Rigorous Uniform Approximation of D-finite Functions Mioara Joldes ∗ Joint work with Alexandre Benoit ∗∗ , Marc Mezzarobba ∗∗ ∗ ´ Ecole Normale Sup´ erieure de Lyon, Ar´ enaire Team, Laboratoire de l’Informatique du Parall´ elisme ∗∗ INRIA Roquencourt, Algorithms Team 1 / 20
D-finite Functions Definition A function y : R → R is D-finite if it is solution of a (homogeneous) linear differential equation with polynomial coefficients: L · y = a r y ( r ) + a r − 1 y ( r − 1) + · · · + a 0 y = 0 , a i ∈ Q [ x ] . (1) 2 / 20
D-finite Functions Definition A function y : R → R is D-finite if it is solution of a (homogeneous) linear differential equation with polynomial coefficients: L · y = a r y ( r ) + a r − 1 y ( r − 1) + · · · + a 0 y = 0 , a i ∈ Q [ x ] . (1) Examples { f ′ − f = 0 , f (0) = 1 } . f ( x ) = exp( x ) ↔ 2 / 20
D-finite Functions Definition A function y : R → R is D-finite if it is solution of a (homogeneous) linear differential equation with polynomial coefficients: L · y = a r y ( r ) + a r − 1 y ( r − 1) + · · · + a 0 y = 0 , a i ∈ Q [ x ] . (1) Examples { f ′ − f = 0 , f (0) = 1 } . f ( x ) = exp( x ) ↔ cos , arccos , Airy functions, Bessel functions, ... 2 / 20
D-finite Functions Definition A function y : R → R is D-finite if it is solution of a (homogeneous) linear differential equation with polynomial coefficients: L · y = a r y ( r ) + a r − 1 y ( r − 1) + · · · + a 0 y = 0 , a i ∈ Q [ x ] . (1) Examples { f ′ − f = 0 , f (0) = 1 } . f ( x ) = exp( x ) ↔ cos , arccos , Airy functions, Bessel functions, ... About 60% of Abramowitz & Stegun 2 / 20
D-finite Functions Definition A function y : R → R is D-finite if it is solution of a (homogeneous) linear differential equation with polynomial coefficients: L · y = a r y ( r ) + a r − 1 y ( r − 1) + · · · + a 0 y = 0 , a i ∈ Q [ x ] . (1) Differential equation + initial conditions = Data Structure Examples { f ′ − f = 0 , f (0) = 1 } . f ( x ) = exp( x ) ↔ cos , arccos , Airy functions, Bessel functions, ... About 60% of Abramowitz & Stegun 2 / 20
D-finite Functions Definition A function y : R → R is D-finite if it is solution of a (homogeneous) linear differential equation with polynomial coefficients: L · y = a r y ( r ) + a r − 1 y ( r − 1) + · · · + a 0 y = 0 , a i ∈ Q [ x ] . (1) Differential equation + initial conditions = Data Structure How can we approximate a D-finite function f ? Polynomial approximation: n � f i x i f ( x ) ≈ i =0 2 / 20
Uniform Approximation of D-finite Functions Problem Given an integer d , and a D-finite function f specified by a differential equation with polynomial coefficients and suitable boundary conditions, find the coefficients of a polynomial p ( x ) of degree d and a “small” bound B such that | p ( x ) − f ( x ) | < B for all x in [ − 1 , 1] . 3 / 20
Uniform Approximation of D-finite Functions Problem Given an integer d , and a D-finite function f specified by a differential equation with polynomial coefficients and suitable boundary conditions, find the coefficients of a polynomial p ( x ) of degree d and a “small” bound B such that | p ( x ) − f ( x ) | < B for all x in [ − 1 , 1] . Applications: Repeated evaluation on a line segment Plot Numerical integration Computation of minimax approximation polynomials using the Remez algorithm 3 / 20
Rigorous Uniform Approximation of D-finite Functions Why? Get the correct answer, not an “almost” correct one Bridge the gap between scientific computing and pure mathematics - speed and reliability 4 / 20
Rigorous Uniform Approximation of D-finite Functions Why? Get the correct answer, not an “almost” correct one Bridge the gap between scientific computing and pure mathematics - speed and reliability How? Use Floating-Point as support for fast computations Bound roundoff, discretization, truncation errors in numerical algorithms Compute enclosures instead of approximations 4 / 20
Rigorous Uniform Approximation of D-finite Functions Why? Get the correct answer, not an “almost” correct one Bridge the gap between scientific computing and pure mathematics - speed and reliability How? Use Floating-Point as support for fast computations Bound roundoff, discretization, truncation errors in numerical algorithms Compute enclosures instead of approximations What? Interval arithmetic 4 / 20
Chebyshev Series vs Taylor Series I Basic properties of Chebyshev polynomials Two approximations of f : by Taylor series T n ( cos ( θ )) = cos( nθ ) + ∞ c n x n , c n = f ( n ) (0) � f = , 0 if m � = n n ! � 1 T n ( x ) T m ( x ) n =0 √ dx = if m = 0 π 1 − x 2 − 1 π otherwise or by Chebyshev series 2 T n +1 = 2 xT n − T n − 1 + ∞ � T 0 ( x ) = 1 f = t n T n ( x ) , n = −∞ T 1 ( x ) = x T 2 ( x ) = 2 x 2 − 1 � 1 t n = 1 f ( t ) T n ( t ) √ 1 − t 2 dt. T 3 ( x ) = 4 x 3 − 3 x π − 1 5 / 20
Chebyshev Series vs Taylor Series I Error of approximation for exp( x ) 0 . 05 Two approximations of f : by Taylor series + ∞ c n x n , c n = f ( n ) (0) � 0 . 025 f = , n ! n =0 or by Chebyshev series x − 1 − 0 . 5 0 0 . 5 1 + ∞ � f = t n T n ( x ) , Taylor expansion of order 3 n = −∞ − 0 . 025 Chebyshev expansion � 1 t n = 1 f ( t ) of order 3 √ T n ( t ) 1 − t 2 dt. π − 1 − 0 . 05 5 / 20
Chebyshev Series vs Taylor Series II Bad approximation outside its circle of convergence 1 . 5 1 0 . 5 x − 1 − 0 . 5 0 0 . 5 1 − 0 . 5 arctan( 2 x ) − 1 Taylor approximation − 1 . 5 6 / 20
Chebyshev Series vs Taylor Series II Approximation of arctan( 2 x ) by Chebyshev expansion of degree 11 1 . 5 1 0 . 5 x − 1 − 0 . 5 0 0 . 5 1 − 0 . 5 arctan( 2 x ) − 1 Taylor approximation − 1 . 5 Chebyshev approximation 6 / 20
Chebyshev Series vs Taylor Series III Convergence Domains : For Taylor series: For Chebyshev series: disc centered at x 0 = 0 which elliptic disc with foci at ± 1 which avoids all the singularities of f avoids all the singularities of f 7 / 20
Chebyshev Series vs Taylor Series III Convergence Domains : For Taylor series: For Chebyshev series: disc centered at x 0 = 0 which elliptic disc with foci at ± 1 which avoids all the singularities of f avoids all the singularities of f Taylor series can not converge over entire [ − 1 , 1] unless all singularities lie outside the unit circle. 7 / 20
Chebyshev Series vs Taylor Series III Convergence Domains : For Taylor series: For Chebyshev series: disc centered at x 0 = 0 which elliptic disc with foci at ± 1 which avoids all the singularities of f avoids all the singularities of f Taylor series can not converge over entire [ − 1 , 1] unless all singularities lie outside the unit circle. � Chebyshev series converge over entire [ − 1 , 1] as soon as there are no real singularities in [ − 1 , 1] . 7 / 20
Chebyshev Series vs Taylor Series IV Truncation Error : Taylor series, Lagrange formula: ∀ x ∈ [ − 1 , 1] , ∃ ξ ∈ [ − 1 , 1] s.t. f ( x ) − T ( x ) = f ( n +1) ( ξ ) ( n + 1)! ( x − x 0 ) n +1 . 8 / 20
Chebyshev Series vs Taylor Series IV Truncation Error : Taylor series, Lagrange formula: ∀ x ∈ [ − 1 , 1] , ∃ ξ ∈ [ − 1 , 1] s.t. f ( x ) − T ( x ) = f ( n +1) ( ξ ) ( n + 1)! ( x − x 0 ) n +1 . Chebyshev series, Bernstein-like formula: ∀ x ∈ [ − 1 , 1] , ∃ ξ ∈ [ − 1 , 1] s.t. f ( x ) − P ( x ) = f ( n +1) ( ξ ) 2 n ( n + 1)! . 8 / 20
Chebyshev Series vs Taylor Series IV Truncation Error : Taylor series, Lagrange formula: ∀ x ∈ [ − 1 , 1] , ∃ ξ ∈ [ − 1 , 1] s.t. f ( x ) − T ( x ) = f ( n +1) ( ξ ) ( n + 1)! ( x − x 0 ) n +1 . Chebyshev series, Bernstein-like formula: ∀ x ∈ [ − 1 , 1] , ∃ ξ ∈ [ − 1 , 1] s.t. f ( x ) − P ( x ) = f ( n +1) ( ξ ) 2 n ( n + 1)! . [ � ] We should have an improvement of 2 n in the width of the Chebyshev truncation error. 8 / 20
Quality of approximation of truncated Chebyshev series compared to best polynomial approximation It is well-known that truncated Chebyshev series π d ( f ) are near-best uniform approximations [Chap 5.5, Mason & Handscomb 2003]. 9 / 20
Quality of approximation of truncated Chebyshev series compared to best polynomial approximation It is well-known that truncated Chebyshev series π d ( f ) are near-best uniform approximations [Chap 5.5, Mason & Handscomb 2003]. Let p ∗ d is the polynomial of degree at most d that minimizes � f − p � ∞ = sup − 1 ≤ x ≤ 1 | f ( x ) − p ( x ) | . 9 / 20
Quality of approximation of truncated Chebyshev series compared to best polynomial approximation It is well-known that truncated Chebyshev series π d ( f ) are near-best uniform approximations [Chap 5.5, Mason & Handscomb 2003]. Let p ∗ d is the polynomial of degree at most d that minimizes � f − p � ∞ = sup − 1 ≤ x ≤ 1 | f ( x ) − p ( x ) | . � 4 � � f − p ∗ � f − π d ( f ) � ∞ � π 2 log d + O (1) d � ∞ (2) � �� � Λ d 9 / 20
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