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# An application of the hyperfunction theory to numerical integration

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The slide of a speech in the conference "ECMI2016" (The 19th European Conference on Mathematics for Industry) held at Santiago de Compostela, Spain in June 2016.

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### An application of the hyperfunction theory to numerical integration

1. 1. 1 / 24 An Application of the Hyperfunction Theory to Numerical Integration ECMI2016 ∗Hidenori Ogata (The University of Electro-Communications, Japan) Hiroshi Hirayama (Kanagawa Institute of Technology, Japan) 17 June 2016
2. 2. Contents 2 / 24 ✓ ✏ We show an application of the hyperfunction theory, a generalized function theory based on complex analysis, to numerical computations, in particular, to numerical integrations. ✒ ✑
3. 3. Contents 2 / 24 ✓ ✏ We show an application of the hyperfunction theory, a generalized function theory based on complex analysis, to numerical computations, in particular, to numerical integrations. ✒ ✑ 1. Hyperfunction theory 2. Hyperfunction method for numerical integrations 3. Hyperfunction method for Hadamard’s ﬁnite parts 4. Numerical examples 5. Summary
4. 4. Contents 3 / 24 1. Hyperfunction theory 2. Hyperfunction method for numerical integrations 3. Hyperfunction method for Hadamard’s ﬁnite parts 4. Numerical examples 5. Summary
5. 5. 1. Hyperfunction theory (M. Sato, 1958) 4 / 24 b a φ(x)δ(x)dx = φ(0). 1 2πi C φ(z) z dz = φ(0). ( a < 0 < b ) Dirac delta function Cauchy integral formula O C
6. 6. 1. Hyperfunction theory (M. Sato, 1958) 4 / 24 b a φ(x)δ(x)dx = φ(0). 1 2πi C φ(z) z dz = φ(0). ( a < 0 < b ) Dirac delta function Cauchy integral formula O C Oa b 1 2πi C φ(z) z dz = − 1 2πi b a φ(x) 1 x + i0 − 1 x − i0 dx.
7. 7. 1. Hyperfunction theory (M. Sato, 1958) 4 / 24 b a φ(x)δ(x)dx = φ(0). 1 2πi C φ(z) z dz = φ(0). ( a < 0 < b ) Dirac delta function Cauchy integral formula O C Oa b b a φ(x)δ(x)dx = 1 2πi C φ(z) z dz = − 1 2πi b a φ(x) 1 x + i0 − 1 x − i0 dx. ∴ δ(x) = − 1 2πi 1 x + i0 − 1 x − i0 .
8. 8. 1. Hyperfunction theory 5 / 24 Hyperfunction theory (M. Sato, 1958)✓ ✏ • A generalized function theory based on complex analysis. • A hyperfunction f(x) on an interval I is the difference of the boundary values of an analytic function F(z). f(x) = [F(z)] ≡ F(x + i0) − F(x − i0). F(z) : the deﬁning function of the hyperfunction f(x) analytic in D I, where D is a complex neighborhood of the interval I. ✒ ✑ D I R
9. 9. 1. Hyperfunction theory: examples 6 / 24 • Dirac delta function δ(x) = − 1 2πiz = − 1 2πi 1 x + i0 − 1 x − i0 . • Heaviside step function H(x) = 1 ( x > 0 ) 0 ( x < 0 ) = − 1 2πi {log(−(x + i0)) − log(−(x − i0))} . ∗ log z is the principal value s.t. −π ≦ arg z < π • Non-integral powers x+ α = xα ( x > 0 ) 0 ( x < 0 ) = − (−(x + i0))α − (−(x − i0))α 2i sin(πα) (α ∈ Z const.). ∗ zα is the principal value s.t. −π ≦ arg z < π.
10. 10. 1. Hyperfunction theory: examples 7 / 24 Heaviside step function H(x) = 1 ( x > 0 ) 0 ( x < 0 ) = F(x+i0)−F(x−i0), F(z) = − 1 2πi log(−z). -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Re z -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Im z -1 -0.5 0 0.5 1 Re F(z) The real part of the deﬁning function F(z) = − 1 2πi log(−z).
11. 11. 1. Hyperfunction theory: examples 7 / 24 Heaviside step function H(x) = 1 ( x > 0 ) 0 ( x < 0 ) = F(x+i0)−F(x−i0), F(z) = − 1 2πi log(−z). -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Re z -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Im z -1 -0.5 0 0.5 1 Re F(z) Many functions with singularities are expressed by analytic functions in the hyperfunction theory.
12. 12. 1. Hyperfunction theory: integral 8 / 24 Integral of a hyperfunction f(x) = F(x + i0) − F(x − i0)✓ ✏ I f(x)dx ≡ − C F(z)dz C : closed path which encircles I in the positive sense and is included in D (F(z) is analytic in D I). ✒ ✑ D C I • The integral in independent of the choise of C by the Cauchy integral theorem.
13. 13. Contents 9 / 24 1. Hyperfunction theory 2. Hyperfunction method for numerical integrations 3. Hyperfunction method for Hadamard’s ﬁnite parts 4. Numerical examples 5. Summary
14. 14. 2. Hyperfunction method for numerical integrations 10 / 24 We consider the evaluation of an integral I f(x)w(x)dx, f(x) : analytic in a domain D s.t. (I ⊂ D ⊂ C), w(x) : weight function. D I R
15. 15. 2. Hyperfunction method for numerical integrations 10 / 24 We consider the evaluation of an integral I f(x)w(x)dx, f(x) : analytic in a domain D s.t. (I ⊂ D ⊂ C), w(x) : weight function. D I R We regard the integrand as a hyperfunction. ✓ ✏ f(x)w(x)χI(x) = − 1 2πi {f(x + i0)Ψ(x + i0) − f(x − i0)Ψ(x − i0)} with χI(x) = 1 (x ∈ I) 0 (x ∈ I) , Ψ(z) = I w(x) z − x dx. ✒ ✑
16. 16. 2. hyperfunction method for numerical integrations 11 / 24 From the deﬁnition of hyperfunction integrals, we have ✓ ✏ I f(x)w(x)dx = 1 2πi C f(z)Ψ(z)dz. = 1 2πi uperiod 0 f(ϕ(u))Ψ(ϕ(u))ϕ′ (u)du, C : z = ϕ(u) ( 0 ≦ u ≦ uperiod ) periodic function of period uperiod. ✒ ✑ D C : z = ϕ(u) I Approximating the r.h.s. by the trapezoidal rule, we have ...
17. 17. 2. Hyperfunction method for numerical integrations 12 / 24 Hyperfunction method✓ ✏ I f(x)w(x)dx ≃ h 2πi N−1 k=0 f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh), with Ψ(z) = b a w(x) z − x dx and h = uperiod N . ✒ ✑ D C : z = ϕ(u), 0 ≦ u ≦ uperiod I
18. 18. 2. Hyperfunction method for numerical integrations 12 / 24 Hyperfunction method✓ ✏ I f(x)w(x)dx ≃ h 2πi N−1 k=0 f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh), with Ψ(z) = b a w(x) z − x dx and h = uperiod N . ✒ ✑ Ψ(z) for typical weight functions w(x) I w(x) Ψ(z) (a, b) 1 log z − a z − b ∗ (0, 1) xα−1(1 − x)β−1 B(α, β)z−1F(α, 1; α + β; z−1)∗∗ ( α, β > 0 ) ∗ log z is the principal value s.t. −π ≦ arg z < π. ∗∗ F(α, 1; α + β, z−1 ) can be evaluated by the continued fraction.
19. 19. 2. Hyperfunction method for numerical integrations 13 / 24 The trapezoidal rule is efﬁcient for integrals of periodic analytic functions.
20. 20. 2. Hyperfunction method for numerical integrations 13 / 24 The trapezoidal rule is efﬁcient for integrals of periodic analytic functions. Theoretical error estimate✓ ✏ If f(ϕ(w)) and ϕ(w) are analytic in | Im w| < d0, |error| ≦ 2uperiod max Im w=±d |f(ϕ(w))Ψ(ϕ(w))ϕ′ (w)| × exp(−(2πd/uperiod)N) 1 − exp(−(2πd/uperiod)N) ( 0 < ∀d < d0 ). . . . Geometric convergence. ✒ ✑
21. 21. Contents 14 / 24 1. Hyperfunction theory 2. Hyperfunction method for numerical integrations 3. Hyperfunction method for Hadamard’s ﬁnite parts 4. Numerical examples 5. Summary
22. 22. 3. Hadamard’s ﬁnite parts 15 / 24 1 0 x−1 f(x)dx ( f(x) : ﬁnite as x → 0+ ) . . . divergent!
23. 23. 3. Hadamard’s ﬁnite parts 15 / 24 1 0 x−1 f(x)dx ( f(x) : ﬁnite as x → 0+ ) . . . divergent! Hadamard’s ﬁnite part✓ ✏ fp 1 0 x−1 f(x)dx ≡ lim ǫ→0+ 1 ǫ x−1 f(x)dx + f(0) log ǫ . ✒ ✑
24. 24. 3. Hadamard’s ﬁnite parts 15 / 24 1 0 x−1 f(x)dx ( f(x) : ﬁnite as x → 0+ ) . . . divergent! Hadamard’s ﬁnite part✓ ✏ fp 1 0 x−1 f(x)dx ≡ lim ǫ→0+ 1 ǫ x−1 f(x)dx + f(0) log ǫ . ✒ ✑ Hadamard’s ﬁnite part (n = 1, 2, . . .)✓ ✏ fp 1 0 x−n f(x)dx ≡ lim ǫ→+0 1 ǫ x−n f(x)dx + n−2 k=0 ǫk+1−n k!(k + 1 − n) f(k) (0) + log ǫ (n − 1)! f(n−1) (0) . ✒ ✑
25. 25. 3. Hadamard’s ﬁnite parts 16 / 24 Hadamard’s ﬁnite parts can be given by hyperfunction integrals. fp 1 0 x−n f(x)dx = 1 0 χ(0,1)x−n f(x)dx hyperfunction integral + n−2 k=0 f(k) (0) k!(k + 1 − n) = 1 2πi C z−n f(z) log z z − 1 dz approximated by the trapezoidal rule + n−2 k=0 f(k) (0) k!(k + 1 − n)
26. 26. Contents 17 / 24 1. Hyperfunction theory 2. Hyperfunction method for numerical integrations 3. Hyperfunction method for Hadamard’s ﬁnite parts 4. Numerical examples 5. Summary
27. 27. 4. Example 1: numerical integration 18 / 24 1 0 ex xα−1 (1−x)β−1 dx = B(α, β)F(α; α+β; 1) with α = β = 10−4 . We evaluated the integral by • the hyperfunction method • the DE formula (efﬁcient for integrals with end-point singularities) and compared the errors of the two methods. • C++ programs, double precision. • integral path for the hyperfunction method z = 0.5 + 2.575 cos u + i2.425 sin u, 0 ≦ u ≦ 2π (ellipse).
28. 28. 4. Example 1: numerical integrations 19 / 24 -16 -14 -12 -10 -8 -6 -4 -2 0 0 5 10 15 20 25 30 log10(relativeerror) N hyperfunction rule DE rule relative errors • the hyperfunction method error = O(0.024N ) (geometric convergence). • The DE formula does not work for this integral.
29. 29. 4. Example 1: Why the hyperfunction method works well? 20 / 24 integrand e z hyperfunction method • (DE rule) The sampling points accumulate at the singularities. • (hyperfunction method) The sampling points are distributed on a curve in the complex plane where the integrand varies slowly.
30. 30. 4. Example 2: Hadamard’s ﬁnite part 21 / 24 fp x 0 x−n ex dx = ∞ k=0(k=n−1) 1 k!(k − n + 1) ( n = 1, 2, . . . ). We computed it by the hyperfunction method. • C++ program & double precision • integral path z = 1 2 + 1 4 ρ + 1 ρ cos u + i 4 ρ − 1 ρ sin u, 0 ≦ u < 2π ( ρ = 10, ellipse ).
31. 31. 4. Example 2: Hadamard’s ﬁnite part 22 / 24 -16 -14 -12 -10 -8 -6 -4 -2 0 0 5 10 15 20 log10(relativeerror) N n=0 n=1 n=2 n=3 n=4 n=5 the relative errors of the hyperfunction method n 1 2 3 4 5 error O(0.021N ) O(0.023N ) O(0.018N ) O(0.034N ) O(0.032N ) ... geometric convergenc
32. 32. Contents 23 / 24 1. Hyperfunction theory 2. Hyperfunction method for numerical integrations 3. Hyperfunction method for Hadamard’s ﬁnite parts 4. Numerical examples 5. Summary
33. 33. 5. Summary 24 / 24 • The hyperfunction theory is a generalized function theory based on complex analysis. • The hyperfunction method approximately computes desired integral by evaluating the complex integrals which deﬁne them as hyperfunction integrals • Numerical examples show that the hyperfunction method is efﬁcient for integral with end-point singularities.
34. 34. 5. Summary 24 / 24 • The hyperfunction theory is a generalized function theory based on complex analysis. • The hyperfunction method approximately computes desired integral by evaluating the complex integrals which deﬁne them as hyperfunction integrals • Numerical examples show that the hyperfunction method is efﬁcient for integral with end-point singularities. functions with singularities (poles, discontinuities, delta functions, ...) ←−←−←− hyperfunction analytic functions
35. 35. 5. Summary 24 / 24 • The hyperfunction theory is a generalized function theory based on complex analysis. • The hyperfunction method approximately computes desired integral by evaluating the complex integrals which deﬁne them as hyperfunction integrals • Numerical examples show that the hyperfunction method is efﬁcient for integral with end-point singularities. functions with singularities (poles, discontinuities, delta functions, ...) ←−←−←− hyperfunction analytic functions We expect that we can apply the hyperfunction theory to a wide range of scientiﬁc computations. ! Gracias!