Algorithmic Arithmetics with DD-Finite Functions Implementation and - PowerPoint PPT Presentation

D-finite functions DD-finite functions Implementation Conclusions Algorithmic Arithmetics with DD-Finite Functions Implementation and Issues Antonio Jiménez-Pastor, Veronika Pillwein ISSAC (July 2018) Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Outline 1 D-finite functions 2 DD-finite functions 3 Implementation of closure properties 4 Conclusions Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Notation Throughout this talk we consider: K : a computable field K [[ x ]]: ring of formal power series over K . Given a field F : V F ( f ) = � f , f ′ , f ′′ , ... � F . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions D-finite functions Definition Let f ∈ K [[ x ]]. We say that f is D-finite (or holonomic ) if there exist d ∈ N and polynomials p 0 ( x ) , ..., p d ( x ) such that: p d ( x ) f ( d ) ( x ) + ... + p 0 ( x ) f ( x ) = 0 . We say that d is the order of f . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Non-D-finite examples There are power series that are not D-finite: Double exponential: f ( x ) = e e x . Tangent: tan( x ) = sin( x ) cos( x ) . Gamma function: f ( x ) = Γ( x + 1). n ≥ 0 p ( n ) x n . Partition Generating Function: f ( x ) = � Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions DD-finite Functions Definition Let f ∈ K [[ x ]]. We say that f is D-finite if there exist d ∈ N and polynomials p 0 ( x ) , ..., p d ( x ) such that: p d ( x ) f ( d ) ( x ) + ... + p 0 ( x ) f ( x ) = 0 . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions DD-finite Functions Definition Let f ∈ K [[ x ]]. We say that f is DD-finite if there exist d ∈ N and D-finite elements r 0 ( x ) , ..., r d ( x ) such that: r d ( x ) f ( d ) ( x ) + ... + r 0 ( x ) f ( x ) = 0 . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Examples The set is bigger than the D-finite functions: f is D-finite ⇒ f is DD-finite , f ( x ) = e e x f ′ ( x ) − e x f ( x ) = 0 , ⇒ cos( x ) 2 f ′′ ( x ) − 2 f ( x ) = 0 , f ( x ) = tan( x ) ⇒ � x 0 J n ( t ) dt f ′ ( x ) − J n ( x ) f ( x ) = 0 f ( x ) = e ⇒ Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Differentially Definable Functions Definition Let f ∈ K [[ x ]]. We say that f is DD-finite if there exist d ∈ N and D-finite elements r 0 ( x ) , ..., r d ( x ) such that: r d ( x ) f ( d ) ( x ) + ... + r 0 ( x ) f ( x ) = 0 . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Differentially Definable Functions Definition Let f ∈ K [[ x ]] and R ⊂ K [[ x ]] a ring. We say that f is differentially definable over R if there exist d ∈ N and elements in R r 0 ( x ) , ..., r d ( x ) such that: r d ( x ) f ( d ) ( x ) + ... + r 0 ( x ) f ( x ) = 0 . D( R ): differentially definable functions over R . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Characterization Theorem The following are equivalent: f ( x ) ∈ D( R ). There are elements r 0 ( x ) , ..., r d ( x ) ∈ R and g ( x ) ∈ D( R ) such: r d ( x ) f ( d ) ( x ) + ... + r 0 ( x ) f ( x ) = g ( x ) . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Characterization Theorem The following are equivalent: f ( x ) ∈ D( R ). There are elements r 0 ( x ) , ..., r d ( x ) ∈ R and g ( x ) ∈ D( R ) such: r d ( x ) f ( d ) ( x ) + ... + r 0 ( x ) f ( x ) = g ( x ) . Let F be the field of fractions of R : dim( V F ( f )) < ∞ Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Closure properties f ( x ) , g ( x ) ∈ D( R ) of order d 1 , d 2 . F the field of fractions of R . a ( x ) algebraic over F of degree p . Is in D( R ) Property Order bound Addition ( f + g ) d 1 + d 2 Product ( fg ) d 1 d 2 f ′ Differentiation d 1 � f d 1 + 1 Integration a ( x ) Be Algebraic p Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Closure properties f ( x ) , g ( x ) ∈ D( R ) of order d 1 , d 2 . F the field of fractions of R . a ( x ) algebraic over F of degree p . Is in D( R ) Property Order bound Addition ( f + g ) d 1 + d 2 Product ( fg ) d 1 d 2 f ′ Differentiation d 1 � f d 1 + 1 Integration a ( x ) Be Algebraic p − → Proof by direct formula − → Proof by linear algebra Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Vector spaces Let R ⊂ K [[ x ]], F its field of fractions and V F ( f ) the F -vector space spanned by f and its derivatives. The Characterization theorem provides f ( x ) ∈ D( R ) ⇔ dim( V F ( f )) < ∞ Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions The ansatz method Specifications Input: A power series h ( x ) ( f ( x ) + g ( x ), f ( x ) g ( x ) or a ( x )) Output: An operator A ∈ R [ ∂ ] such that A h = 0 Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions The ansatz method Specifications Input: A power series h ( x ) ( f ( x ) + g ( x ), f ( x ) g ( x ) or a ( x )) Output: An operator A ∈ R [ ∂ ] such that A h = 0 Method 1 Compute W ⊂ K [[ x ]] such that dim( W ) < ∞ and V F ( h ) ⊂ W . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions The ansatz method Specifications Input: A power series h ( x ) ( f ( x ) + g ( x ), f ( x ) g ( x ) or a ( x )) Output: An operator A ∈ R [ ∂ ] such that A h = 0 Method 1 Compute W ⊂ K [[ x ]] such that dim( W ) < ∞ and V F ( h ) ⊂ W . 2 Compute generators Φ = { φ 1 , ..., φ n } of W . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions The ansatz method Specifications Input: A power series h ( x ) ( f ( x ) + g ( x ), f ( x ) g ( x ) or a ( x )) Output: An operator A ∈ R [ ∂ ] such that A h = 0 Method 1 Compute W ⊂ K [[ x ]] such that dim( W ) < ∞ and V F ( h ) ⊂ W . 2 Compute generators Φ = { φ 1 , ..., φ n } of W . 3 For i = 0 , ..., n , compute vectors v i ∈ F n such that: h ( i ) ( x ) = � n j =0 v ij φ j . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions The ansatz method Specifications Input: A power series h ( x ) ( f ( x ) + g ( x ), f ( x ) g ( x ) or a ( x )) Output: An operator A ∈ R [ ∂ ] such that A h = 0 Method 4 Set up the ansatz: α 0 h ( x ) + ... + α n h ( n ) = 0 . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions The ansatz method Specifications Input: A power series h ( x ) ( f ( x ) + g ( x ), f ( x ) g ( x ) or a ( x )) Output: An operator A ∈ R [ ∂ ] such that A h = 0 Method 4 Set up the ansatz: α 0 h ( x ) + ... + α n h ( n ) = 0 . 5 Solve the induced F -linear system for the variables α k . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions The ansatz method Specifications Input: A power series h ( x ) ( f ( x ) + g ( x ), f ( x ) g ( x ) or a ( x )) Output: An operator A ∈ R [ ∂ ] such that A h = 0 Method 4 Set up the ansatz: α 0 h ( x ) + ... + α n h ( n ) = 0 . 5 Solve the induced F -linear system for the variables α k . 6 Return A = α n ∂ n + ... + α 1 ∂ + α 0 . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions The ansatz method Specifications Input: A power series h ( x ) ( f ( x ) + g ( x ), f ( x ) g ( x ) or a ( x )) Output: An operator A ∈ R [ ∂ ] such that A h = 0 Method 1 Compute W ⊂ K [[ x ]] such that dim( W ) < ∞ and V F ( h ) ⊂ W . 2 Compute generators Φ = { φ 1 , ..., φ n } of W . 3 For i = 0 , ..., n , compute vectors v i ∈ F n such that: h ( i ) ( x ) = � n j =0 v ij φ j . Algorithmic Arithmetics with DD-Finite Functions

D-finite functions DD-finite functions Implementation Conclusions Derivation matrices Let V be an F -vector space with derivation ∂ and Φ be n generators of V . Derivation matrix M ∈ F n × n is a derivation matrix w.r.t Φ if α ′ α 1 α 1       1 . . . . . .  = M  + ∂       . . .     α ′ α n α n n Algorithmic Arithmetics with DD-Finite Functions