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# A Numerical Analytic Continuation and Its Application to Fourier Transform

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# A Numerical Analytic Continuation and Its Application to Fourier Transform

It is a slide for a talk given in the conference "ApplMath18" (9th Conference on Applied Mathematics and Scientific Computing, 17-20 September, 2018, Solaris, Sibenik, Croatia). We propose a numerical method of analytic continuation using continued fraction. From theoretical analysis and numerical examples, our method is so effective that it shows exponential convergence. We also apply our method to the computation of Fourier transforms.

It is a slide for a talk given in the conference "ApplMath18" (9th Conference on Applied Mathematics and Scientific Computing, 17-20 September, 2018, Solaris, Sibenik, Croatia). We propose a numerical method of analytic continuation using continued fraction. From theoretical analysis and numerical examples, our method is so effective that it shows exponential convergence. We also apply our method to the computation of Fourier transforms.

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### A Numerical Analytic Continuation and Its Application to Fourier Transform

1. 1. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary A Numerical Analytic Continuation and Its Application to Fourier Transform Hidenori Ogata Dept. Computer and Network Engineering, The Graduate School of Informatics and Engineering, The Univerisity of Electro-Communications, Tokyo, Japan 18 September, 2018 Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
2. 2. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary Contents 1 Proposition of a numerical method of the analytic continuation of analytic functions using continued fractions. 2 Application of the proposed method of analytic continuation to the computation of Fourier transforms. 3 Numerical examples which show the eﬀectiveness of the proposed method 4 Summary Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
3. 3. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analytic continuation using continued fractions Let’s consider an analytic function f (z) given in a Taylor series f (z) = ∞ n=0 cnzn and its analytic continuation. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
4. 4. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analytic continuation using continued fractions We get an analytic continuation of the analytic function f (z) by transforming it into a continued fraction. f (z) = ∞ n=0 cnzn ⇒ f (z) = a1 1 + a2z 1 + a3z 1 + ... . Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
5. 5. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analytic continuation using continued fractions We get an analytic continuation of the analytic function f (z) by transforming it into a continued fraction. f (z) = ∞ n=0 cnzn ⇒ f (z) = a1 1 + a2z 1 + a3z 1 + ... . Why continued fractions? The reasons are as follows. 1 It converges in a wide region. 2 It is easy to compute. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
6. 6. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analytic continuation using continued fractions 1 A continued fraction converges in a wide region. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
7. 7. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analytic continuation using continued fractions 1 A continued fraction converges in a wide region. Taylor series f (z) = ∞ n=0 cnzn . R converges in a disk |z| < R Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
8. 8. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analytic continuation using continued fractions 1 A continued fraction converges in a wide region. The Taylor series can be transformed into a continued fraction under some condition. continued fraction f (z) = a1 1 + a2z 1 + a3z 1 + ... R converges∗ in a plane with a cut ∗ There is a possibility that f (z) = ∞. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
9. 9. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analytic continuation using continued fractions 1 A continued fraction converges in a wide region. continued fraction f (z) = a1 1 + a2z 1 + a3z 1 + ... R analytic continuation disk → plane with a cut∗ ∗ There is a possibility that f (z) = ∞. We can get an analytic continuation of f (z) by transforming it into a continued fraction. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
10. 10. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analytic continuation using a continued fraction 2 A continued fraction is easy to compute. We can easily compute the continued fraction by the recurrence formula for the approximants fn(z) = pn(z) qn(z) ≡ a1 1 + a2z 1 + a3z 1 + · · · + anz 1 , p0 = 0, q0 = 1, p1 = a1, q1 = 1, pn(z) = anz pn−2(z) + pn−1(z) qn(z) = anz qn−2(z) + qn−1(z), ( n = 2, 3, . . .). The approximant converges fast, i.e., fn(z) → f (z) as n → ∞, exponentially. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
11. 11. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analyticity and convergence of the continued fraction Once we obtain the continued fraction, it satisﬁes the following theorem on its analyticity and convergence. f (z) = a1 1 + a2z 1 + a3z 1 + · · · , a = lim n→∞ an = 0, S : a plane with a cut as in the ﬁgure S a −1/(4a) O Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
12. 12. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Analyticity and convergence of the continued fraction f (z) = a1 1 + a2z 1 + a3z 1 + · · · , a = lim n→∞ an = 0, S a −1/(4a) O 1 The continued fraction is meromorphic in S. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
13. 13. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1.Analyticity and convergence of the continued fraction 2 Let T(⊂ S) be an arbitrary compact set. For arbitrary ǫ > 0, there exists m ∈ N s.t. |f ∗ m,n(z) − f ∗ m(z)| ≦ C(θT + ǫ)n ( ∀n > m, ∀z ∈ T ), S a −1/(4a) O T where f ∗ m(z) = amz 1 + am+1z 1 + · · · a tail of the continued fraction , f ∗ m,n(z) = amz 1 + am+1z 1 + · · · + anz 1 , θT ( 0 < θT < 1 ) : const. depending on T only. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
14. 14. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1.Analyticity and convergence of the continued fraction 2 Let T(⊂ S) be an arbitrary compact set. For arbitrary ǫ > 0, there exists m ∈ N s.t. |f ∗ m,n(z) − f ∗ m(z)| ≦ C(θT + ǫ)n ( ∀n > m, ∀z ∈ T ), exponential convergence S a −1/(4a) O T where f ∗ m(z) = amz 1 + am+1z 1 + · · · a tail of the continued fraction , f ∗ m,n(z) = amz 1 + am+1z 1 + · · · + anz 1 , θT ( 0 < θT < 1 ) : const. depending on T only. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
15. 15. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Numerical analytic continuation As seen above, once we obtain the continued fraction corresponding to the analytic function f (z), it gives an analytic continuation of f (z) which we can easily compute. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
16. 16. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Numerical analytic continuation As seen above, once we obtain the continued fraction corresponding to the analytic function f (z), it gives an analytic continuation of f (z) which we can easily compute. How can we obtain the continued fraction? Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
17. 17. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Numerical analytic continuation: QD algorithm How can we get the continued fraction corresponding to f (z) = cnzn Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
18. 18. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Numerical analytic continuation: QD algorithm How can we get the continued fraction corresponding to f (z) = cnzn Quotient-diﬀerence (QD) algorithm We generate the sequences e (n) k and q (n) k by e (n) 0 = 0, q (n) 1 = cn+1/cn ( n = 0, 1, 2, . . . ), e (n) k = q (n+1) k − q (n) k + e (n+1) k−1 , q (n) k+1 = (e (n+1) k /e (n) k )q (n+1) k ( n = 0, 1, . . .; k = 1, 2, . . . ). f (z) = c0 1 − q (0) 1 z 1 − e (0) 1 z 1 − q (0) 2 z 1 − e (0) 2 z 1 − · · · . Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
19. 19. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 1. Numerical analytic continuation: QD algorithm How can we get the continued fraction corresponding to f (z) = cnzn Quotient-diﬀerence (QD) algorithm We generate the sequences e (n) k and q (n) k by e (n) 0 = 0, q (n) 1 = cn+1/cn ( n = 0, 1, 2, . . . ), e (n) k = q (n+1) k − q (n) k + e (n+1) k−1 , q (n) k+1 = (e (n+1) k /e (n) k )q (n+1) k ( n = 0, 1, . . .; k = 1, 2, . . . ). f (z) = c0 1 − q (0) 1 z 1 − e (0) 1 z 1 − q (0) 2 z 1 − e (0) 2 z 1 − · · · . The QD algorithm is numerically unstable. ⇒ multiple precision arithmetics Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
20. 20. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 2. Numerical Fourier transform We consider a Fourier transform F[f ](ξ) = ∞ −∞ f (x)e−2πiξx dx. Familiar and important in science and engineering. It is diﬃcult to compute one by conventional numerical quadrature rules especially for slowly decaying functions f (x). We can compute F[f ](ξ) by the proposed method of analytic continuation. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
21. 21. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 2. Numerical Fourier transform Rewrite F[f ](ξ) = ∞ −∞ f (x)e−2πiξx dx (1) ⇓ F[f ](ξ) = lim ǫ→0+ 0 −∞ f (x)e−2πi(ξ+iǫ)x dx + +∞ 0 f (x)e−2πi(ξ−iǫ)x dx . e−2πǫ|x| : convergence factor We can deﬁne F[f ](ξ) even if the integral (1) does not exist in the conventional sense. ∗ deﬁnition of a Fourier transform in hyperfunction theory. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
22. 22. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 2. Numerical Fourier transform By rewriting F[f ](ξ), the problem of the Fourier transform is reduced to that of an analytic continuation. F[f ](ξ) = ∞ −∞ f (x)e−2πiξx dx = lim ǫ→0+ {F+(ξ + iǫ) − F−(ξ − iǫ)} ( ξ ∈ R ), where F+(ζ) = 0 −∞ f (x)e−2πiζx dx analytic in Im ζ > 0, F−(ζ) = − ∞ 0 f (x)e−2πiζx dx analytic in Im ζ < 0. We can get F[f ](ξ) by the analytic continuation of F±(ζ) onto R. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
23. 23. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 2. Numerical Fourier transform Actually, we get F[f ](ξ) by the following algorithm. 1 We compute F±(ζ) in ± Im ζ > 0. First, we choose ζ (±) 0 s.t. ± Im ζ (±) 0 > 0. F±(ζ) = ∞ n=0 c(±) n (ζ − ζ (±) 0 )n ( ± Im ζ (±) 0 > 0 ), c(±) n = 1 n! F (n) ± (ζ (±) 0 ) = ± 1 n! ∞ 0 (±2πix)n f (∓x)e±2πiζ (±) 0 x exponential decay dx. We can compute c (±) n by conventional numerical quadrature rules. 2 We get F[f ](ξ) by the analytic continuation of F±(ζ) onto R. F[f ](ξ) = lim ǫ→0+ {F+(ξ + iǫ) − F−(ξ − iǫ)} ( ξ ∈ R ). Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
24. 24. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 3. Numerical examples: Analytic continuation The analytic continuation of f (z) = 1 − z 2 + z2 3 − · · · = log(1 + z) z ( |z| < 1 ). Throughout this study, we carried out all the computations using C++ programs in 100 decimal digit precision (using exﬂib) Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
25. 25. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 3. Numerical examples: Analytic continuation The analytic continuation of f (z) = 1 − z 2 + z2 3 − · · · = log(1 + z) z ( |z| < 1 ). f (z) = a1 1 + a2z 1 + a3z 1 + · · · , n an n an 1 1.0000 . . . 8 0.28571 42857 14285 . . . 2 0.5000 . . . 9 0.22222 . . . 3 0.16666 . . . 10 0.27777 . . . 4 0.33333 . . . 11 0.22727 27272 72727 . . . 5 0.20000 . . . 12 0.27272 72727 27272 . . . 6 0.30000 . . . 13 0.23076 92307 69230 . . . 7 0.21428 57142 85714 . . . 14 0.26923 07692 30769 . . . Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
26. 26. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 3. Numerical examples: Analytic continuation The analytic continuation of f (z) = 1 − z 2 + z2 3 − · · · = log(1 + z) z ( |z| < 1 ). The error of the analytic continuation of f (z). -3 -2 -1 0 1 2 3Re(z) -3 -2 -1 0 1 2 3 Im(z) 1.0e-50 1.0e-40 1.0e-30 1.0e-20 1.0e-10 1.0e+00 |error| 1.0e-60 1.0e-50 1.0e-40 1.0e-30 1.0e-20 1.0e-10 1.0e+00 -3 -2 -1 0 1 2 3 |error| Re(z) Im(z)=0 Im(z)=0.5 Im(z)=1 We can compute the analytic continuation in S = C {x ≦ −1}. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
27. 27. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 3. Numerical examples: Analytic continuation (1) f (z) = 1 − z/2 + z2 /3 − · · · (= z−1 log(1 + z)) ( |z| < 1 ), (2) f (z) = z − z3 /3 + z5 /5 − · · · (= arctan z) ( |z| < 1 ). -35 -30 -25 -20 -15 -10 -5 0 0 5 10 15 20 25 30 35 40 log10(error) n z=1 z=2 z=1+i -35 -30 -25 -20 -15 -10 -5 0 0 5 10 15 20 25 30 35 40 log10(error) n z=1 z=2 z=1+i (1) (2) Errors with the continued fractions truncated at the n-th term. ( vertical axis: log10(error), horizontal axis: n ) Our method gives exponential convergence. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
28. 28. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 3. Numerical examples: Zeta functions We can also apply the numerical analytic continuation to the compution of ζ(s). ζ(s) = ∞ n=1 1 ns = fs (1) 1 − 21−s , where fs (z) = ∞ n=0 (−1)n (n + 1)s zn (|z| < 1 ). We compute fs(1) by the numerical analytic continuation. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
29. 29. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 3. Numerical examples: Zeta functions We can also apply the numerical analytic continuation to the compution of ζ(s). ζ(s) = ∞ n=1 1 ns = fs (1) 1 − 21−s , where fs (z) = ∞ n=0 (−1)n (n + 1)s zn (|z| < 1 ). We compute fs(1) by the numerical analytic continuation. Results with the continued fractions truncated at the n-th term. n ζ(3) error 5 1.2020 46303 07249 1.1e-05 10 1.2020 56904 61243 1.5e-09 15 1.2020 56903 15940 2.0e-13 20 1.2020 56903 15959 2.9e-17 · · · exact value 1.2020 56903 15959 . . . Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
30. 30. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 3. Numerical examples: Fourier transforms The Fourier transforms (1) F[tanh(πx)](ξ), (2) F[(1+x2 )−2 ](ξ), (3) F[log |x|](ξ). 1.0e-45 1.0e-40 1.0e-35 1.0e-30 1.0e-25 1.0e-20 1.0e-15 1.0e-10 1.0e-05 1.0e+00 -4 -2 0 2 4 |error| xi (1) (2) (3) vertical axis: error horizontal axis: ξ Our method works well (especially for (1)). Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
31. 31. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 3. Numerical examples Comparison with the previous methods 1 Sugihara’s method using the DE rule & the Richardson extrapolation (1987) 2 DE-type rule for oscillatory integrals by T. Ooura & Mori (1991) Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
32. 32. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary 3. Numerical examples Comparison with the previous methods 1 Sugihara’s method using the DE rule & the Richardson extrapolation (1987) 2 DE-type rule for oscillatory integrals by T. Ooura & Mori (1991) F[tanh(πx)](ξ = 1) = −i cosechπ. number of the evaluations of f (x) = tanh(πx) error our method 666 7.4e-43 (1) DE & Richardson 17156 7.8e-21 (2) DE for oscillatory integrals 1892 1.5e-46 Our method is superior to the previous methods for this example. Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
33. 33. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary Summary 1 We proposed a numerical method of analytic continuation using continued fractions 2 We applied the method to Fourier transforms 3 Numerical examples show the eﬀectiveness of our method. Problem for future study 1 reduce the cost of transforming a given analytic function into a continued fraction (The present method needs multiple precision arithmetics due to the instability of the QD algorithm). Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier
34. 34. Numerical analytic continuation Numerical Fourier transform Numerical examples Summary Summary 1 We proposed a numerical method of analytic continuation using continued fractions 2 We applied the method to Fourier transforms 3 Numerical examples show the eﬀectiveness of our method. Problem for future study 1 reduce the cost of transforming a given analytic function into a continued fraction (The present method needs multiple precision arithmetics due to the instability of the QD algorithm). Thank you very much! Hidenori Ogata A Numerical Analytic Continuation and Its Application to Fourier