Exact real arithmetical algorithms in binary continued fractions - PowerPoint PPT Presentation

Exact real arithmetical algorithms in binary continued fractions Petr K˚ urka Center for Theoretical Study Academy of Sciences and Charles University in Prague 22nd IEEE Symposium on Computer Arithmetic Lyon, june 24, 2015 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. .

Exact real computation: arbitrary precision The first k digits of x + y , xy , . . . depend only on the first n k digits of x and y Exact real computation is possible in redundant number systems whose digits represent M¨ obius transformations. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

M¨ obius transformations M ( z ) = az + b cz + d , det( M ) = ad − bc ̸ = 0 M : R → R = R ∪ {∞} the extended real line The space M ( R ) of transformations is a group. F = { F a : R → R : a ∈ A } M¨ obius iterative system F u = F u 0 F u 1 · · · F u n − 1 , u ∈ A n . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

The convergence space X F = { u ∈ A ω : n →∞ F u [0 , n ) ( i ) ∈ R } lim u ∈ X F ⇐ ⇒ ∀ z ∈ C \ R , lim n →∞ F u [0 , n ) ( z ) ∈ R Φ : X F → R the value map Φ( u ) = lim n →∞ F u [0 , n ) ( i ) ∈ R , u ∈ X F M¨ obius number system ( F , Σ): Σ ⊆ X F is a subshift Φ : Σ → R is continuous and surjective. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. .

Subshifts D ⊆ A ∗ : forbidden words Σ D = { u ∈ A N : ∀ v ∈ D , v ̸⊑ u } the language of a subshift: L (Σ) = { v ∈ A ∗ : ∃ u ∈ Σ , v ⊑ u } Σ is a sofic subshift if L (Σ) is regular. L (Σ): labels of paths of a finite labelled graph . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

Positional systems β > 1: base A = [ p , p + 1 , . . . , q ] ⊂ Z alphabet F a ( z ) = z + a transformations β F u ( z ) = u 0 β + · · · + u n − 1 β n + z β n ∞ [ ] u n p q β n +1 , Φ : A ω → ∑ Φ( u ) = β − 1 , β − 1 n =0 F 0 ( x ) = β x : number systems for R ∞ n u ) = ∑ u k β n − k − 1 Φ(0 k =0 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

Simple continued fractions A = { 0 , 1 } z 1 F 1 ( z ) = z + 1, F 0 ( z ) = z +1 = 1+ 1 z 1 F a 1 ( z ) = a + z , F a 0 ( z ) = a + 1 z u = 1 a 0 0 a 1 1 a 2 · · · 1 a 2 n , a i > 0 for i > 0 1 F u ( x ) = a 0 + a 1 +... + 1 a 2 n + x = a 0 + 1 1 1 a 1 + a 2 + · · · + a 2 n + x . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. .

Simple continued fractions u = 1 a 0 0 a 1 1 a 2 · · · 0 a 2 n +1 , a i > 0 for i > 0 1 F u ( x ) = a 0 + a 1 +... + 1 a 2 n +1 + 1 x = a 0 + 1 1 1 1 a 1 + a 2 + · · · + a 2 n +1 + x u = 1 a 0 0 a 1 1 a 2 0 a 3 · · · , a i > 0 for i > 0 Φ( u ) = a 0 + 1 1 a 2 + · · · , a 1 + Φ : { 0 , 1 } ω → [0 , ∞ ] . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

Symmetric continued fractions A = { 0 , 1 , 2 , 3 } Σ = { 0 , 1 } ω ∪ { 2 , 3 } ω x F 0 ( x ) = x + 1 , F 1 ( x ) = x + 1 , x F 2 ( x ) = 1 − x , F 3 ( x ) = x − 1 V + = [0 , ∞ ] = F 0 ( V + ) ∪ F 1 ( V + ) = [0 , 1] ∪ [1 , ∞ ] V − = [ ∞ , 0] = F 3 ( V − ) ∪ F 2 ( V − ) = [ ∞ , − 1] ∪ [ − 1 , 0] V λ = R = F 0 ( V + ) ∪ F 1 ( V + ) ∪ F 2 ( V − ) ∪ F 3 ( V − ) 2,3 0,1 λ 2,3 - + 0,1 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

Sofic number system ( F , G , V ) over A F = { F a : a ∈ A } : iterative system G = ( B , E , i ), E ⊆ B × A × B : labelled graph Σ G : labels of paths in G a ( p , a , q ) ∈ E ⇐ ⇒ p → q − V = { V p : p ∈ B } closed intervals ∪ V p = { F a ( V q ) : p a → q } , V i = R − If p a → q then F a is contractive on V q − . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

The unary graph ( X , p , q ) ∈ M ( R ) × B × B : X ( V p ) ⊆ V q vertices ( X , p , q ) a ,λ a → ( XF a , r , q ) if p → r − − absorption edge ( X , p , q ) λ, a → ( F − 1 a a X , p , r ) if q → r , X ( V p ) ⊆ F a ( V r ) − − emission edge If ( X , i , i ) u , v → then Φ( v ) = X (Φ( u )). − . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

Modular systems: det( F a ) = 1 Theorem[Raney] In symmetric continued fractions, the unary algorithm can be computed by a finite state transducer. Theorem[Delacourt and K˚ urka] In modular systems the norm of the state matrix of the unary algorithm remains bounded. Theorem[K˚ urka and V´ avra] M¨ obius transformations are the only analytic functions computable by finite state transducers. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. .

Fractional bilinear functions P ( x , y ) = axy + bx + cy + d exy + fx + gy + h , M ( x ) = ax + b cx + d ( P ∗ x )( y ) = P ( x , y ), ( P ∗ y )( x ) = P ( x , y ) P ∗ x , P ∗ y are transformations. ( P ∗ M )( x , y ) = P ( M ( x ) , y ), ( P ∗ M )( x , y ) = P ( x , M ( y )), ( MP )( x , y ) = M ( P ( x , y )) P ∗ M , P ∗ M , MP are fractional bilinear functions. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. .

The bilinear graph ( X , p , q , r ) : X ( V p , V q ) ⊆ V r : vertices ( X ∗ F a , p ′ , q , r ) if p → p ′ a ,λ,λ a ( X , p , q , r ) − → − ( X ∗ F a , p , q ′ , r ) if q → q ′ λ, a ,λ a ( X , p , q , r ) − → − ( F − 1 a X ∗ , p , q , r ′ ) if r → r ′ , λ,λ, a a ( X , p , q , r ) − → − X ( V p , V q ) ⊆ F a ( V ′ r ) If ( X , i , i , i ) u , v , w → then Φ( w ) = X (Φ( u ) , Φ( v )). − . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. .

Modular systems are not redundant and converge slowly 1/0 -3/1 3/1 -2/1 2/1 33 3 11 1 32 3 1 10 3 1 1/1 -1/1 2 0 23 2 0 01 2 22 00 0 1/2 -1/2 1/3 -1/3 0/1 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

The binary and prefix codes b : N → { 0 , 1 } + : the binary code p : { 1 , 2 , . . . , ∞} → { 0 , 1 } + ∪ { 1 ω } n = 2 k + 2 k − 1 u 0 + · · · u k − 1 p ( n ) = 1 k 0 u , | p ( n ) | = 2 ⌈ ln n ⌉ b ( n ) p ( n ) n n b ( n ) p ( n ) 5 101 11001 0 0 − 6 110 11010 1 1 0 7 111 11011 2 10 100 8 1000 1110000 3 11 101 9 1001 1110001 4 100 11000 − 1 ω ∞ . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. .

The compression code is continuous and one-to-one c : { 0 , 1 } ω ∪ { 2 , 3 } ω → { 0 , 1 } ω c (0 a 0 1 a 1 0 a 2 · · · ) = 00 p ( a 0 ) p ( a 1 ) p ( a 2 ) · · · c (1 a 0 0 a 1 1 a 2 · · · ) = 01 p ( a 0 ) p ( a 1 ) p ( a 2 ) · · · c (2 a 0 3 a 1 2 a 2 · · · ) = 10 p ( a 0 ) p ( a 1 ) p ( a 2 ) · · · c (3 a 0 2 a 1 3 a 2 · · · ) = 11 p ( a 0 ) p ( a 1 ) p ( a 2 ) · · · ∀ i , a i > 0 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .