De part en retraite de Pierre

Transcription

1 Introduction Combinatorics Generating Series Oracle Conclusion De part en retraite de Pierre Villetaneuse, fe vrier / 41 De part en retraite de Pierre

2 Introduction Combinatorics Generating Series Oracle Conclusion My only paper with Pierre Pierre Nicode me,, and Philippe Flajolet Motif statistics Theoretical Computer Science 287 (2002), no. 2, / 41 De part en retraite de Pierre

3 Introduction Combinatorics Generating Series Oracle Conclusion My only paper with Pierre Pierre Nicode me,, and Philippe Flajolet Motif statistics Theoretical Computer Science 287 (2002), no. 2, will be mentioned by Julien this afternoon. 2 / 41 De part en retraite de Pierre

4 Newton Iteration in Computer Algebra and Combinatorics Inria AriC Project, LIP ENS Lyon Joint work with Carine Pivoteau and Michèle Soria, Journal of Combinatorial Theory, Series A 119 (2012), Villetaneuse, Pierre s retirement, Feb / 41

5 I Introduction 4 / 41

6 Motivation: Random Generation Random generation of large objects = simulation in the discrete world. It helps evaluate the order of magnitude of quantities of interest; differentiate exceptional values from statistically expected ones; compare models; test software. 5 / 41

7 Framework: Constructible Species A small set of species 1, Z,, +, Seq, Set, Cyc, cardinality constraints that are finite unions of intervals, used recursively. Examples: Regular languages Unambiguous context-free languages Trees (B = Z + Z B 2, T = Z Set(T )) Mappings,... 6 / 41

8 Framework: Constructible Species A small set of species 1, Z,, +, Seq, Set, Cyc, cardinality constraints that are finite unions of intervals, used recursively (when it makes sense). Examples: Regular languages Unambiguous context-free languages Trees (B = Z + Z B 2, T = Z Set(T )) Mappings,... 6 / 41

9 Framework: Constructible Species A small set of species 1, Z,, +, Seq, Set, Cyc, cardinality constraints that are finite unions of intervals, used recursively (when it makes sense). Examples: Regular languages Unambiguous context-free languages Trees (B = Z + Z B 2, T = Z Set(T )) Mappings,... Two related problems: 1 Enumeration: number of objects of size n for n = 0, 1, 2, Random generation: all objects of size n with the same proba. Two contexts: labelled/unlabelled. 6 / 41

10 Recursive Method B = Z + Z B 2 DrawBinTree(n) = { If n = 1 return Z Else { U := Uniform([0, 1]); k := 0; S := 0; while (S < U){k := k + 1; S := S + b k b n k 1 /b n ; } return Z DrawBinTree(n k 1) DrawBinTree(k)}} [Nijenhuis and Wilf; Flajolet, Zimmermann, Van Cutsem] 7 / 41

11 Recursive Method B = Z + Z B 2 DrawBinTree(n) = { If n = 1 return Z Else { U := Uniform([0, 1]); k := 0; S := 0; while (S < U){k := k + 1; S := S + b k b n k 1 /b n ; } return Z DrawBinTree(n k 1) DrawBinTree(k)}} Generalizes to all constructible species. Requires b 0,..., b n. [Nijenhuis and Wilf; Flajolet, Zimmermann, Van Cutsem] 7 / 41

12 Boltzmann Samplers Principle (Duchon, Flajolet, Louchard, Schaeffer 2004) Generate each t T with probability x t /T (x), where: x > 0 fixed; T (z) := t T z t = generating series of T ; t = size. Same size, same probability Expected size xt (x)/t (x) increases with x. Complexity linear in t when the values T (x) are available. 8 / 41

13 Boltzmann Samplers Principle (Duchon, Flajolet, Louchard, Schaeffer 2004) Generate each t T with probability x t /T (x), where: x > 0 fixed; T (z) := t T z t = generating series of T ; t = size. Singleton Easy. Same size, same probability Expected size xt (x)/t (x) increases with x. Complexity linear in t when the values T (x) are available. 8 / 41

14 Boltzmann Samplers Principle (Duchon, Flajolet, Louchard, Schaeffer 2004) Generate each t T with probability x t /T (x), where: x > 0 fixed; T (z) := t T z t = generating series of T ; t = size. Singleton Easy. Same size, same probability Expected size xt (x)/t (x) increases with x. Cartesian Product C = A B Generate a A; b B; Return (a, b). Proof. C(x) = (a,b) x a + b = A(x)B(x); x a + b C(x) = x a A(x) B(x). Complexity linear in t when the values T (x) are available. x b 8 / 41

15 Boltzmann Samplers Principle (Duchon, Flajolet, Louchard, Schaeffer 2004) Generate each t T with probability x t /T (x), where: x > 0 fixed; T (z) := t T z t = generating series of T ; t = size. Singleton Easy. Same size, same probability Expected size xt (x)/t (x) increases with x. Cartesian Product C = A B Generate a A; b B; Return (a, b). Disjoint Union C = A B Proof. Draw b = Bernoulli(A(x)/C(x)); If b = 1 then generate a A else generate b B. x a C(x) = x a A(x) A(x) C(x). Complexity linear in t when the values T (x) are available. 8 / 41

16 Boltzmann Samplers Principle (Duchon, Flajolet, Louchard, Schaeffer 2004) Generate each t T with probability x t /T (x), where: x > 0 fixed; T (z) := t T z t = generating series of T ; t = size. Singleton Easy. Same size, same probability Expected size xt (x)/t (x) increases with x. Cartesian Product C = A B Generate a A; b B; Return (a, b). Disjoint Union C = A B Draw b = Bernoulli(A(x)/C(x)); If b = 1 then generate a A else generate b B. Use recursively (e.g., binary trees B = Z Z B B) Also: sets, cycles,... ; Complexity linear in t when the values T (x) are available. 8 / 41

17 Boltzmann Samplers Principle (Duchon, Flajolet, Louchard, Schaeffer 2004) Generate each t T with probability x t /T (x)/ t!, where: x > 0 fixed; T (z) := t T z t / t! = generating series of T ; t = size. Singleton Easy. Same size, same probability Expected size xt (x)/t (x) increases with x. Cartesian Product C = A B Generate a A; b B; Return (a, b). Disjoint Union C = A B Draw b = Bernoulli(A(x)/C(x)); If b = 1 then generate a A else generate b B. Use recursively (e.g., binary trees B = Z Z B B) Also: sets, cycles,... ; labelled case Complexity linear in t when the values T (x) are available. 8 / 41

18 Oracle: Large Systems that are Interesting to Solve The generating series are given by systems of equations. We need: only one solution; the right one; only numerically. In the worst case, these requirements would make no difference. But these systems inherit structure from combinatorics. 9 / 41

19 Results (1/2): Fast Enumeration Theorem (Enumeration in Quasi-Optimal Complexity) First N coefficients of gfs of constructible species in 1 arithmetic complexity: O(N log N) (both ogf and egf); 2 binary complexity: O(N 2 log 2 N log log N) (ogf); O(N 2 log 3 N log log N) (egf). 10 / 41

20 Results (2/2): Oracle 1 The egfs and the ogfs of constructible species are convergent in the neighborhood of 0; 2 A numerical iteration converging to Y(α) in the labelled case (inside the disk); 3 A numerical iteration converging to the sequence Y(α), Y(α 2 ), Y(α 3 ),... for in the unlabelled case (inside the disk). 11 / 41

21 Examples (I): Polynomial Systems [Darrasse 2008] Random generation following given XML grammars Grammar nb eqs max deg nb sols oracle (s.) FGb (s.) rss PNML xslt relaxng xhtml-basic mathml xhtml xhtml-strict xmlschema SVG >1.5Go docbook >1.5Go OpenDoc / 41

22 Introduction Combinatorics Generating Series Oracle Conclusion Example (II): A Non-Polynomial System Unlabelled rooted trees: 1 1 f (x) = x exp(f (x) + f (x 2 ) + f (x 3 ) + ) / 41

23 II Combinatorics 14 / 41

24 Mini-Introduction to Species Species F: Examples: 0, Z, 1; Set; Seq, Cyc. 15 / 41

25 Mini-Introduction to Species Species F: Examples: 0, Z, 1; Set; Seq, Cyc. Composition F G: 15 / 41

26 Mini-Introduction to Species Species F: Examples: 0, Z, 1; Set; Seq, Cyc. Composition F G: Y = H(Z, Y) 15 / 41

27 Derivative Introduction Combinatorics Generating Series Oracle Conclusion H : 16 / 41

28 Derivative Introduction Combinatorics Generating Series Oracle Conclusion H : (F G) : 16 / 41

29 Derivative Introduction Combinatorics Generating Series Oracle Conclusion H : (F G) : Huet s zipper 16 / 41

30 Derivative Introduction Combinatorics Generating Series Oracle Conclusion H : (F G) : species derivative A + B A + B A B A B + A B Seq(B) Seq(B) B Seq(B) Cyc(B) Seq(B) B Set(B) Set(B) B 16 / 41

31 Derivative Introduction Combinatorics Generating Series Oracle Conclusion H : (F G) : species derivative A + B A + B A B A B + A B Seq(B) Seq(B) B Seq(B) Cyc(B) Seq(B) B Set(B) Set(B) B Example: H(G, S, P) := (S + P, Seq >0 (Z + P), Set >1 (Z + S)). 1 1 H Y = Seq(Z + P) 1 Seq(Z + P) Set >0 (Z + S) 1 16 / 41

32 Joyal s Implicit Species Theorem Theorem If H(0, 0) = 0 and H/ Y(0, 0) is nilpotent, then Y = H(Z, Y) has a unique solution, limit of Y [0] = 0, Y [n+1] = H(Z, Y [n] ) (n 0). Def. A = k B if they coincide up to size k (contact k). Key Lemma If Y [n+1] = k Y [n], then Y [n+p+1] = k+1 Y [n+p], (p = dimension). 17 / 41

33 Joyal s Implicit Species Theorem Theorem If H(0, 0) = 0 and H/ Y(0, 0) is nilpotent, then Y = H(Z, Y) has a unique solution, limit of Y [0] = 0, Y [n+1] = H(Z, Y [n] ) (n 0). Def. A = k B if they coincide up to size k (contact k). Key Lemma If Y [n+1] = k Y [n], then Y [n+p+1] = k+1 Y [n+p], (p = dimension). We prove an iff when no 0 coordinate. 17 / 41

34 Newton Iteration for Binary Trees Y = 1 Z Y 2 18 / 41

35 Newton Iteration for Binary Trees Y = 1 Z Y 2 Y 0 = Y 1 = 2 Y 2 = Y 3 = Y [Décoste, Labelle, Leroux 1982] 18 / 41

36 Newton Iteration for Binary Trees Y = 1 Z Y 2 Y n+1 = Y n Seq(Z Y n Z Y n ) (1 Z Yn 2 \ Y n ). Y 0 = Y 1 = 2 Y 2 = Y 3 = Y [Décoste, Labelle, Leroux 1982] 18 / 41

37 Combinatorial Newton Iteration Theorem (essentially Labelle) For any well-founded system Y = H(Z, Y), if A has contact k with the solution and A H(Z, A), then A + i 0 has contact 2k + 1 with it. ( ) H i (Z, A) (H(Z, A) A) Y A+ 19 / 41

38 Combinatorial Newton Iteration Theorem (essentially Labelle) For any well-founded system Y = H(Z, Y), if A has contact k with the solution and A H(Z, A), then A + i 0 has contact 2k + 1 with it. ( ) H i (Z, A) (H(Z, A) A) Y A+ Generation by increasing Strahler numbers. 19 / 41

39 Newton Iteration for Series-Parallel Graphs 0 0 ( ) ( ) S [n+1] S [n] P [n+1] = P [n] + ( ) 0 Seq 2 (Z + P [n] k ( ) ) 1 Set k 0 >0 (Z + S [n] Seq >1 (Z + P [n] ) S [n] ) 0 Set >0 (Z + S [n] ) P [n]. 20 / 41

40 Linear Species and Ordered Structures The underlying sets are ordered F: min 21 / 41

41 Linear Species and Ordered Structures The underlying sets are ordered Examples F: increasing trees: Y = Z + F(Y); min 21 / 41

42 Linear Species and Ordered Structures The underlying sets are ordered Examples F: increasing trees: Y = Z + F(Y); min alternating permutations (odd/even): A e = A e A o, A o = Z + A 2 o; 21 / 41

43 Linear Species and Ordered Structures The underlying sets are ordered Examples F: increasing trees: Y = Z + F(Y); min alternating permutations (odd/even): A e = A e A o, A o = Z + A 2 o; cycles: Cyc(A) = Seq(A)A ; 21 / 41

44 Linear Species and Ordered Structures The underlying sets are ordered Examples F: increasing trees: Y = Z + F(Y); min alternating permutations (odd/even): A e = A e A o, A o = Z + A 2 o; cycles: Cyc(A) = Seq(A)A ; sets: Set(A) = 1 + Set(A)A. 21 / 41

45 Linear Species and Ordered Structures F: min Examples increasing trees: Y = Z + F(Y); alternating permutations (odd/even): A e = A e A o, A o = Z + A 2 o; cycles: Cyc(A) = Seq(A)A ; sets: Set(A) = 1 + Set(A)A. Theorem (Enumeration in Quasi-Optimal Complexity) First N coefficients of the solution of Y(Z) = H(Z, Y(Z)) + Z 0 G(T, Y(T )) dt with H and G constructible, in O(N log N) operations. 21 / 41

46 III Newton Iteration for Power Series 22 / 41

47 Newton Did It in 1671! 23 / 41

48 Generating Series: a Simple Dictionary ogf := t T z t, egf := t T z t t!. Language and Gen. Fcns (labelled) A B A B A(z) + B(z) A(z) B(z) 1 Seq(C) 1 C(z) A A (z) 1 Cyc(C) log 1 C(z) Set(C) exp(c(z)) Consequence: Combinatorial Newton iteration Newton iteration for GFs 24 / 41

49 Generating Series: a Simple Dictionary ogf := t T z t, egf := t T z t t!. Language and Gen. Fcns (labelled) (unlabelled) A B A(z) + B(z) A(z) + B(z) A B A(z) B(z) A(z) B(z) Seq(C) 1 1 C(z) 1 1 C(z) A A (z) 1 φ(k) 1 Cyc(C) log 1 C(z) k 1 k log 1 C(z k ) Set(C) exp(c(z)) exp( C(z i )/i) Consequence: Combinatorial Newton iteration Newton iteration for GFs 24 / 41

50 Newton Iteration for Power Series has Good Complexity To solve φ(y) = 0, iterate y [n+1] = y [n] u [n+1], φ (y [n] )u [n+1] = φ(y [n] ). Quadratic convergence Divide-and-Conquer 25 / 41

51 Newton Iteration for Power Series has Good Complexity To solve φ(y) = 0, iterate y [n+1] = y [n] u [n+1], φ (y [n] )u [n+1] = φ(y [n] ). Quadratic convergence Divide-and-Conquer To solve at precision N 1 Solve at precision N/2; 2 Compute φ and φ there; 3 Solve for u [n+1]. Cost(y [n] ) = constant Cost(last step). 25 / 41

52 Newton Iteration for Power Series has Good Complexity To solve φ(y) = 0, iterate y [n+1] = y [n] u [n+1], φ (y [n] )u [n+1] = φ(y [n] ). Quadratic convergence Divide-and-Conquer To solve at precision N 1 Solve at precision N/2; 2 Compute φ and φ there; 3 Solve for u [n+1]. Cost(y [n] ) = constant Cost(last step). Useful in conjunction with fast multiplication (quasi-linear): power series at order N: O(N log N) ops on the coefficients; N-bit integers: O(N log N log log N) bit ops. 25 / 41

53 Example: Series-Parallel Graphs G = S + P, S = Seq >0 (Z + P), P = Set >1 (Z + S). 1 1 H Y = Seq 2 (Z + P) Set >0 (Z + S) 26 / 41

54 Example: Series-Parallel Graphs (labelled case) G = S + P, S = Seq >0 (Z + P), P = Set >1 (Z + S). 1 1 H Y = Seq 2 (Z + P) Set >0 (Z + S) translates into G = S + P, S = (1 z P) 1 1, P = e z+s 1 z S H Y = 0 0 (1 z P) 2 0 e z+s / 41

55 Example: Series-Parallel Graphs (labelled case) translates into G = S + P, S = (1 z P) 1 1, P = e z+s 1 z S. Newton iteration: Y [n] := ( G [n] H Y = 0 0 (1 z P) 2 0 e z+s 1 0 S [n] P [n] ), ( Y [n+1] = Y [n] + Id H ) 1 ( Y (Y[n] ) H(Y [n] ) Y [n]). 26 / 41

56 Example: Series-Parallel Graphs (labelled case) translates into G = S + P, S = (1 z P) 1 1, P = e z+s 1 z S. Newton iteration: Y [n] := ( G [n] H Y = 0 0 (1 z P) 2 0 e z+s 1 0 S [n] P [n] ), ( Y [n+1] = Y [n] + Id H ) 1 ( Y (Y[n] ) H(Y [n] ) Y [n]) mod z 2n / 41

57 Example: Series-Parallel Graphs (labelled case) translates into G = S + P, S = (1 z P) 1 1, P = e z+s 1 z S. Newton iteration: Y [n] := ( G [n] H Y = 0 0 (1 z P) 2 0 e z+s 1 0 S [n] P [n] ), ( Y [n+1] = Y [n] + Id H ) 1 ( Y (Y[n] ) H(Y [n] ) Y [n]) mod z 2n+1. Wanted: efficient matrix inverse, efficient exp. 26 / 41

58 Example: Unlabelled Rooted Trees 1 Combinatorial equation: Y = Z Set(Y) =: H(Z, Y); 27 / 41

59 Example: Unlabelled Rooted Trees 1 Combinatorial equation: Y = Z Set(Y) =: H(Z, Y); 2 Combinatorial Newton iteration: Y [n+1] = Y [n] + Seq(H(Y [n] )) (H(Y [n] ) \ Y [n] ) 27 / 41

60 Example: Unlabelled Rooted Trees 1 Combinatorial equation: Y = Z Set(Y) =: H(Z, Y); 2 Combinatorial Newton iteration: Y [n+1] = Y [n] + Seq(H(Y [n] )) (H(Y [n] ) \ Y [n] ) 3 OGF equation: Ỹ (z) = H(z, Ỹ (z)) Ỹ (z) = z exp(ỹ (z) + 1 2Ỹ (z2 ) + 1 3Ỹ (z3 ) + ) 27 / 41

61 Example: Unlabelled Rooted Trees 1 Combinatorial equation: Y = Z Set(Y) =: H(Z, Y); 2 Combinatorial Newton iteration: Y [n+1] = Y [n] + Seq(H(Y [n] )) (H(Y [n] ) \ Y [n] ) 3 OGF equation: Ỹ (z) = H(z, Ỹ (z)) Ỹ (z) = z exp(ỹ (z) + 1 2Ỹ (z2 ) + 1 3Ỹ (z3 ) + ) 4 Newton for OGF: Ỹ [n+1] = Ỹ [n] + H(z, Ỹ [n] ) Ỹ [n] 1 H(z, Ỹ [n] ) 0, z + z 2 + z 3 + z 4 +, z + z 2 + 2z 3 + 4z 4 + 9z z / 41

62 Newton Iteration for Inverses φ(y) = a 1/y 1/φ (y) = y 2 y [n+1] = y [n] y [n] (ay [n] 1). Works for: 1 Numerical inversion; 2 Reciprocal of power series; 3 Inversion of matrices. Cost: a small number of multiplications Applications: Seq (I H Y ) 1 [Schulz 1933; Cook 1966; Sieveking 1972; Kung 1974] 28 / 41

63 Inverses for Series-Parallel Graphs G = S + P, S = (1 z P) 1 1, P = e z+s 1 z S H Y = 0 0 (1 z P) 2 0 e z+s 1 0 Newton iteration: ( ) U [n+1] = U [n] + U [n] H Y (Y[n] ) U [n] + Id U [n] mod z 2n, ( Y [n+1] = Y [n] + U [n+1] H(Y [n] ) Y [n]) mod z 2n+1. Can be lifted combinatorially. Also a numerical iteration! 29 / 41

64 Inverses for Series-Parallel Graphs G = S + P, S = (1 z P) 1 1, P = e z+s 1 z S H Y = 0 0 (1 z P) 2 0 e z+s 1 0 Newton iteration: ( ) U [n+1] = U [n] + U [n] H Y (Y[n] ) U [n] + Id U [n] mod z 2n, ( Y [n+1] = Y [n] + U [n+1] H(Y [n] ) Y [n]) mod z 2n+1. Can be lifted combinatorially. Also a numerical iteration! Wanted: efficient exp. 29 / 41

65 From the Inverse to the Exponential 1 Logarithm of power series: log f = (f /f ) (recall Cyc(A) = A Seq(A)) 2 exponential of power series: φ(y) = a log y. e [n+1] = e [n] a log e[n] + 1/e [n] mod z 2n+1, ) = e [n] + e (a [n] e [n] /e [n] mod z 2n+1. And 1/e [n] is computed by Newton iteration too! [Brent 1975; Hanrot-Zimmermann 2002] 30 / 41

66 Application: Power Sums F = t N + a N 1 t N a 0 S i = F (α)=0 α i, i = 0,..., N. 31 / 41

67 Application: Power Sums F = t N + a N 1 t N a 0 S i = F (α)=0 α i, i = 0,..., N. Fast conversion using the generating series: rev(f ) rev(f ) = ( S i+1 t i rev(f ) = exp ) S i i ti. i 0 31 / 41

68 Application: Power Sums F = t N + a N 1 t N a 0 S i = F (α)=0 α i, i = 0,..., N. Fast conversion using the generating series: rev(f ) rev(f ) = ( S i+1 t i rev(f ) = exp ) S i i ti. i 0 Application: composed product and sums (F, G) (t αβ) or F (α)=0,g(β)=0 F (α)=0,g(β)=0 (t (α + β)). 31 / 41

69 Application: Power Sums F = t N + a N 1 t N a 0 S i = F (α)=0 α i, i = 0,..., N. Fast conversion using the generating series: rev(f ) rev(f ) = ( S i+1 t i rev(f ) = exp ) S i i ti. i 0 Application: composed product and sums (F, G) (t αβ) or F (α)=0,g(β)=0 F (α)=0,g(β)=0 (t (α + β)). Easy in Newton representation: α s β s = (αβ) s and (α + β) s ( ) ( α t s s ) β = t s s t s. s! s! s! [Schönhage 1982; Bostan, Flajolet, Salvy, Schost 2006] 31 / 41

70 Timings Introduction Combinatorics Generating Series Oracle Conclusion Applications (crypto): over finite fields, degree > expected Bivariate resultant computation 60 Our algorithm Timings in seconds vs. output degree N, over F p, 26 bits prime p 32 / 41

71 Conclusion for Series-Parallel Graphs G = S + P, S = Seq >0 (Z + P), P = Set >1 (Z + S) compiles into the Newton iteration: i [n+1] = i [n] i [n] (e [n] i [n] 1), v [n+1] = v [n] v [n] ((1 z P [n] )v [n] 1), U [n+1] = U [n] + U [n] 0 0 v [n+1]2 U [n] + Id U [n], e [n+1] = e [n] e [n] ( 1 + d dz S [n] ( d dz e[n] )i [n]), G [n+1] S [n+1] P [n+1] = G [n] S [n] P [n] 0 e [n+1] 1 0 S [n] + P [n] G [n] + U [n+1] v [n+1] S [n] mod z 2n+1. e [n+1] P [n] Computation reduced to products and linear ops. 33 / 41

72 Linear Differential Equations of Arbitrary Order Given a linear differential equation with power series coefficients, a r (t)y (r) (t) + + a 0 (t)y(t) = 0, compute the first N terms of a basis of power series solutions. 34 / 41

73 Linear Differential Equations of Arbitrary Order Given a linear differential equation with power series coefficients, a r (t)y (r) (t) + + a 0 (t)y(t) = 0, compute the first N terms of a basis of power series solutions. Algorithm 1 Convert into a system Φ : Y Y A(t)Y (DΦ = Φ); 2 DΦ Y (U) = Φ(Y ) rewrites U AU = Y AY ; 3 Variation of constants: U = Y Y 1 (Y AY ); 4 Y 1 by Newton iteration too. Special case: recover good exponential. [Bostan, Chyzak, Ollivier, Salvy, Schost, Sedoglavic 2007] 34 / 41

74 Linear Differential Equations of Arbitrary Order Given a linear differential equation with power series coefficients, a r (t)y (r) (t) + + a 0 (t)y(t) = 0, compute the first N terms of a basis of power series solutions. Algorithm 1 Convert into a system Φ : Y Y A(t)Y (DΦ = Φ); 2 DΦ Y (U) = Φ(Y ) rewrites U AU = Y AY ; 3 Variation of constants: U = Y Y 1 (Y AY ); 4 Y 1 by Newton iteration too. Special case: recover good exponential. Lifts to integral equations for linear species. [Bostan, Chyzak, Ollivier, Salvy, Schost, Sedoglavic 2007] 34 / 41

75 Timings Introduction Combinatorics Generating Series Oracle Conclusion "MatMul.dat" "Newton.dat" time (in seconds) time (in seconds) precision order precision order 9 10 Polynomial matrix multiplication vs. solving Y = AY. 35 / 41

76 Non-Linear Differential Equations Example from cryptography: φ : y (x 3 + Ax + B)y 2 (y 3 + Ãy + B). 36 / 41

77 Non-Linear Differential Equations Example from cryptography: φ : y (x 3 + Ax + B)y 2 (y 3 + Ãy + B). Differential: Dφ y : u 2(x 3 + Ax + B)y u (3y 2 + Ã)u. Solve the linear differential equation at each iteration. Dφ y u = φ(y) Again, quasi-linear complexity. [Bostan, Morain, Salvy, Schost 2008] 36 / 41

78 IV Oracle 37 / 41

79 Exponential Generating Series Combinatorial Specification Combinatorial Newton iteration for Y Newton iteration for the gf Y (z) ((y 0,..., y N ) fast) Numerical Newton iteration starting from 0 converges to the value of Y (x). 38 / 41

80 Exponential Generating Series Combinatorial Specification Combinatorial Newton iteration for Y Newton iteration for the gf Y (z) ((y 0,..., y N ) fast) Numerical Newton iteration starting from 0 converges to the value of Y (x). 38 / 41

81 Ordinary Generating Function on an Example 1 Combinatorial Equation: Y = Z Set(Y) =: H(Z, Y); 39 / 41

82 Ordinary Generating Function on an Example 1 Combinatorial Equation: Y = Z Set(Y) =: H(Z, Y); 2 Combinatorial Newton iteration: Y [n+1] = Y [n] + Seq(H(Y [n] )) (H(Y [n] ) \ Y [n] ) 39 / 41

83 Ordinary Generating Function on an Example 1 Combinatorial Equation: Y = Z Set(Y) =: H(Z, Y); 2 Combinatorial Newton iteration: Y [n+1] = Y [n] + Seq(H(Y [n] )) (H(Y [n] ) \ Y [n] ) 3 OGF equation: Ỹ (z) = H(z, Ỹ (z)) = z exp(ỹ (z) + 1 2Ỹ (z2 ) + 1 3Ỹ (z3 ) + ) 39 / 41

84 Ordinary Generating Function on an Example 1 Combinatorial Equation: Y = Z Set(Y) =: H(Z, Y); 2 Combinatorial Newton iteration: Y [n+1] = Y [n] + Seq(H(Y [n] )) (H(Y [n] ) \ Y [n] ) 3 OGF equation: Ỹ (z) = H(z, Ỹ (z)) = z exp(ỹ (z) + 1 2Ỹ (z2 ) + 1 3Ỹ (z3 ) + ) 4 Newton for OGF: Ỹ [n+1] = Ỹ [n] + H(z,Ỹ [n] ) Ỹ [n] 1 H(z,Ỹ [n] ) 39 / 41

85 Ordinary Generating Function on an Example 1 Combinatorial Equation: Y = Z Set(Y) =: H(Z, Y); 2 Combinatorial Newton iteration: Y [n+1] = Y [n] + Seq(H(Y [n] )) (H(Y [n] ) \ Y [n] ) 3 OGF equation: Ỹ (z) = H(z, Ỹ (z)) = z exp(ỹ (z) + 1 2Ỹ (z2 ) + 1 3Ỹ (z3 ) + ) 4 Newton for OGF: Ỹ [n+1] = Ỹ [n] + H(z,Ỹ [n] ) Ỹ [n] 1 H(z,Ỹ [n] ) 5 Numerical iteration: n Ỹ [n] (0.3) Ỹ [n] (0.3 2 ) Ỹ [n] (0.3 3 ) e e e e e e e e e e-1 39 / 41

86 V Conclusion 40 / 41

87 Conclusion Introduction Combinatorics Generating Series Oracle Conclusion Summary: Newton iteration has good complexity; Oracle: numerical Newton iteration that gives the values of... power series that are the gfs of... combinatorial iterates. 41 / 41

88 Conclusion Introduction Combinatorics Generating Series Oracle Conclusion Summary: Newton iteration has good complexity; Oracle: numerical Newton iteration that gives the values of... power series that are the gfs of... combinatorial iterates. Read the paper for: Well-defined systems (with 1); Majorant species; PowerSet (it is not a species). 41 / 41

89 Conclusion Introduction Combinatorics Generating Series Oracle Conclusion Summary: Newton iteration has good complexity; Oracle: numerical Newton iteration that gives the values of... power series that are the gfs of... combinatorial iterates. Read the paper for: Well-defined systems (with 1); Majorant species; PowerSet (it is not a species). THE END 41 / 41