Introduction Computation of the coefficients Validation Conclusion
Rigorous Uniform Approximation of D-finite Functions Alexandre Benoit (INRIA), Joint work with
Mioara Jolde¸s (ENS Lyon, INRIA) and Marc Mezzarobba (INRIA)
March 16, 2011
1 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
I Introduction
2 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Approximation of D-finite Functions Definition A function is D-finite if it is solution of a linear differential equation with polynomial coefficients.
Examples About 60% of Abramowitz & Stegun cos, arccos, Airy functions, Bessel functions, ...
3 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Approximation of D-finite Functions Definition A function is D-finite if it is solution of a linear differential equation with polynomial coefficients.
Examples About 60% of Abramowitz & Stegun cos, arccos, Airy functions, Bessel functions, ... How can we approximate a D-finite function f ? Polynomial approximation: f (x) ≈
n X
fi x i
i=0
3 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
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].
4 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
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
4 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
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
5 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
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
5 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
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
5 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Chebyshev Series vs Taylor Series I Basic properties of Chebyshev polynomials Two approximations of f : by Taylor series f =
+∞ X
Tn (cos(θ)) = cos(nθ)
cn x n , cn =
n=0
f
(n)
(0) , n!
+∞ X
tn =
1 π
2
tn Tn (x),
1
−1
0 Tn (x)Tm (x) √ π dx = π 1 − x2
if m 6= n if m = 0 otherwise
Tn+1 = 2xTn − Tn−1
n=−∞
Z
1
−1
or by Chebyshev series f =
Z
Tn (t) √
T0 (x) = 1 T1 (x) = x
f (t) dt. 1 − t2
T2 (x) = 2x 2 − 1 T3 (x) = 4x 3 − 3x
6 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Chebyshev Series vs Taylor Series I Error of approximation for exp(x) 0.05
Two approximations of f : by Taylor series f =
+∞ X
cn x n , cn =
n=0
f (n) (0) , n!
0.025
or by Chebyshev series +∞ X
f =
x −1
tn Tn (x),
n=−∞
1 tn = π
Z
−0.5
0
0.5
1
Taylor expansion of order 3
−0.025
1
f (t) Tn (t) √ dt. 1 − t2 −1
Chebyshev expansion of order 3
−0.05 6 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Chebyshev Series vs Taylor Series II Approximation of arctan(2x) by Taylor expansion of degree 1 1
0.5
−0.5
−0.25
0
0.25
x 0.5
−0.5 arctan(2x) −1
Taylor approximation 7 / 19
Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
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 −1
arctan(2x) Taylor approximation
−1.5
7 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Chebyshev Series vs Taylor Series II
Aspproximation of arctan(2x) by Chebyshev expansion of degree 1 1.5 1 0.5 x −1
−0.5
0
0.5
1
−0.5 −1
arctan(2x) Taylor approximation
−1.5
Chebyshev approximation 7 / 19
Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Previous Work
Computation of the Chebyshev coefficients for D-finite functions Using a relation between coefficients Clenshaw (1957) Using the recurrence relation between the coefficients Fox-Parker (1968) The tau method of Lanczos (1938), Ortiz (1969-1993)
Validation: Kaucher-Miranker (1984)
8 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Our Work
Given a linear differential equation with polynomial coefficients, boundary conditions and an integer d Compute a polynomial approximation p on [−1, 1] of degree d of the solution f in the Chebyshev basis in O(d) arithmetic operations. Compute a sharp bound B such that |f (x) − p(x)| < B, x ∈ [−1, 1] in O(d) arithmetic operations.
9 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
II Computation of the coefficients
10 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Chebyshev Series of D-finite Functions Theorem (60’s, BenoitJoldesMezzarobba11) P un Tn (x) is solution of a linear differential equation with polynomial coefficients iff the sequence un is cancelled by a linear recurrence with polynomial coefficients.
11 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Chebyshev Series of D-finite Functions Theorem (60’s, BenoitJoldesMezzarobba11) P un Tn (x) is solution of a linear differential equation with polynomial coefficients iff the sequence un is cancelled by a linear recurrence with polynomial coefficients. Recurrence relation + good initial conditions ⇒ Fast numerical computation of the coefficients P 1 n Taylor: exp = x n! Rec: u(n + 1) = u(n) n+1 u(0) = 1
1/0! = 1
u(1) = 1
1/1! = 1
u(2) = 0, 5
1/2! = 0, 5
.. . u(50) ≈ 3, 28.10−65
.. . 1/50! ≈ 3, 28.10−65 11 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Chebyshev Series of D-finite Functions Theorem (60’s, BenoitJoldesMezzarobba11) P un Tn (x) is solution of a linear differential equation with polynomial coefficients iff the sequence un is cancelled by a linear recurrence with polynomial coefficients. Recurrence relation + good initial conditions ⇒ Fast numerical computation of the coefficients P 1 n Taylor: exp = x n! Rec: u(n + 1) = u(n) n+1
P Chebyshev: exp = In (1)Tn (x) Rec: u(n + 1) = −2nu(n) + u(n − 1) u(0) = 1, 266
I0 (1) ≈ 1, 266
u(0) = 1
1/0! = 1
u(1) = 1
1/1! = 1
u(1) = 0, 565
I1 (1) ≈ 0, 565
1/2! = 0, 5
u(2) ≈ 0, 136
I2 (1) ≈ 0, 136
u(2) = 0, 5 .. . u(50) ≈ 3, 28.10−65
.. .
.. .
.. .
1/50! ≈ 3, 28.10−65 11 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Chebyshev Series of D-finite Functions Theorem (60’s, BenoitJoldesMezzarobba11) P un Tn (x) is solution of a linear differential equation with polynomial coefficients iff the sequence un is cancelled by a linear recurrence with polynomial coefficients. Recurrence relation + good initial conditions ⇒ Fast numerical computation of the coefficients P 1 n Taylor: exp = x n! Rec: u(n + 1) = u(n) n+1
P Chebyshev: exp = In (1)Tn (x) Rec: u(n + 1) = −2nu(n) + u(n − 1) u(0) = 1, 266
I0 (1) ≈ 1, 266
u(0) = 1
1/0! = 1
u(1) = 1
1/1! = 1
u(1) = 0, 565
I1 (1) ≈ 0, 565
1/2! = 0, 5
u(2) ≈ 0, 136
I2 (1) ≈ 0, 136
u(2) = 0, 5 .. . u(50) ≈ 3, 28.10−65
.. .
.. . 1/50! ≈ 3, 28.10−65
u(50) ≈ 4, 450.1067
.. . I50 (1) ≈ 2, 934.10−80 11 / 19
Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Convergent and Divergent Solutions of the Recurrence
Study of the Chebyshev recurrence If u(n) is solution, then there exists 1 another solution v (n) ∼ u(n)
Newton polygon of a Chebyshev recurrence
n
κ−2 = · · · = κ2 κ−3 κ3 S
12 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Convergent and Divergent Solutions of the Recurrence
Study of the Chebyshev recurrence
Newton polygon of a Chebyshev recurrence
n
If u(n) is solution, then there exists 1 another solution v (n) ∼ u(n)
κ−2 = · · · = κ2 κ−3 κ3 S
For the recurrence u(n + 1) + 2nu(n) − u(n − 1) Two independent solutions are In (1) ∼
1 (2n)!
and Kn (1) ∼ (2n)!
12 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Convergent and Divergent Solutions of the Recurrence
Study of the Chebyshev recurrence
Newton polygon of a Chebyshev recurrence
n
If u(n) is solution, then there exists 1 another solution v (n) ∼ u(n)
κ−2 = · · · = κ2 κ−3 κ3 S
For the recurrence u(n + 1) + 2nu(n) − u(n − 1) Two independent solutions are In (1) ∼
1 (2n)!
and Kn (1) ∼ (2n)!
Miller’s algorithm To compute the first N coefficients of the most convergent solution of a recurrence relation of order 2 Initialize u(N) = 0 and u(N − 1) = 1 and compute the first coefficients using the recurrence backwards Normalize u with the initial condition of the recurrence 12 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Algorithm for Computing the Coefficients
Algorithm Input: a differential equation of order r with boundary conditions Output: a polynomial approximation of degree N of the solution compute the Chebyshev recurrence of order 2s ≥ 2r for i from 1 to s using the recurrence relation backwards, compute the first N coefficients of the sequence u [i] starting with the initial conditions u [i] (N + 2s), · · · , u [i] (N + i), · · · , u [i] (N + 1) = (0, · · · , 1, · · · , 0)
combine the s sequences u [i] according to the r boundary conditions and the s − r symmetry relations
13 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Example: Back to exp
u(52) = 0
I52 (1) ≈ 2, 77.10−84
u(51) = 1
I51 (1) ≈ 2, 88.10−82
u(50) = −102
I50 (1) ≈ 2, 93.10−80
.. .
.. .
u(2) ≈ −4, 72.1080
I2 (1) ≈ 0, 14
81
u(1) ≈ 1, 96.10
u(0) ≈ −4, 4.10
I1 (1) ≈ −0, 57
81
I0 (1) ≈ 1, 27
14 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Example: Back to exp
u(52) = 0
I52 (1) ≈ 2, 77.10−84
u(51) = 1
I51 (1) ≈ 2, 88.10−82
u(50) = −102
I50 (1) ≈ 2, 93.10−80
.. .
.. .
u(2) ≈ −4, 72.1080
I2 (1) ≈ 0, 14
81
u(1) ≈ 1, 96.10
u(0) ≈ −4, 4.10 C =
50 X
I1 (1) ≈ −0, 57
81
I0 (1) ≈ 1, 27
u(n)Tn (0) ≈ −3, 48.1081
n=−50
14 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Example: Back to exp u(52) =0 C u(51) ≈ −2, 88.10−82 C u(50) ≈ 2, 93.10−80 C .. .
I52 (1) ≈ 2, 77.10−84 I51 (1) ≈ 2, 88.10−82 I50 (1) ≈ 2, 93.10−80 .. .
u(2) ≈ 0, 14 C u(1) ≈ −0, 57 C u(0) ≈ 1.27 C 50 X C = u(n)Tn (0) ≈ −3, 48.1081
I2 (1) ≈ 0, 14 I1 (1) ≈ −0, 57 I0 (1) ≈ 1, 27
n=−50
14 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
III Validation
15 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Fixed Point Theorem Applied to a Differential Equation f is solution of y 0 (x) − a(x)y (x) = 0, with y (0) = y0 , if and only if f is a fixed point of τ defined by Z t τ (y )(t) = y0 + a(x)y (x)dx. 0
16 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Fixed Point Theorem Applied to a Differential Equation f is solution of y 0 (x) − a(x)y (x) = 0, with y (0) = y0 , if and only if f is a fixed point of τ defined by Z t τ (y )(t) = y0 + a(x)y (x)dx. 0
For all rational functions a(x), there exists i such that τ i is a contraction map from the space of continuous functions to itself. We deduce: τ i (B(p, R)) ⊂ B(p, R) =⇒ f ∈ B(p, R)
16 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Fixed Point Theorem Applied to a Differential Equation f is solution of y 0 (x) − a(x)y (x) = 0, with y (0) = y0 , if and only if f is a fixed point of τ defined by Z t τ (y )(t) = y0 + a(x)y (x)dx. 0
For all rational functions a(x), there exists i such that τ i is a contraction map from the space of continuous functions to itself. We deduce: τ i (B(p, R)) ⊂ B(p, R) =⇒ f ∈ B(p, R)
Goal: Computation of R Given p, find i, pi and Ri such that τ i (B(p, R)) ⊂ B(pi , Ri ) ⊂ B(p, R) 16 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Algorithm for a Differential Equation of Order 1 Given p, find i, pi and Ri such that τ i (B(p, R)) ⊂ B(pi , Ri ) ⊂ B(p, R) Algorithm (Find R) p0 := p i
while i! < kak∞ Compute pi (t) a “good” approximation of y0 + Mi = kτ (pi−1 ) − pi k∞
Return R=
kpi − pk∞ + 1−
Pi
j=1
Mj
Rt 0
a(x)pi−1 (x)dx
kakj∞ j!
kaki∞ i!
17 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Algorithm for a Differential Equation of Order 1 Given p, find i, pi and Ri such that τ i (B(p, R)) ⊂ B(pi , Ri ) ⊂ B(p, R) Algorithm (Find R) p0 := p i
while i! < kak∞ Compute pi (t) a “good” approximation of y0 + Mi = kτ (pi−1 ) − pi k∞
Return R=
kpi − pk∞ + 1−
Pi
j=1
Mj
Rt 0
a(x)pi−1 (x)dx
kakj∞ j!
kaki∞ i!
f ∈ B(p, R) =⇒ f = τ i (f ) ∈ B(pi , Ri ), with Ri =
i j X kak∞ Mj |t|i i + kak∞ R j! i! j=1
17 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Algorithm for a Differential Equation of Order 1 Given p, find i, pi and Ri such that τ i (B(p, R)) ⊂ B(pi , Ri ) ⊂ B(p, R) Algorithm (Find R) p0 := p i
while i! < kak∞ Compute pi (t) a “good” approximation of y0 + Mi = kτ (pi−1 ) − pi k∞
Return R=
kpi − pk∞ + 1−
Pi
j=1
Mj
Rt 0
a(x)pi−1 (x)dx
kakj∞ j!
kaki∞ i!
f ∈ B(p, R) =⇒ f = τ i (f ) ∈ B(pi , Ri ), with Ri =
i j X kak∞ Mj |t|i i + kak∞ R j! i! j=1
kf − pk∞ ≤ kf − pi k∞ + kp − pi k∞ ≤ Ri + kp − pi k∞ = R Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
17 / 19
Introduction Computation of the coefficients Validation Conclusion
IV Conclusion
18 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
Introduction Computation of the coefficients Validation Conclusion
Final Algorithm
Algorithm INPUT: Differential equation with boundary conditions and a degree d OUTPUT: a polynomial approximation of degree d and a bound Compute an approximation P of degree d of the solution with the first algorithm Compute the bound B of the approximation with the second algorithm. return the pair P, B
19 / 19 Alexandre Benoit
Rigorous Uniform Approximation of D-finite Functions.
