Chebyshev Expansions for Solutions of Linear Differential Equations - PowerPoint PPT Presentation

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works Chebyshev Expansions for Solutions of Linear Differential Equations Alexandre Benoit, Joint work with Bruno Salvy INRIA October 28, 2009 1 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works I Introduction 2 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works How to evaluate a function f in [ − 1 , 1]? Two representations of f : in Taylor series + ∞ c n x n , c n = f ( n ) (0) � f = , n ! n =0 or in Chebyshev series + ∞ � f = t n T n ( x ) , n =0 � 1 t n = 2 f ( t ) T n ( t ) √ 1 − t 2 dt . π − 1 3 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works How to evaluate a function f in [ − 1 , 1]? Basic properties of Chebyshev polynomials Two representations of f : in Taylor series T n ( cos ( θ )) = cos( n θ ) + ∞ 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 = π if m = 0 1 − x 2 π − 1 otherwise or in Chebyshev series  2 T n +1 = 2 xT n − T n − 1 + ∞ � T 0 ( x ) = 1 f = t n T n ( x ) , n =0 T 1 ( x ) = x T 2 ( x ) = 2 x 2 − 1 � 1 t n = 2 f ( t ) T n ( t ) √ 1 − t 2 dt . T 3 ( x ) = 4 x 3 − 3 x π − 1 3 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works How to evaluate a function f in [ − 1 , 1]? Two representations of f : in Taylor series + ∞ c n x n , c n = f ( n ) (0) � f = , n ! n =0 or in Chebyshev series + ∞ � f = t n T n ( x ) , n =0 � 1 t n = 2 f ( t ) T n ( t ) √ 1 − t 2 dt . π − 1 Projects using Chebyshev series to represent functions in Matlab : Chebfun, Miscfun. 3 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works How to evaluate a function f in [ − 1 , 1]? Two representations of f : in Taylor series + ∞ c n x n , c n = f ( n ) (0) � f = , n ! n =0 or in Chebyshev series + ∞ � f = t n T n ( x ) , n =0 � 1 t n = 2 f ( t ) T n ( t ) √ 1 − t 2 dt . π − 1 Projects using Chebyshev series to represent functions in Matlab : Chebfun, Miscfun. How to compute t n ? General case: numerical computation of the integral. Slow. 3 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works Computation of Coefficients with Recurrences Theorem (60’s) If f is solution of a linear differential equation with polynomial coefficients, then the Chebyshev coefficients are cancelled by a linear recurrence with polynomial coefficients. Applications: Numerical computation of the coefficients. 4 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works Computation of Coefficients with Recurrences Theorem (60’s) If f is solution of a linear differential equation with polynomial coefficients, then the Chebyshev coefficients are cancelled by a linear recurrence with polynomial coefficients. Applications: Numerical computation of the coefficients. 4 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works Computation of Coefficients with Recurrences Theorem (60’s) If f is solution of a linear differential equation with polynomial coefficients, then the Chebyshev coefficients are cancelled by a linear recurrence with polynomial coefficients. Applications: Numerical computation of the coefficients. Computation of closed-form for coefficients. Example ( f ( x ) = arctan( x / 2)) 4 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works State of the Art Clenshaw (1957): numerical scheme to compute the Chebyshev coefficients without computing all these integrals. Fox and Parker (1968): method for the computation of the Chebyshev recurrence relations for differential equations of small orders. Paszkowski (1975): algorithm for computing the Chebyshev recurrence relation. Lewanowicz (1976): algorithm for computing a smaller order Chebyshev recurrence relation in some cases. Rebillard (1998): new algorithm for computing the Chebyshev recurrence relation. Rebillard and Zakrajˇ sek (2006): algorithm for computing a smaller order Chebyshev recurrence relation compared with Lewanowicz algorithm. 5 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works New Results (2009) Simple unified presentation of these algorithms using fractions of recurrence operators. Complexity analysis of the existing algorithms (order k , degree k ) Paszkowski’s and Lewanowicz’s algorithms: O ( k 4 ) arithmetic operations in Q . Rebillard’s algorithm: O ( k 5 ) arithmetic operations in Q . New fast algorithm: O ( k ω ) arithmetic operations. Here, ω is a feasible exponent for matrix multiplication with coefficients in Q ( ω ≤ 3). Implementation of algorithm in Maple. 6 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works II Fractions of Recurrence Operators 7 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works Morphisms of Rings of Operators ( S · u n = u n +1 ) := � c n x n ) Taylor series ( f c n x n +1 = X X c n − 1 x n , xf = f ′ = nc n x n − 1 = X X ( n + 1) c n +1 x n 8 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works Morphisms of Rings of Operators ( S · u n = u n +1 ) := � c n x n ) Taylor series ( f c n x n +1 = X X c n − 1 x n , xf = nc n x n − 1 = f ′ = X X ( n + 1) c n +1 x n x �→ X := S − 1 , d dx �→ D := ( n + 1) S . 8 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works Morphisms of Rings of Operators ( S · u n = u n +1 ) := � c n x n ) Taylor series ( f c n x n +1 = X X c n − 1 x n , xf = f ′ = nc n x n − 1 = X X ( n + 1) c n +1 x n x �→ X := S − 1 , d dx �→ D := ( n + 1) S . „ d « 2 (4 + x 2 ) + 2 x d dx dx �→ (4+ S − 2 )( n +1)( n +2) S 2 +2 S − 1 ( n + 1) S 4( n + 2) S 2 + n ` ´ = ( n + 1) 4( n + 2) c n +2 + nc n = 0 8 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works Morphisms of Rings of Operators ( S · u n = u n +1 ) Monomial Basis x n = M n ( x ) Chebyshev series xT n ( x ) =1 / 2 ( T n +1 ( x ) + T n − 1 ( x )) xM n ( x ) = M n +1 ( x ) , n ( x ) = n ( T n − 1 ( x ) − T n +1 ( x )) ( M n ( x )) ′ = nM n − 1 ( x ) . T ′ . 2(1 − x 2 ) x �→ X := S + S − 1 x �→ X := S − 1 , , 2 d dx �→ D := ( n + 1) S − ( n − 1) S − 1 d 2 n dx �→ D := ( n + 1) S . = S − 1 − S . 2(1 − X 2 ) „ d « 2 + 2 x d (4 + x 2 ) ( n + 2) S 2 + 18 n + ( n − 2) S − 2 ´ ` ( n − 1)( n + 1) dx dx , (( n − 1) S 2 − 2 n + ( n + 1) S − 2 ) �→ (4+ S − 2 )( n +1)( n +2) S 2 +2 S − 1 ( n +1) S 4( n + 2) S 2 + n ` ´ = ( n + 1) ( n + 2) t n +2 + 18 nt n + ( n − 2) t n − 2 = 0 . 4( n + 2) c n +2 + nc n = 0 8 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations

Introduction Fractions of Recurrence Operators Algorithms Conclusion and Future Works Morphisms of Rings of Operators ( S · u n = u n +1 ) Monomial Basis x n = M n ( x ) Chebyshev series xT n ( x ) =1 / 2 ( T n +1 ( x ) + T n − 1 ( x )) xM n ( x ) = M n +1 ( x ) , n ( x ) = n ( T n − 1 ( x ) − T n +1 ( x )) ( M n ( x )) ′ = nM n − 1 ( x ) . T ′ . 2(1 − x 2 ) x �→ X := S + S − 1 , x �→ X := S − 1 , 2 dx �→ D := ( n + 1) S − ( n − 1) S − 1 d 2 n d dx �→ D := ( n + 1) S . = S − 1 − S . 2(1 − X 2 ) „ d « 2 + 2 x d (4 + x 2 ) ( n + 2) S 2 + 18 n + ( n − 2) S − 2 ´ ( n − 1)( n + 1) ` dx dx , (( n − 1) S 2 − 2 n + ( n + 1) S − 2 ) �→ (4+ S − 2 )( n +1)( n +2) S 2 +2 S − 1 ( n +1) S 4( n + 2) S 2 + n ` ´ = ( n + 1) ( n + 2) t n +2 + 18 nt n + ( n − 2) t n − 2 = 0 . 4( n + 2) c n +2 + nc n = 0 8 / 19 Alexandre Benoit Chebyshev Expansions for Solutions of Linear Differential Equations