Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Composition of Power Series, Change of Basis and Orthogonal Polynomials Bruno Salvy Bruno.Salvy@inria.fr Algorithms Project, Inria June 2nd, 2008 Joint work with Alin Bostan and ´ Eric Schost arXiv:0804.2337 (ISSAC’08) and arXiv:0804.2373 . 1 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion I Introduction 2 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Univariate Composition Problem (Composition of Power Series) Input: two power series f and g at precision N , with g (0) = 0. Output: f ( g ) at precision N . Measure of complexity: number of arithmetic operations. Hypothesis for the talk: characteristic 0. General algorithms: ıve: O ( N 3 ); Na¨ Na¨ ıve with fast multiplication: O ( N M( N )); Brent & Kung (1978): O ( √ N log N M( N )). 3 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Univariate Composition Problem (Composition of Power Series) Input: two power series f and g at precision N , with g (0) = 0. Output: f ( g ) at precision N . Measure of complexity: number of arithmetic operations. Hypothesis for the talk: characteristic 0. General algorithms: O ( √ N log N M( N )); Fast Multiplication of Polynomials of degree N ıve: M( N ) = O ( N 2 ); Na¨ Karatsuba (1963): M( N ) = O ( N 1 . 59 ); FFT: M( N ) = O ( N log N ) (Sch¨ onhage & Strassen 71). 3 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Univariate Composition Problem (Composition of Power Series) Input: two power series f and g at precision N , with g (0) = 0. Output: f ( g ) at precision N . Measure of complexity: number of arithmetic operations. Hypothesis for the talk: characteristic 0. General algorithms: O ( √ N log N M( N )); Nothing better known in general → exploit structure Special f ’s by Newton iteration: O (M( N )) for inverse (Sieveking 72, Kung 74), log, exp, power (Brent 75); 3 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Univariate Composition Problem (Composition of Power Series) Input: two power series f and g at precision N , with g (0) = 0. Output: f ( g ) at precision N . Measure of complexity: number of arithmetic operations. Hypothesis for the talk: characteristic 0. General algorithms: O ( √ N log N M( N )); Nothing better known in general → exploit structure f ∈ { Inv , log , exp , Pow } : O (M( N )); 3 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Univariate Composition Problem (Composition of Power Series) Input: two power series f and g at precision N , with g (0) = 0. Output: f ( g ) at precision N . Measure of complexity: number of arithmetic operations. Hypothesis for the talk: characteristic 0. General algorithms: O ( √ N log N M( N )); Nothing better known in general → exploit structure f ∈ { Inv , log , exp , Pow } : O (M( N )); Special g ’s by Divide-and-Conquer: O (M( N ) log N ); polynomials (Brent & Kung 78); algebraic series (van der Hoeven 02); 3 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Univariate Composition Problem (Composition of Power Series) Input: two power series f and g at precision N , with g (0) = 0. Output: f ( g ) at precision N . Measure of complexity: number of arithmetic operations. Hypothesis for the talk: characteristic 0. General algorithms: O ( √ N log N M( N )); Nothing better known in general → exploit structure f ∈ { Inv , log , exp , Pow } : O (M( N )); Special g ’s by Divide-and-Conquer: O (M( N ) log N ); f a polynomial, special g ’s: x + a : O (M( N )) (Aho, Steiglitz, Ullman 75); ( ax + b ) / ( cx + d ): O (M( N )) (Pan 98); NEW: several other functions in O (M( N )); NEW: exp and log in O (M( N ) log N ). 3 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Bivariate Evaluation Problem (Bivariate Evaluation) � � F i , j x i t j = G j ( x ) t j . F ( x , t ) = i , j j Input: a 0 , . . . , a N ; Output: � j a j G j ( x ) mod x N +1 . Examples: G j = g j : univariate composition; G j polynomial of degree j : change of basis from ( G j ) to ( x j ). Our Result Good complexity for this map and its inverse for “nice” F . 4 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Special Cases with Fast Change of Basis Very fast ( O (M( N ))) 4 F 3 (4) Wilson Racah Orthogonal: Jacobi, Legendre, Chebyshev U , T , Gegenbauer, Laguerre, Hermite. Continuous Continuous 3 F 2 (3) Hahn Dual Hahn dual Hahn Hahn Other: Fibonacci, Bernoulli, Euler, Mott, Spread, Bessel,. . . Meixner Fast ( O (M( N ) log N )) 2 F 1 (2) - Jacobi Meixner Krawtchouk Pollaczek Orthogonal: Meixner, Meixner-Pollaczek, Krawtchouk, Charlier; 1 F 1 (1) / 2 F 0 (1) Laguerre Charlier Other: Falling factorial, Bell, Actuarial, Narumi, Peters,. . . 2 F 0 (0) Hermite 5 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Special Cases with Fast Change of Basis Very fast ( O (M( N ))) 4 F 3 (4) Wilson Racah Orthogonal: Jacobi, Legendre, Chebyshev U , T , Gegenbauer, Laguerre, Hermite. Continuous Continuous 3 F 2 (3) Hahn Dual Hahn dual Hahn Hahn Other: Fibonacci, Bernoulli, Euler, Mott, Spread, Bessel,. . . Meixner Fast ( O (M( N ) log N )) 2 F 1 (2) - Jacobi Meixner Krawtchouk Pollaczek Orthogonal: Meixner, Meixner-Pollaczek, Krawtchouk, Charlier; 1 F 1 (1) / 2 F 0 (1) Laguerre Charlier Other: Falling factorial, Bell, Actuarial, Narumi, Peters,. . . Not all of Sheffer type. 2 F 0 (0) Hermite 5 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Special Cases with Fast Change of Basis Very fast ( O (M( N ))) 4 F 3 (4) Wilson Racah Orthogonal: Jacobi, Legendre, Chebyshev U , T , Gegenbauer, Laguerre, Hermite. Continuous Continuous 3 F 2 (3) Hahn Dual Hahn dual Hahn Hahn Other: Fibonacci, Bernoulli, Euler, Mott, Spread, Bessel,. . . Meixner Fast ( O (M( N ) log N )) 2 F 1 (2) - Jacobi Meixner Krawtchouk Pollaczek Orthogonal: Meixner, Meixner-Pollaczek, Krawtchouk, Charlier; 1 F 1 (1) / 2 F 0 (1) Laguerre Charlier Other: Falling factorial, Bell, Actuarial, Narumi, Peters,. . . Not all of Sheffer type. 2 F 0 (0) Hermite Other algorithm giving all orthogonal pols in O (M( N ) log N ) 5 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion II Very Fast Composition ( O ( M ( N )) operations) 6 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Shift is Cheap (Aho, Steiglitz, Ullman 1975) Problem (Polynomial Shift) Input: P ( x ) polynomial of degree N , a a point. Output: P ( x + a ). Horner: O ( N 2 ) operations Taylor: i P ( N − i + j ) (0) a j � P ( N − i ) ( a ) = j ! . j =0 Consequence: generating series N n a j x j � � � x i + · · · = x N − k × P ( N − i ) ( a ) P ( k ) (0) j ! . � �� � � �� � i =0 k =0 j ≥ 0 coeffs of P ( x + a ) coeffs of P ( x ) M( N ) + O ( N ) operations. 7 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
Introduction Very Fast Fast Change of Basis Orthogonal Polynomials Conclusion Warm-up: Euler Transform for 3 Multiplications Problem (Euler Transform) Input: P ( x ) polynomial of degree N (or truncated series); � � 1 x Output: First N coefficients of 1 − x P . 1 − x Algorithm: S 1 ( x ) := P ( x − 1); S 2 ( x ) := x N S 1 (1 / x ); S 3 ( x ) := S 2 ( x + 1); S 4 ( x ) := S 3 ( − x ); return (1 − x ) − N − 1 S 4 ( x ) . 8 / 31 Alin Bostan, Bruno Salvy, ´ Eric Schost Composition, Change of Basis, Orthogonal Polynomials
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