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