction approach Almkvist and Zeilberger’s approach Non-minimal telescopers Implementation and Application Conclusion Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Introduction 5 / 29 Definite integration for special functions Definite Integration: Creative telescoping (CT): � b by − → L ( x , D x )( f ) = D y ( g ) F ( x ) = f ( x , y ) dy a L : telescoper g : certificate y → a g ( x , y ) = lim lim y → b g ( x , y ) = ⇒ L ( x , D x )( F ) = 0 . Example: An integral of a product of four Bessel functions [GlaMon1994] � + ∞ yJ 1 ( xy ) I 1 ( xy ) Y 0 ( y ) K 0 ( y ) dy = − ln(1 − x 4 ) 2 π x 2 0 L = x 3 ( x 4 − 1) D 4 x + · · · and g = poly. in Bessel functions Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Introduction 6 / 29 Previous works: General special functions � P ( x , y , D x )( f ) = 0 + Ini. cond. = D -finite data structure Q ( x , y , D y )( f ) = 0 Existence: If f is D -finite, then there exists ( L , g ) s.t. L ( f ) = D y ( g ). ◮ Holonomic D-modules: Bernstein (1971) ◮ Closure property of diagonal operation: Lipshitz (1988) Algorithms and implementations: ◮ Slow algo. for general holonomic inputs: Zeilberger (1990) ◮ Fast algo. for hyperexponential inputs: AlmkvistZeilberger (1990) ◮ Gr¨ obner-bases approach: Takayama (1992) ◮ Fast algo. for general holonomic inputs: Chyzak (1997) ◮ Non-holonomic generalization: Chyzak-Kauers-Salvy (2009) ◮ Mgfun (Chyzak1997, Pech2010), HolonomicFunctions (Koutschan2009) Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Introduction 7 / 29 Motivation for our work 1. No complexity analysis for CT algorithms yet 2. Algorithms for special functions are often slow in practice 3. Interesting applications of rational-function telescoping 3.1 Differential equations for algebraic functions [BCLSS2007] � yD y ( P ) � P ( x , α ) = 0 → L = D y ( g ) ⇒ L ( α ) = 0 P 3.2 Differential equations for diagonals [PemantleWilson2008] � f ( y , x / y ) � L = D y ( g ) ⇒ L (diag( f )) = 0 y Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Introduction 8 / 29 Our work: Bivariate rational functions Problem (CT for bivariate rational functions) f ∈ k ( x , y ) , construct ( L , g ) ∈ k ( x ) � D x � \ { 0 } × k ( x , y ) such that L ( x , D x )( f ) = D y ( g ) (Telescoping equation) Example: An integral of a bivariate rational function � + ∞ dy � xy � x + ( x 2 + 1) D x , − F ( x ) := x 2 + y 2 + 1 → ( L , g ) = x 2 + y 2 + 1 0 xF + ( x 2 + 1) D x ( F ) = 0 and F (0) = π π 2 → F ( x ) = √ . x 2 + 1 2 Focus: Compute a telescoper of minimal order (minimal telescoper). Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Introduction 9 / 29 Main results Theorem (Complexity for rational-function telescoping) CT for bivariate rational functions has polynomial complexity. f = P Q ∈ k ( x , y ) → L ( x , D x )( f ) = D y ( g ) d : The max. of total degrees of P and Q in x and y . ω : Any feasible exponent of matrix multiplication (2 ≤ ω ≤ 3). Method deg x ( L ) deg D x ( L ) deg x ( g ) deg y ( g ) Cost Expon. ˜ O ( d 3 ) O ( d 3 ) O ( d 2 ) O ( d ω +4 ) Minimal Hermite ≤ d 6.80 ˜ O ( d 3 ) O ( d 3 ) O ( d 2 ) O ( d 2 ω +3 ) Telescoper RatAZ ≤ d 8.61 O ( d 2 ) O ( d 2 ) O ( d 3 ) O ( d 3 ) O ( d 6 ω ) Non-mini. Lipshitz 16.8 O ( d 2 ) O ( d 2 ) O ( d 2 ) O ( d 4 ω ) Telescoper Cubic O ( d ) 11.2 (Complexity is in terms of arithmetic operations in k ) Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Introduction 10 / 29 Linear systems in different methods Non-linear problem: L ( f ) = D y ( g ) − → Linear problem: M · x = 0 Method System size Coeff. deg. Cost ˜ O ( id 2 ) O ( d ω +4 ) Minimal Hermite i × d ˜ O ( d 2 ω +3 ) Telescoper RatAZ id × id O ( id ) O ( d 6 ) × O ( d 6 ) O ( d 6 ω ) Non-mini. Lipshitz 0 O ( d 4 ) × O ( d 4 ) O ( d 4 ω ) Telescoper Cubic 0 (For mini. telescoper, costs take account of a loop over i = 1 , . . . , d ) Theorem (StorjohannVillard2005) Given M ∈ k [ x ] m × n ≤ d , its rank and a basis of its null space can be computed using ˜ ˜ O ( nmr ω − 2 d ) ops. O ( n ω d ) Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Minimal telescopers 11 / 29 Introduction Minimal telescopers Hermite reduction approach Almkvist and Zeilberger’s approach Non-minimal telescopers Implementation and Application Conclusion Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Minimal telescopers 12 / 29 Two approaches for constructing minimal telescopers Aim: Given f = P / Q ∈ k ( x , y ), find L := � ρ i =0 η i ( x ) D i x ∈ k [ x ] � D x � \ { 0 } and g ∈ k ( x , y ), s.t. L ( x , D x )( f ) = D y ( g ) and deg D x ( L ) is minimal . Almkvist and Zeilberger’s approach: g = Rf for R ∈ k ( x , y ) L − D y ( R ) ≡ 0 mod Ann( f ) � ODE in R parametrized by η i Hermite reduction approach: D i x ( f ) ≡ r i mod D y ( k ( x , y )) � Linear system in η i Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Minimal telescopers 13 / 29 Hermite reduction for indefinite integration Additive decomposition: For f ∈ K ( y ), decompose f into f = D y ( g ) + a b , deg y ( a ) < deg y ( b ) and b square-free. 1. Hermite (1872): Algorithm for computing ( g , a / b ) by GCD only! Key idea: Reduction of pole order for A / Q m with Q square-free � � Q m = sQ + tD y ( Q ) A = (1 − m ) s − D y ( t ) t + D y Q m (1 − m ) Q m − 1 (1 − m ) Q m − 1 2. Ostrogradsky (1845) and Horowitz (1971): Algorithm by linear solver. 3. Yun (1977): Complexity analysis of Hermite reduction (quasi-linear). Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Minimal telescopers 14 / 29 Bivariate Hermite reduction (BHR) Horowitz-Ostrogradsky’s method: Given f = P / Q ∈ k ( x , y ), � A P � + a Q ∗ := Q 1 · · · Q m and Q − := Q Q = D y Q ∗ , Q ∗ . Q − � A � H ∈ k [ x ] d y × d y H.O. System: P = H , ≤ d x a Output size: d x := max { deg x P , deg x Q } , d y := max { deg y P , deg y Q } Cramer’s rule → deg x ( A ) , deg x ( a ) ∈ O ( d x d y ) Eval-Interp algorithm: BHR with quasi-optimal complexity ˜ O ( d x d 2 y ) BHR = (Eval. f ( x 0 , y ) + UHR on f ( x 0 , y )) × O ( d x d y ) + Rat.interp. Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Minimal telescopers 15 / 29 Hermite reduction for creative telescoping Key idea: For f = P / Q ∈ k ( x , y ), d ∗ y := deg y ( Q ∗ ), → D y ( g i ) + a i HR y D i a i ∈ k ( x )[ y ] and deg y ( a i ) < d ∗ x ( f ) − − Q ∗ , y . Lemma: a 0 , a 1 , . . . , a d ∗ y are linearly dependent over k ( x ). Furthermore, � ρ ρ ρ � � � � η i ( x ) D i η i ( x ) a i = 0 ⇐ ⇒ x ( f ) = D y η i g i . i =0 i =0 i =0 Theorem 1. There exists a telescoper for f of order at most d ∗ y . 2. If � ρ i =0 η i a i = 0 for smallest ρ ∈ N , then � ρ i =0 η i D i x is a minimal telescoper for f with certificate � ρ i =0 η i g i . Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Minimal telescopers 16 / 29 Algorithm and complexity I Algorithm (HermiteTelescoping) 1. BHR: f = D y ( g 0 ) + a 0 / Q ∗ ; 2. For i from 1 to d ∗ y do 2.1 BHR: D i x ( f ) = D y ( g i ) + a i / Q ∗ ; 2.2 If � i j =0 η j a j = 0 for η j ∈ k ( x ) , not all zero, then return ( � i x , � i j =0 η j D j j =0 η j g j ) . Incremental strategy: ( g i , a i ) → ( g i +1 , a i +1 ) x ( f ) = D y ( g i ) + a i ( f ) = D y ( D x ( g i )) + D x ( a i ) − a i D x ( Q ∗ ) D i Q ∗ ⇒ D i +1 x Q ∗ 2 Q ∗ − a i D x ( Q ∗ ) g i +1 ) + ˜ a i +1 g i +1 ) + D x ( a i ) + ˜ a i +1 Q ∗ ⇒ D i +1 = D y (˜ ( f ) = D y ( D x ( g i ) + ˜ x Q ∗ 2 Q ∗ Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
Minimal telescopers 17 / 29 Algorithm and complexity II Theorem (Complexity for HermiteTelescoping) For f = P / Q ∈ k ( x , y ) of bidegree ( d x , d y ) , HermiteTelescoping computes ( L , g ) in ˜ O ( d x d ω +3 ) ops. y Degree bounds on g i and a i : deg y ( g i ) ≤ id y , and deg y ( a i ) ≤ d ∗ deg x ( g i ) , deg x ( a i ) ∈ O ( id x d y ) , y . Cost estimate for Step i ≥ 1: Hermite reduction + linear system solving x ( f ): ˜ 2.1 Hermite reduction on D i O ( i 2 d x d 2 y ); 2.2 Finding linear dependence of a i ’s: ˜ O ( i ω d x d 2 y ). i SV � − ˜ O ( i ω d x d 2 η j a j = 0 � deg x ∈ O ( id x d y ) ← − y ) . j =0 i × d ∗ y Shaoshi Chen Creative Telescoping for Bivariate Rational Functions
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