Formulas for Continued Fractions: an Automated Guess and Prove - PowerPoint PPT Presentation

Formulas for Continued Fractions: an Automated Guess and Prove Approach ´ S´ ebastien Maulat, ENS de Lyon (France) Bruno Salvy, INRIA (France) ISSAC 2015, Bath (UK) July 8, 2015 S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 0 / 10

Introduction Rational approximation The Taylor and Pad´ e approximants at order 4 for the logarithm are: ln(1 + x ) = x − x 2 2 + x 3 3 − x 4 x + x 2 / 2 4 + O ( x 5 ) = 1 + x + x 2 / 6 + O ( x 5 ) . Convergence for x ∈ I R , and for x ∈ C . (order 2) Taylor at order 2 n : convergence for | x | < 1, Pad´ e at order n / n : convergence for 1 + x / ∈ I R − . S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 1 / 10

Introduction Rational approximation The Taylor and Pad´ e approximants at order 4 for the logarithm are: ln(1 + x ) = x − x 2 2 + x 3 3 − x 4 x + x 2 / 2 4 + O ( x 5 ) = 1 + x + x 2 / 6 + O ( x 5 ) . Convergence for x ∈ I R , and for x ∈ C . (order 2, 4) Taylor at order 2 n : convergence for | x | < 1, Pad´ e at order n / n : convergence for 1 + x / ∈ I R − . S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 1 / 10

Introduction Rational approximation The Taylor and Pad´ e approximants at order 4 for the logarithm are: ln(1 + x ) = x − x 2 2 + x 3 3 − x 4 x + x 2 / 2 4 + O ( x 5 ) = 1 + x + x 2 / 6 + O ( x 5 ) . Convergence for x ∈ I R , and for x ∈ C . (order 2, 4, etc.) Taylor at order 2 n : convergence for | x | < 1, Pad´ e at order n / n : convergence for 1 + x / ∈ I R − . S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 1 / 10

Introduction Rational approximation The Taylor and Pad´ e approximants at order 4 for the logarithm are: ln(1 + x ) = x − x 2 2 + x 3 3 − x 4 x + x 2 / 2 4 + O ( x 5 ) = 1 + x + x 2 / 6 + O ( x 5 ) . Convergence for x ∈ I R , and for x ∈ C . Areas with 10 correct digits: (order 2, 4, etc.) (order 16) Taylor at order 2 n : convergence for | x | < 1, Pad´ e at order n / n : convergence for 1 + x / ∈ I R − . S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 1 / 10

Introduction Rational approximation The Taylor and Pad´ e approximants at order 4 for the logarithm are: ln(1 + x ) = x − x 2 2 + x 3 3 − x 4 x + x 2 / 2 4 + O ( x 5 ) = 1 + x + x 2 / 6 + O ( x 5 ) . Convergence for x ∈ I R , and for x ∈ C . Areas with 10 correct digits: (order 2, 4, etc.) (order 16, 24) Taylor at order 2 n : convergence for | x | < 1, Pad´ e at order n / n : convergence for 1 + x / ∈ I R − . S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 1 / 10

Introduction Rational approximation The Taylor and Pad´ e approximants at order 4 for the logarithm are: ln(1 + x ) = x − x 2 2 + x 3 3 − x 4 x + x 2 / 2 4 + O ( x 5 ) = 1 + x + x 2 / 6 + O ( x 5 ) . Convergence for x ∈ I R , and for x ∈ C . Areas with 10 correct digits: (order 2, 4, etc.) (order 16, 24, 36) Taylor at order 2 n : convergence for | x | < 1, Pad´ e at order n / n : convergence for 1 + x / ∈ I R − . S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 1 / 10

Introduction Rational approximation The Taylor and Pad´ e approximants at order 4 for the logarithm are: ln(1 + x ) = x − x 2 2 + x 3 3 − x 4 x + x 2 / 2 4 + O ( x 5 ) = 1 + x + x 2 / 6 + O ( x 5 ) . Convergence for x ∈ I R , and for x ∈ C . Areas with 10 correct digits: (order 2, 4, etc.) (order 16, 24, 36. . . ) Taylor at order 2 n : convergence for | x | < 1, Pad´ e at order n / n : convergence for 1 + x / ∈ I R − . S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 1 / 10

Introduction Explicit formulas Diagonal Pad´ e approximants for ln(1 + x ) are viewed as corresponding fractions : x + x 2 / 2 x + x 2 + 11 x 3 / 60 x . . . 1 + 3 x / 2 + 3 / 5 x 2 + x 3 / 20 1 + x + x 2 / 6 1 + x / 2  a 1 = x x x x a 1 ( x )   m x a 2 m = = = , . 4 m − 2 x / 2 x / 2 a 2 ( x ) 1 + x / 2 m x  a 2 m +1 = 1 + 1 + 1 +  4 m +2 x / 6 x / 6 a 3 ( x ) 1 + 1 + 1 + 1 + x / 3 x / 3 ... 1 + x / 5 1 + 1+ 1 + 3 x / 10 1 + a 2 m ( x ) with a m monomial in x . This leads to many formulas: for Bessel ratios J ν +1 ( x ) for x exp( x 2 ) erf( x ), for tan( x ), J ν ( x ) . . . − 2(2 m − 1) x 2 − x 2 − x 2 a m = a 2 m = a m = 4( ν + m − 1)( ν + m ) . (2 m − 1)(2 m − 3) (4 m − 3)(4 m − 1) 4 mx 2 a 2 m +1 = (4 m − 1)(4 m +1) S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 2 / 10

Introduction Explicit formulas Diagonal Pad´ e approximants for ln(1 + x ) are viewed as corresponding fractions : x + x 2 / 2 x + x 2 + 11 x 3 / 60 x . . . 1 + 3 x / 2 + 3 / 5 x 2 + x 3 / 20 1 + x + x 2 / 6 1 + x / 2  a 1 = x x x x a 1 ( x )   m x a 2 m = = = , . 4 m − 2 x / 2 x / 2 a 2 ( x ) 1 + x / 2 m x  a 2 m +1 = 1 + 1 + 1 +  4 m +2 x / 6 x / 6 a 3 ( x ) 1 + 1 + 1 + 1 + x / 3 x / 3 ... 1 + x / 5 1 + 1+ 1 + 3 x / 10 1 + a 2 m ( x ) with a m monomial in x . This leads to many formulas: for Bessel ratios J ν +1 ( x ) for x exp( x 2 ) erf( x ), for tan( x ), J ν ( x ) . . . − 2(2 m − 1) x 2 − x 2 − x 2 a m = a 2 m = a m = 4( ν + m − 1)( ν + m ) . (2 m − 1)(2 m − 3) (4 m − 3)(4 m − 1) 4 mx 2 a 2 m +1 = (4 m − 1)(4 m +1) � We study C-fractions with ( a 2 m ) m ≥ 1 and ( a 2 m +1 ) m ≥ 0 rational in m . S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 2 / 10

Introduction Automating the proofs Chapter 11 > > chapter := 11; (2.1.1) (2.1.1) > > labels(chapter); (2.1.2) (2.1.2) "11.2.2" > > N:=20; [Abramowitz & Stegun, 1964] (2.1.3) (2.1.3) Generic expansion, with finite series as input. "11.4.8" "11.7.1" The order of the series is doubled when the C-fraction coefficients are in z^2. > > for formula in labels(chapter) do print( sprintf(" %a.%a" ,chapter,formula) ); f := Cuyt_Cfrac[chapter][formula]: [+the online DLMF] i f f o r m u l a i n { 1 . 3 , 2 . 2 , 7 . 1 , 7 . 2 } t h e n S : = M u l t i S e r i e s : - s e r i e s ( f , z , N ) : e l i f f o r m u l a = 7 . 4 t h e n "11.2.4" S : = s e r i e s ( f , z , 2 * N ) ; e l s e S : = M u l t i S e r i e s : - s e r i e s ( f , z , 2 * N ) "11.5.5" f i ; "11.7.2" � R i c c a t i _ t o _ C f r a c ( S , y , z ) ; od; "11.1.2" "11.3.7" "11.6.4" [Cuyt & others, 2008] "11.7.4" "11.1.3" "11.4.5" "11.6.9" DEMO S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 3 / 10

Introduction Automating the proofs Chapter 11 > > chapter := 11; (2.1.1) (2.1.1) > > labels(chapter); (2.1.2) (2.1.2) "11.2.2" > > N:=20; [Abramowitz & Stegun, 1964] (2.1.3) (2.1.3) Generic expansion, with finite series as input. "11.4.8" "11.7.1" The order of the series is doubled when the C-fraction coefficients are in z^2. > > for formula in labels(chapter) do print( sprintf(" %a.%a" ,chapter,formula) ); f := Cuyt_Cfrac[chapter][formula]: [+the online DLMF] i f f o r m u l a i n { 1 . 3 , 2 . 2 , 7 . 1 , 7 . 2 } t h e n S : = M u l t i S e r i e s : - s e r i e s ( f , z , N ) : e l i f f o r m u l a = 7 . 4 t h e n "11.2.4" S : = s e r i e s ( f , z , 2 * N ) ; S : = M u l t i S e r i e s : - s e r i e s ( f , z , 2 * N ) e l s e "11.5.5" f i ; "11.7.2" � R i c c a t i _ t o _ C f r a c ( S , y , z ) ; od; "11.1.2" "11.3.7" "11.6.4" [Cuyt & others, 2008] "11.7.4" "11.1.3" "11.4.5" "11.6.9" Human proofs for corresponding fractions are: � k , � 1 + x clever, e.g. exp( x ) = lim k →∞ k hard to automate: generalization/specialization. They all concern solutions of: Riccati equations y ′ ( x ) = py 2 + qy + r with rational coefficients, and Riccati-like equations (difference and q -difference equations). These are invariant under the building block: y �→ ax / (1 + y ) for a � = 0. S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 3 / 10

Automatic expansions A generic procedure > with(gfun: -ContFrac) : > riccati2cfrac( { y ′ = 1 + y 2 , y (0) = 0 } , y , x ); x y ( x ) = 0 + − x 2 / 3 1 + . . . 1 + − x 2 (formula and proof) (2 m − 1)(2 m − 3) 1 + . . . Input: a Riccati equation y ′ ( x ) = py 2 + qy + r with p , q , r ∈ C ( x ), Guessing: a finite expansion at a small order gives the conjecture − x 2 a 1 = x , a m = (2 m − 1)(2 m − 3) , m > 0. Proving: do these coefficients lead to y ′ = 1 + y 2 ? Todo: prove this. S´ ebastien Maulat (Lyon, France) Guessing and Proving Continued Fractions July 8, 2015 4 / 10