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- 1. 1 / 29 Numerical Integration based on the Hyperfunction Theory The 6th China-Japan-Korea Joint Conference on Numerical Mathematics ∗Hidenori Ogata (The University of Electro-Communications, Japan) joint work with Hiroshi Hirayama (Kanaga Institute of Technology, Japan) 23 August, 2016
- 2. Contents 2 / 29 An application of the hyperfunction theory to numerical analysis
- 3. Contents 2 / 29 An application of the hyperfunction theory to numerical analysis hyperfunction theory✓ ✏ generalized function theory proposed by M. Sato (1958) based on the complex function theory function with singularities pole discontinuity delta function, ... ←− complex analytic functions familiar in numerical computations ✒ ✑
- 4. Contents 2 / 29 An application of the hyperfunction theory to numerical analysis hyperfunction theory✓ ✏ generalized function theory proposed by M. Sato (1958) based on the complex function theory function with singularities pole discontinuity delta function, ... ←− complex analytic functions familiar in numerical computations ✒ ✑ In this speech, we show a numerical integration method based on the hyperfunction theory, which is expected to be efﬁcient for integrals with singularities.
- 5. Contents 3 / 29 1. Hyperfunction theory 2. Numerical integration over a ﬁnite interval 3. Numerical integration over an inﬁnite interval 4. Numerical examples 5. Summary
- 6. Contents 4 / 29 1. Hyperfunction theory 2. Numerical integration over a ﬁnite interval 3. Numerical integration over an inﬁnite interval 4. Numerical examples 5. Summary
- 7. 1. Hyperfunction theory (an example) 5 / 29 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 D φ(z) : analytic in D
- 8. 1. Hyperfunction theory (an example) 5 / 29 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 D Oa b φ(z) : analytic in D 1 2πi C φ(z) z dz = − 1 2πi b a φ(x) 1 x + i0 − 1 x − i0 dx.
- 9. 1. Hyperfunction theory (an example) 5 / 29 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 D Oa b φ(z) : analytic in D 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 .
- 10. 1. Hyperfunction theory 6 / 29 Hyperfunction theory (M. Sato, 1958)✓ ✏ • 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
- 11. 1. Hyperfunction theory: examples 7 / 29 • 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.
- 12. 1. Hyperfunction theory: examples 8 / 29 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).
- 13. 1. Hyperfunction theory: examples 8 / 29 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.
- 14. 1. Hyperfunction theory: integral 9 / 29 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.
- 15. 1. Hyperfunction theory: integral 9 / 29 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 I I f(x)dx = I {F(x + i0) − F(x − i0)} dx.
- 16. Contents 10 / 29 1. Hyperfunction theory 2. Numerical integration over a ﬁnite interval 3. Numerical integration over an inﬁnite interval 4. Numerical examples 5. Summary
- 17. 2. Numerical integration over a ﬁnite interval 11 / 29 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
- 18. 2. Numerical integration over a ﬁnite interval 11 / 29 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 Hilbert transform ✒ ✑
- 19. 2. Numerical integration over a ﬁnite interval 12 / 29 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 ...
- 20. 2. Numerical integration over a ﬁnite interval 13 / 29 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
- 21. 2. Numerical integration over a ﬁnite interval 13 / 29 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 easily evaluated by the continued fraction.
- 22. 2. Numerical integration over a ﬁnite interval 13 / 29 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 . ✒ ✑ In the hyperfunction method, we numerically evaluate the complex integral which deﬁnes the desired integral as a hyperfunction integral.
- 23. 2. Numerical integration over a ﬁnite interval 14 / 29 If f(x) is real valued for real x, we can reduce the number of sampling points N by half by the reﬂection principle. I f(x)w(x)dx ≃ h π Im 1 2 f(ϕ(0))Ψ(ϕ(0))ϕ′ (0) + 1 2 f(ϕ(Nh))Ψ(ϕ(Nh))ϕ′ (Nh) + N′ k=1 f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh) , where z = ϕ(u) is a parameterization of C s.t. ϕ(−u) = ϕ(u) and h = uperiod 2N′ . D C : z = ϕ(u) I
- 24. 2. Numerical integration over a ﬁnite interval (error estimate) 15 / 29 The hyperfunction method is very accurate. ∵ The trapezoidal rule is efﬁcient for the integrals of periodic analytic functions.
- 25. 2. Numerical integration over a ﬁnite interval (error estimate) 15 / 29 The hyperfunction method is very accurate. ∵ The trapezoidal rule is efﬁcient for the integrals of periodic analytic functions. Theoretical error estimate✓ ✏ If f(ϕ(w)) and ϕ(w) are analytic in | Im w| < d0, the error of the hyperfunction method (with N reduction by half) is bounded by the inequality |error| ≦ 2uperiod max Im w=±d |f(ϕ(w))Ψ(ϕ(w))ϕ′ (w)| × exp(−(4πd/uperiod)N) 1 − exp(−(4πd/uperiod)N) ( 0 < ∀d < d0 ). . . . Geometric convergence. ✒ ✑
- 26. Contents 16 / 29 1. Hyperfunction theory 2. Numerical integration over a ﬁnite interval 3. Numerical integration over an inﬁnite interval 4. Numerical examples 5. Summary
- 27. 3. Numerical integration over an inﬁnite interval 17 / 29 We consider the evaluation of an integral over the inﬁnite interval (0, +∞) ∞ 0 f(x)w(x)dx, f(x) : analytic in a domain D s.t. [0, +∞) ⊂ D ⊂ C, w(x) : weight function. D R O
- 28. 3. Numerical integration over an inﬁnite interval 17 / 29 We consider the evaluation of an integral over the inﬁnite interval (0, +∞) ∞ 0 f(x)w(x)dx, f(x) : analytic in a domain D s.t. [0, +∞) ⊂ D ⊂ C, w(x) : weight function. D R O We regard the integrand as a hyperfunction. ✓ ✏ f(x)w(x)H(x) = − 1 2πi {f(x + i0)Ψ(x + i0) − f(x − i0)Ψ(x − i0)} with H(x) = 1 (x > 0) 0 (x < 0) (Heaviside’s step function), w(x) xα−1 ( α > 0, α ∈ Z ) 1 Ψ(z) −πzα−1/ sin(πα) − log(−z) ✒ ✑
- 29. 3. Numerical integration over an inﬁnite interval 18 / 29 From the deﬁnition of hyperfunction integrals, we have ✓ ✏ ∞ 0 f(x)w(x)dx = 1 2πi C f(z)Ψ(z)dz = 1 2πi +∞ −∞ f(ϕ(u))Ψ(ϕ(u))ϕ′ (u)du, C : z = ϕ(u) ( −∞ < u < +∞ ). ✒ ✑ R D C : z = ϕ(u) O Approximating the r.h.s. by the trapezoidal rule, we have ...
- 30. 3. Numerical integration over an inﬁnite interval 19 / 29 Hyperfunction method✓ ✏ ∞ 0 f(x)w(x)dx ≃ h 2πi ∞ k=−∞ f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh) ≃ h 2πi N1 k=−N2 f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh). ✒ ✑ R D C : z = ϕ(u) O ∗ Actually, we use the DE transform to make the convergence of the inﬁnite sum fast.
- 31. 3. Numerical integration over an inﬁnite interval 19 / 29 Hyperfunction method✓ ✏ ∞ 0 f(x)w(x)dx ≃ h 2πi ∞ k=−∞ f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh) ≃ h 2πi N1 k=−N2 f(ϕ(kh))Ψ(ϕ(kh))ϕ′ (kh). ✒ ✑ R D C : z = ϕ(u) O ∗ We can reduce the number of sampling points N by half by the reﬂection principle also in the cases of inﬁnite intervals.
- 32. Contents 20 / 29 1. Hyperfunction theory 2. Numerical integration over a ﬁnite interval 3. Numerical integration over an inﬁnite interval 4. Numerical examples 5. Summary
- 33. 4. Example 1: numerical integration over a ﬁnite interval 21 / 29 1 0 ex xα−1 (1 − x)β−1 dx = B(α, β)F(α; α + β; 1) ( α, β > 0 ). We evaluated the integral by • the hyperfunction method (with N reduction by half) • 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 = 1 2 + 1 4 ρ + 1 ρ cos u + i 4 ρ − 1 ρ sin u ( ρ = 10 ) = 0.5 + 2.575 cos u + i2.425 sin u.
- 34. 4. Example 1: numerical integration over a ﬁnite interval 22 / 29 -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(relativeerror) N hyperfunction rule DE rule -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(relativeerror) N hyperfunction rule DE rule α = 0.5 α = 10−4 (very strong singularity) The errors of the hyperfunction method and the DE rule. hyperfunction method DE rule α = 0.5 O(0.025N ) O(0.36N ) α = 10−4 O(0.025N ) —
- 35. 4. Example 1: numerical integration over a ﬁnite interval 22 / 29 -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(relativeerror) N hyperfunction rule DE rule -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(relativeerror) N hyperfunction rule DE rule α = 0.5 α = 10−4 (very strong singularity) The errors of the hyperfunction method and the DE rule. hyperfunction method DE rule α = 0.5 O(0.025N ) O(0.36N ) α = 10−4 O(0.025N ) — The DE rule does not work if the end-point singularities are very strong.
- 36. 4. Example 1: numerical integration over a ﬁnite interval 22 / 29 -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(relativeerror) N hyperfunction rule DE rule -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(relativeerror) N hyperfunction rule DE rule α = 0.5 α = 10−4 (very strong singularity) The errors of the hyperfunction method and the DE rule. hyperfunction method DE rule α = 0.5 O(0.025N ) O(0.36N ) α = 10−4 O(0.025N ) — The convergence of the hyperfunction method is not affected by the end-point singularities.
- 37. 4. Example 2: numerical integration over a ﬁnite interval 23 / 29 1 0 xα−1(1 − x)β−1 1 + x2 dx = B(α, β) Re{F(α, 1; α + β; i)} ( α, β > 0 ). • integral path for the hyperfunction method z = 1 2 + 1 4 ρ + 1 ρ cos u + i 4 ρ − 1 ρ sin u ( ρ = 2 ) = 0.5 + 0.625 cos u + i0.375 sin u.
- 38. 4. Example 2: numerical integration over a ﬁnite interval 23 / 29 1 0 xα−1(1 − x)β−1 1 + x2 dx = B(α, β) Re{F(α, 1; α + β; i)} ( α, β > 0 ). -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(relativeerror) N hyperfunction method DE rule -16 -14 -12 -10 -8 -6 -4 -2 0 0 10 20 30 40 50 60 log10(relativeerror) N hyperfunction method DE rule α = 0.5 α = 10−4 (very strong singularity) The errors of the hyperfunction method and the DE rule.
- 39. 4. Example 2: numerical integration over a ﬁnite interval 23 / 29 1 0 xα−1(1 − x)β−1 1 + x2 dx = B(α, β) Re{F(α, 1; α + β; i)} ( α, β > 0 ). The errors of the hyperfunction method and the DE rule. hyperfunction method DE rule α = 0.5 O(0.18N ) O(0.54N ) α = 10−4 O(0.25N ) — • The DE rule does not work if the end-point singularities are very strong. • The convergence of the hyperfunction method is not affected by the end-point singularities.
- 40. 4. Example: Why the hyperfunction method works well? 24 / 29 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.
- 41. 4. Example: Why the hyperfunction method works well? 24 / 29 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. Thus, the hyperfunction method is not affected by the end-point singularities.
- 42. 4. Example 3: Numerical integration over an inﬁnite interval 25 / 29 ∞ 0 xα−1 1 + x2 dx = π/2 sin(πα/2) We computed it by the hyperfunction method (with N reduction by half) and the DE rule. • C++ program & double precision • integral path z = ϕ(u) = w(u) iπ log 1 + iw(u) 1 − iw(u) , w = sinh(sinh u) DE transform +0.5i. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Imz Re z
- 43. 4. Example 3: Numerical integration over an inﬁnite interval 26 / 29 -16 -14 -12 -10 -8 -6 -4 -2 0 0 20 40 60 80 100 log10(relativeerror) N hyperfunction method DE rule -16 -14 -12 -10 -8 -6 -4 -2 0 0 20 40 60 80 100 log10(relativeerror) N hyperfunction method DE rule α = 0.5 α = 10−4 (very strong singularity) The error of the hyperfunction method and the DE rule. hyperfunction method DE rule α = 0.5 O(0.51N ) O(0.34N ) α = 10−4 O(0.46N ) O(0.57N )
- 44. 4. Example 3: Numerical integration over an inﬁnite interval 26 / 29 -16 -14 -12 -10 -8 -6 -4 -2 0 0 20 40 60 80 100 log10(relativeerror) N hyperfunction method DE rule -16 -14 -12 -10 -8 -6 -4 -2 0 0 20 40 60 80 100 log10(relativeerror) N hyperfunction method DE rule α = 0.5 α = 10−4 (very strong singularity) The error of the hyperfunction method and the DE rule. hyperfunction method DE rule α = 0.5 O(0.51N ) O(0.34N ) α = 10−4 O(0.46N ) O(0.57N ) The convergence rate of the hyperfunction method is not affected by the end-point singularity.
- 45. 4. Example 4: Numerical integration over an inﬁnite interval 27 / 29 ∞ 0 xα−1 e−x dx = Γ(α) ( α > 0 ). • integral path for the hyperfunction method z = ϕ(u) = w(u) iπ log 1 + iw(u) 1 − iw(u) , w = sinh u DE transform +0.5i. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Imz Re z
- 46. 4. Example 4: Numerical integration over an inﬁnite interval 27 / 29 ∞ 0 xα−1 e−x dx = Γ(α) ( α > 0 ). -16 -14 -12 -10 -8 -6 -4 -2 0 0 20 40 60 80 100 log10(error) N alpha=0.5 alpha=0.1 alpha=0.01 alpha=1.0e-4 -16 -14 -12 -10 -8 -6 -4 -2 0 0 20 40 60 80 100 log10(error) N d=0.5 d=0.1 d=0.01 d=1.0e-4 hyperfunction method DE rule The erros of the hyperfunction method and the DE rule.
- 47. 4. Example 4: Numerical integration over an inﬁnite interval 27 / 29 ∞ 0 xα−1 e−x dx = Γ(α) ( α > 0 ). The errors of the hyperfunction method and the DE rule. α 0.5 0.1 0.01 10−4 hyperfunction rule O(0.46N ) O(0.46N ) O(0.46N ) O(0.46N ) error DE rule O(0.39N ) O(0.47N ) O(0.53N ) O(0.55N ) The convergence of the hyperfunction method is not affected by the end-point singularities.
- 48. Contents 28 / 29 1. Hyperfunction theory 2. Numerical integration over a ﬁnite interval 3. Numerical integration over an inﬁnite interval 4. Numerical examples 5. Summary
- 49. 5. Summary 29 / 29 • The hyperfunction theory is a generalized function theory based on the complex function theory. • 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.
- 50. 5. Summary 29 / 29 • The hyperfunction theory is a generalized function theory based on the complex function theory. • 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
- 51. 5. Summary 29 / 29 • The hyperfunction theory is a generalized function theory based on the complex function theory. • 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 Hyperfunctions connect singular functions with analytic functions. We expect that we can apply the hyperfunction theory to a wide range of scientiﬁc computations.
- 52. 5. Summary 29 / 29 • The hyperfunction theory is a generalized function theory based on the complex function theory. • 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 Hyperfunctions connect singular functions with analytic functions. We expect that we can apply the hyperfunction theory to a wide range of scientiﬁc computations. Thank you!