convergence acceleration and continued fraction ...luminy09/slides/cuyt.pdf · of continued...

$: Convergence acceleration and Continued fraction ...luminy09/SLIDES/Cuyt.pdf · of continued fraction representations for these functions. This handbook is the result of such an endeavour$
Convergence acceleration andContinued fraction representations

Team:

A. Cuyt,V.B. Petersen, J. Van Deun, B. Verdonk, H. Waadeland,

F. Backeljauw, S. Becuwe, M. Colman,W.B. Jones

Universiteit Antwerpen

Sør-Trøndelag University College

Norwegian University of Science and Technology

University of Colorado at Boulder

1 / 63

. Project

Book

Maple

C++

Motivation

Related to large-scale projects:

I dlmf.nist.gov

“the NBS handbook rewritten with regard to the needsof today”

2 / 63

. Project

Book

Maple

C++

3 / 63

. Project

Book

Maple

C++

Motivation


I dlmf.nist.gov


I functions.wolfram.com

“the world´ s largest collection of formulas and graphicsavailable on the web”

4 / 63

. Project

Book

Maple

C++

5 / 63

. Project

Book

Maple

C++

Motivation


I dlmf.nist.gov


I functions.wolfram.com

“the world´ s largest collection of formulas and graphicsavailable on the web”

I Askey-Bateman project“will be an encyclopedia of special functions (Ismail /Van Assche)”

6 / 63

. Project

Book

Maple

C++

1 3

Handbook of Continued

Fractions for Special Functions

Special functions are pervasive in all fields of science and

industry. The most well-known application areas are in physics,

engineering, chemistry, computer science and statistics.

Because of their importance, several books and websites and a

large collection of papers have been devoted to these functions.

Of the standard work on the subject, namely the Handbook

of Mathematical Functions with Formulas, Graphs and

Mathematical Tables edited by Milton Abramowitz and Irene

Stegun, the American National Institute of Standards claims to

have sold over 700 000 copies!

But so far no project has been devoted to the systematic study

of continued fraction representations for these functions.

This handbook is the result of such an endeavour. We emphasise

that only 10% of the continued fractions contained in this book,

can also be found in the Abramowitz and Stegun project or at

special functions websites!

ISBN 978-1-4020-6948-2

Handbook ofContinued Fractions for Special Functions

11

Cuyt | Petersen | VerdonkW

aadeland | Jones

A. CuytV. Brevik PetersenB. Verdonk H. Waadeland W. B. Jones

7 / 63

. Project

Book

Maple

C++

Project

Deliverables:

I Book: formulas with tables and graphsI Maple library: all fractionsI C++ library: all functionsI Web version to explore: all of the above

(www.cfsf.ua.ac.be)

8 / 63

. Project

Book

Maple

C++

A lot of well-known constants in mathematics, physics andengineering, as well as elementary and special functions enjoyvery nice and rapidly converging series or continued fractionrepresentations.

“Algorithms with strict bounds on truncation and roundingerrors are not generally available for special functions ” (DanLozier)

Only 10% of the CF representations in CFSF handbook arealso found in the NBS handbook or at the Wolfram site!

9 / 63

. Project

Book

Maple

C++

special function: f(z)

continued fraction: b0(z) +K∞m=1

am(z)

bm(z)

f(z) = b0(z) +a1(z)

b1(z) +a2(z)

b2(z) +a3(z)

b3(z) + · · ·

= b0(z) +a1(z)

b1(z) +

a2(z)

b2(z) + · · ·

10 / 63

. Project

Book

Maple

C++

I SF: limit-periodic representationI CF: larger convergence domain

Example:

Atan(z) =

∞∑m=0

(−1)m+1

2m+ 1z2m+1,

|z| < 1

=z

1 +

∞Km=2

(m− 1)2z2

2m− 1,

iz 6∈ (−∞,−1) ∪ (1,+∞)

11 / 63

. Project

Book

Maple

C++

tail: tn(z) = K∞m=n+1

am(z)

bm(z)

Example:√1 + 4x− 1

2=

∞Km=1

x

1, x > −

14, tn(x) = f(x)

√2 − 1 =

11 +

21 +

11 +

21 +

. . . , t2n →√2 − 1

t2n+1 →√2

1 =∞Km=1

m(m+ 2)1

, tn →∞12 / 63

. Project

Book

Maple

C++

approximant:

fn(z; 0) = b0(z) +n

Km=1

am(z)

bm(z)

modified approximant:

fn(z;wn) = b0(z) +n−1

Km=1

am(z)

bm(z) +

an(z)

bn(z) +wn

13 / 63

Project

Book. Part IPart IIPart IIIWeb

Maple

C++

Part I: Basic theory

I BasicsI Continued fraction representation of functionsI Convergence criteriaI Padé approximantsI Moment theory and orthogonal polynomials

14 / 63

Project

BookPart I. Part IIPart IIIWeb

Maple

C++

Part II: Numerics

I Continued fraction constructionI Truncation error boundsI Continued fraction evaluation

15 / 63

Project

BookPart IPart II. Part IIIWeb

Maple

C++

Part III: Special functions

I Elementary functionsI Incomplete gamma and relatedI Error function and relatedI Exponential integralsI Hypergeometric 2F1(a,n; c; z), n ∈ ZI Confluent hypergeometric 1F1(n;b; z), n ∈ ZI Parabolic cylinder functionsI Legendre functionsI Bessel functions of integer and fractional orderI Coulomb wave functionsI Zeta function and related

16 / 63

Project

BookPart IPart IIPart III. Web

Maple

C++

Formulas

Ln(1 + z) =z

1 +

∞

Km=2

(amz

1

), |Arg(1 + z)| < π, (11.2.2) – – –

– – –

– – – AS

a2k =k

2(2k − 1), a2k+1 =

k

2(2k + 1)

=2z

2 + z +

∞

Km=2

(−(m− 1)2z2

(2m− 1)(2 + z)

), (11.2.3) – – –

– – –

– – –

|Arg(1− z2/(2 + z)2

)| < π

Ln(

1 + z

1− z

)=

2z

1 +

∞

Km=1

(amz2

1

), |Arg(1− z2)| < π, (11.2.4) – – –

– – –

– – – AS

am =−m2

(2m− 1)(2m + 1).

17 / 63

Project


Maple

C++

Tables

Table 11.2.3: Relative error of 20th partial sum and 20th approximants.

x Ln(1 + x) (11.2.1) (11.2.2) (11.2.3) (6.8.8)−0.9 −2.302585e+00 1.5e−02 2.8e−06 5.8e−12 1.5e−06

−0.4 −5.108256e−01 2.5e−10 1.8e−18 2.2e−36 2.4e−19

0.1 9.531018e−02 4.4e−23 5.3e−33 1.9e−65 1.3e−34

0.5 4.054651e−01 1.8e−08 1.9e−20 2.3e−40 2.0e−21

1.1 7.419373e−01 2.4e−01 2.8e−15 5.3e−30 5.5e−16

5 1.791759e+00 1.0e+13 4.2e−08 1.3e−15 2.0e−08

10 2.397895e+00 1.8e+19 5.3e−06 2.1e−11 3.3e−06

100 4.615121e+00 1.0e+40 1.8e−02 3.6e−04 2.5e−02

18 / 63

Project


Maple

C++

Modification:

since in (11.2.2), limm→∞ amz = z/4 and

limm→∞ am+1 − 1

4

am − 14

= −1,

we find

w(z) =−1 +

√1 + z

2and {

w(1)2k (z) = w(z) + kz

2(2k+1) − z4

w(1)2k+1(z) = w(z) +

(k+1)z2(2k+1) − z

4

19 / 63

Project


Maple

C++

Table 11.2.4: Relative error of 20th (modified) approximants.

x Ln(1 + x) (11.2.2) (11.2.2) (11.2.2)−0.9 −2.302585e+00 2.8e−06 1.9e−07 2.9e−10

−0.4 −5.108256e−01 1.8e−18 9.3e−20 4.3e−22

0.1 9.531018e−02 5.3e−33 2.7e−34 3.2e−37

0.5 4.054651e−01 1.9e−20 9.5e−22 5.5e−24

1.1 7.419373e−01 2.8e−15 1.5e−16 1.8e−18

5 1.791759e+00 4.2e−08 2.6e−09 1.1e−10

10 2.397895e+00 5.3e−06 3.7e−07 2.8e−08

100 4.615121e+00 1.8e−02 2.9e−03 9.2e−04

20 / 63

Project


Maple

C++

Graphs

Figure 11.2.1: Complex region where f8(z; 0) of (11.2.2) guarantees ksignificant digits for Ln(1 + z) (from light to dark k = 6, 7, 8, 9).

21 / 63

Project


Maple

C++

Figure 11.2.2: Number of significant digits guaranteed by the nth classicalapproximant of (11.2.2) (from light to dark n = 5, 6, 7) and the 5th modifiedapproximant evaluated with w

(1)5 (z) (darkest).

22 / 63

Project


Maple

C++

Web interface

2F1(1/2, 1; 3/2; z) =1

2√zLn(1 +√z

1 −√z

)I series representationI continued fraction representations:

I C-fraction (15.3.7)I M-fraction (15.3.12)I Nörlund fraction (15.3.17)

23 / 63

Project


Maple

C++

z 2F1 (1/2, 1; 3/2; z) =∞

Km=1

(cmz

1

), z ∈ C \ [1,+∞) (15.3.7a) – – –

– – –

– – –

with

c1 = 1, cm =−(m− 1)2

4(m− 1)2 − 1, m ≥ 2. (15.3.7b)

2F1(1/2, 1; 3/2; z) =1/2

1/2 + z/2 −z

3/2 + 3z/2 −4z

5/2 + 5z/2 − . . .(15.3.12) – – –

– – –

– – –

2F1(1/2, 1; 3/2; z) =1

1− z +z(1− z)

3/2 − 5/2z +

∞

Km=2

(m(m− 1/2)z(1− z)

(m + 1/2)− (2m + 1/2)z

),

<z < 1/2.(15.3.17) – – –

– – –

– – –

24 / 63

Project


Maple

C++

Table 15.3.1: Relative error of the 5th partial sum and 5th (modified)approximants. More details can be found in the Examples 15.3.1, 15.3.2and 15.3.3.

x 2F1(1/2, 1; 3/2;x) (15.3.7) (15.3.12) (15.3.17)0.1 1.035488e+00 1.9e−08 1.4e−05 6.1e−06

0.2 1.076022e+00 7.9e−07 4.4e−04 3.4e−04

0.3 1.123054e+00 8.0e−06 3.1e−03 4.9e−03

0.4 1.178736e+00 4.7e−05 1.2e−02 4.3e−02

x 2F1(1/2, 1; 3/2;x) (15.3.7) (15.3.7) (15.3.7)0.1 1.035488e+00 1.9e−08 2.0e−10 1.6e−12

0.2 1.076022e+00 7.9e−07 8.7e−09 1.5e−10

0.3 1.123054e+00 8.0e−06 9.4e−08 2.7e−09

0.4 1.178736e+00 4.7e−05 5.9e−07 2.5e−08

. . .

25 / 63

Project


Maple

C++

Table 15.3.2: Relative error of the 20th partial sum and 20th (modified)approximants. More details can be found in the Examples 15.3.1, 15.3.2and 15.3.3.

x 2F1(1/2, 1; 3/2;x) (15.3.7) (15.3.12) (15.3.17)0.1 1.035488e+00 4.0e−32 1.5e−20 1.5e−20

0.2 1.076022e+00 1.3e−25 1.5e−14 1.7e−13

0.3 1.123054e+00 1.4e−21 4.8e−11 7.6e−09

0.4 1.178736e+00 1.8e−18 1.4e−08 5.0e−05

x 2F1(1/2, 1; 3/2;x) (15.3.7) (15.3.7) (15.3.7)0.1 1.035488e+00 4.0e−32 2.6e−35 6.5e−38

0.2 1.076022e+00 1.3e−25 8.8e−29 4.8e−31

0.3 1.123054e+00 1.4e−21 1.1e−24 9.6e−27

0.4 1.178736e+00 1.8e−18 1.4e−21 1.9e−23

. . .

26 / 63

Project


Maple

C++

Function categories

Elementary functions

Gamma function and related

functions

Error function and related

integrals

Exponential integrals and

related functions

Hypergeometric functions

2F1(a,b;c;z)

2F1(a,n;c;z)

2F1(a,1;c;z)

2F1(a,b;c;z) /

2F1(a,b+1;c+1;z)

2F1(a,b;c;z) /

2F1(a+1,b+1;c+1;z)

2F1(1/2,1;3/2;z)

2F1(2a,1;a+1;1/2)

Confluent hypergeometric

functions

Bessel functions

Hypergeometric functions » 2F1(1/2,1;3/2;z) » tabulate

Representation

Continue

Special functions :continued fraction and series representations

Handbook

A. CuytW. B. JonesV. PetersenB. VerdonkH. Waadeland

Software

F. BackeljauwS. BecuweM. ColmanA. CuytT. Docx

© Copyright 2008 Universiteit Antwerpen [email protected]

27 / 63

Project


Maple

C++

Function categories



functions


integrals


related functions


2F1(a,b;c;z)

2F1(a,n;c;z)

2F1(a,1;c;z)

2F1(a,b;c;z) /

2F1(a,b+1;c+1;z)

2F1(a,b;c;z) /

2F1(a+1,b+1;c+1;z)

2F1(1/2,1;3/2;z)

2F1(2a,1;a+1;1/2)


functions

Bessel functions


Representation

Input

parameters none

base 1010

approximant (1 ≤ approximant ≤ 999)

z

output in tables relative error absolute error

Output

10th approximant (relative error)

z 2F1(1/2,1;3/2;z) (HY.1.4) (HY.3.7) (HY.3.7) (HY.3.7) (HY.3.12) (HY.3.12) (HY.3.12) (HY.3.17) (HY.3.17) (HY.3.17)

0.1 1.035488e+00 4.6e−13 2.4e−16 6.4e−19 3.0e−21 1.5e−10 4.5e−13 8.5e−15 7.5e−11 3.9e−12 5.4e−14

0.15 1.055046e+00 4.1e−11 1.8e−14 5.0e−17 3.6e−19 8.3e−09 2.8e−11 8.4e−13 7.6e−09 4.0e−10 8.6e−12

0.2 1.076022e+00 1.0e−09 4.3e−13 1.2e−15 1.2e−17 1.4e−07 5.4e−10 2.3e−11 2.4e−07 1.3e−08 4.0e−10

0.25 1.098612e+00 1.2e−08 5.4e−12 1.6e−14 2.1e−16 1.3e−06 5.5e−09 3.1e−10 4.2e−06 2.5e−07 9.8e−09

0.3 1.123054e+00 9.5e−08 4.6e−11 1.4e−13 2.3e−15 7.9e−06 3.8e−08 2.7e−09 5.1e−05 3.3e−06 1.7e−07

0.35 1.149640e+00 5.4e−07 3.0e−10 9.1e−13 1.9e−14 3.6e−05 1.9e−07 1.7e−08 4.9e−04 3.6e−05 2.4e−06

0.4 1.178736e+00 2.5e−06 1.6e−09 5.1e−12 1.3e−13 1.3e−04 8.3e−07 9.1e−08 4.0e−03 3.8e−04 3.4e−05

0.45 1.210806e+00 9.4e−06 7.5e−09 2.5e−11 7.4e−13 4.1e−04 3.0e−06 4.0e−07 3.1e−02 4.6e−03 6.7e−04

Continue


Handbook


Software



10

.1,.15,.2,.25,.3,.35,.4,.45

28 / 63

Project


Maple

C++

Function categories



functions


integrals


related functions


2F1(a,b;c;z)

2F1(a,n;c;z)

2F1(a,1;c;z)

2F1(a,b;c;z) /

2F1(a,b+1;c+1;z)

2F1(a,b;c;z) /

2F1(a+1,b+1;c+1;z)

2F1(1/2,1;3/2;z)

2F1(2a,1;a+1;1/2)


functions

Bessel functions


Representation

Input

parameters none

base 1010


z

output in tables relative error absolute error

Output

z = 1/4 (relative error)

n 2F1(1/2,1;3/2;z) (HY.1.4) (HY.3.7) (HY.3.7) (HY.3.7) (HY.3.12) (HY.3.12) (HY.3.12) (HY.3.17) (HY.3.17) (HY.3.17)

5 1.098612e+00 2.2e−05 2.8e−06 3.1e−08 7.3e−10 1.3e−03 2.1e−05 2.0e−06 1.4e−03 1.7e−04 1.2e−05

10 1.098612e+00 1.2e−08 5.4e−12 1.6e−14 2.1e−16 1.3e−06 5.5e−09 3.1e−10 4.2e−06 2.5e−07 9.8e−09

15 1.098612e+00 8.4e−12 1.0e−17 1.3e−20 1.2e−22 1.3e−09 2.5e−12 9.7e−14 1.4e−08 5.5e−10 1.5e−11

20 1.098612e+00 6.3e−15 2.0e−23 1.4e−26 1.0e−28 1.3e−12 1.4e−15 4.2e−17 5.1e−11 1.5e−12 3.0e−14

25 1.098612e+00 5.0e−18 3.8e−29 1.8e−32 1.0e−34 1.2e−15 8.7e−19 2.1e−20 1.9e−13 4.3e−15 7.2e−17

Continue


Handbook


Software



5,10,15,20,25

.25

29 / 63

Project

Book

Maple. LibraryWeb

C++

Maple library

symbolic:

f(z) = c0 + c1z+ c2z2 + . . .

f(z) = b0(z) +a1(z)

b1(z) +a2(z)

b2(z) +a3(z)

b3(z) + · · ·

30 / 63

Project

Book

MapleLibrary. Web

C++

Web interface

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions

Error function and related integrals » erfc » approximate

Representation

Continue


Handbook


Software



31 / 63

Project

Book

MapleLibrary. Web

C++

CFHB live | Error function and related integrals » erfc http://www.cfhblive.ua.ac.be/evaluate/index.php

1 of 1 09/12/2008 03:11 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010

digits (5 ≤ digits ≤ 999)


z

tail estimatenone standard improved

user defined (simregular form)

Continue


Handbook


Software



42

13

6.5

32 / 63

Project

Book

MapleLibrary. Web

C++


1 of 1 09/12/2008 03:12 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Output

approximant 3.84214832712064746987580452585280852276469e-20

absolute error 1.792e-46

relative error 4.663e-27

tail estimate 0

Continue


Handbook


Software



42

13

6.5

33 / 63

Project

Book

MapleLibrary. Web

C++


1 of 1 09/12/2008 03:13 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Continue


Handbook


Software



42

13

6.5

34 / 63

Project

Book

MapleLibrary. Web

C++


1 of 1 09/12/2008 03:14 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Output

approximant 3.84214832712064746987580500914386200233213e-20



tail estimate -6.67500000000000000000000000000000000000000e+01

Continue


Handbook


Software



42

13

6.5

35 / 63

Project

Book

MapleLibrary. Web

C++

∣∣∣∣f(6.5) − fn(6.5;wn)

f(6.5)

∣∣∣∣ 6 5× 10−40

I n = 25 and wn = 0I n = 20 and wn ∈ [−0.062489,−0.062487]I n = 15 andwn ∈ [−0.044392096002,−0.044392096001]

36 / 63

Project

Book

Maple

. C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombineWeb

C++ library

numeric:

f(x) = ±d0.d1 . . .dt−1 × βe

u(t) =12β−t+1,

∣∣∣∣f(x) − f(x)

f(x)

∣∣∣∣ 6 2αu(t), f(x) > 0

⇒f(x)

1 + 2αu(t)6 f(x) 6

f(x)

1 − 2αu(t)

f(x) ≈ F(x) ≈ F(x) = f(x)

37 / 63

Project

Book

Maple


truncation error:

f(x) ≈ F(x)

|f(x) − F(x)|

|f(x)|6 ?

round-off error:F(x) −→ F(x)

+,−,×,÷ ⊕,,⊗,�

|F(x) − F(x)||f(x)|

6 ?

38 / 63

Project

Book

Maple


Example:

√π

2erf(x) =

∞∑m=0

(−1)m

m! (2m+ 1)x2m+1, 0 6 x 6 1

|c2m+1|↘ 0

= x

(1 + x2 c3

c1

(1 + x2 c5

c3(1 + . . .)

))c2m+1

c2m−1=

2m− 1m(2m+ 1)

, 1 6 m 6 231 − 1

β = 10, t = 40,α = 1, x = 0.8↙ ↘∣∣∣∣f(x) − F(x)

f(x)

∣∣∣∣ 6 1210−39

∣∣∣∣F(x) − F(x)f(x)

∣∣∣∣ 6 1210−39

39 / 63

Project

Book

Maple


Example:

exp(x2)√π erfc(x) =

2x2x2+1

1 +

∞Km=1

am(x)

1, x > 0

=

2x2x2+1

1 +

∞Km=1

−2m(2m−1)(2x2+1+4m)(2x2+1+4(m−1))

1

0 > am(x)↘ −1/4

β = 10, t = 40,α = 1, x = 6.5↙ ↘∣∣∣∣f(x) − F(x)

f(x)

∣∣∣∣ 6 1210−39

∣∣∣∣F(x) − F(x)f(x)

∣∣∣∣ 6 1210−39

40 / 63

Project

Book

Maple

C++. Type S.IType S.IISeriesType F.IType F.IIFractionsCombineWeb

Type S.I

monotonely decreasing am > 0

f(x) =

∞∑m=0

amxm

Example:

exp(x) =

∞∑m=0

xm

m!, x ∈ R

41 / 63

Project

Book

Maple

C++Type S.I. Type S.IISeriesType F.IType F.IIFractionsCombineWeb

Type S.II

alternating am with |am| monotonely decreasing

Example:

erf(x) =2√π

∞∑m=0

(−1)mx2m+1

(2m+ 1)m!, x ∈ R

42 / 63

Project

Book

Maple

C++Type S.IType S.II. SeriesType F.IType F.IIFractionsCombineWeb

F(x) =

n∑m=0

cmxm:

0 6 cmxm ↘ 0 :

∣∣∣∣∣∞∑

m=n+1

cmxm

∣∣∣∣∣ 6 cnxn

1 − R,

cmx

cm−16 R < 1

0 6 c0, |cmxm|↘ 0 :

∣∣∣∣∣∞∑

m=n+1

cmxm

∣∣∣∣∣ 6 |cn+1xn+1|, n odd

|f(x)| > |F(x)|⇒∣∣∣∣f(x) − F(x)

f(x)

∣∣∣∣ 6{

cn|xn|/(1−R)|F(x)|

|cn+1xn+1|

|F(x)|

43 / 63

Project

Book

Maple


Example:

√π

2erf(x) =

∞∑m=0

(−1)m

m! (2m+ 1)x2m+1

|F(x)| > |x− x3/3| > 0.62

|c1x| = 0.80, |c3x3| = 0.17↘ |c61x61| 6 7.58× 10−41

n = 29 (2n+ 1 = 59),∣∣∣∣f(x) − F(x)

f(x)

∣∣∣∣ 6 1.27× 10−40

44 / 63

Project

Book

Maple


F(x) = c0

(1 + c1

c0x(1 + c2

c1x(1 + . . . + cn

cn−1x). . .))

:

r0 = c0, r0 = r0(1 + ρ0), |ρ0| 6ε(r0)u(s)

1−ε(r0)u(s)

rm =cm

cm−1x, rm = rm(1 + ρm), |ρm| 6 ε(rm)u(s)

1−ε(rm)u(s)

∣∣∣∣F(x) − F(x)f(x)

∣∣∣∣ 6 ε(F)u(s)

1 − ε(F)u(s)

F+(|x|)

|F(x)|

F+(x) =

n∑m=0

|cm|xm,F+(|x|)

|F(x)|> 1

ε(F) = 1 + ε(r0) + n

(2 +

nmaxm=1

ε(rm)

)

45 / 63

Project

Book

Maple


Example:

√π

2erf(x) =

∞∑m=0

c2m+1x2m+1

r0 = x, ε(r0) = 0, rm =x2(2m− 1)m(2m+ 1)

, ε(rm) = 4

n = 29, ε(F) = 175,F+(|x|)

|F(x)|6 2

ε(F)β−s+1

1 − ε(F)β−s+1/26

1210−39 ⇒ s > 43

46 / 63

Project

Book

Maple

C++Type S.IType S.IISeries. Type F.IType F.IIFractionsCombineWeb

Type F.I

monotone behaviour of am(x)

f(x) =∞Km=1

am(x)

1

=a1(x)

1 +a2(x)

1 +a3(x)

1 + · · ·

47 / 63

Project

Book

Maple

C++Type S.IType S.IISeries. Type F.IType F.IIFractionsCombineWeb

Example:

erfc(x) =∞Km=1

am(x)

1, x > 0

=

2x exp(−x2)√π(2x2+1)

1 +

∞Km=1

−2m(2m−1)(2x2+1+4m)(2x2+1+4(m−1))

1

am(x)↘ −1/4, tn(x)↘ −1/2

48 / 63

Project

Book

Maple

C++Type S.IType S.IISeriesType F.I. Type F.IIFractionsCombineWeb

Type F.II

interval bound for am(x) from m > µ on

f(x) =∞Km=1

am(x)

1

crude tµ interval → finer tn interval

49 / 63

Project

Book

Maple

C++Type S.IType S.IISeriesType F.I. Type F.IIFractionsCombineWeb

Example:

f(x) =2F1(a,k; c; x)

2F1(a,k+ 1; c+ 1; x)= 1 +

∞Km=1

amx

1,

x < 1

a2m+1 =−(a+m)(c− k+m)

(c+ 2m)(c+ 2m+ 1), m > 0

a2m =−(k+m)(c− a+m)

(c+ 2m− 1)(c+ 2m), m > 1

50 / 63

Project

Book

Maple

C++Type S.IType S.IISeriesType F.IType F.II. FractionsCombineWeb

F(x) = fn(x;wn),bm(z) = 1:

Ln 6 tn ≈ wn 6 Rn

|f(x) − F(x)|

|f(x)|6Rn − Ln

1 + Ln

n−1∏m=1

Mm

Mm = max{∣∣∣∣ Lm

1 + Lm

∣∣∣∣ , ∣∣∣∣ Rm

1 + Rm

∣∣∣∣} , m = 1, . . . ,n− 1

51 / 63

Project

Book

Maple


Example:


2x2x2 + 1 +

∞Km=2

am(x)

1

tn =∞K

m=n+1

am(x)

1, n > 1

Ln =∞K

m=n+1

−1/4

1= −

12

Rn =∞K

m=n+1

an+1(x)

1=

−1 +√

1 + 4an+1(x)

2

n = 13

M1 6 2.62× 10−4 ↗M12 6 3.43× 10−2

2(Rn − Ln) 6 2.64× 10−16,12∏m=1

Mm 6 1.27× 10−25

52 / 63

Project

Book

Maple


F(x) = fn(z;wn),bm(z) = 1:

am = am(1 + δm), max(|δ1|, . . . , |δn|) 6 ∆ u(s)

|F(x) − F(x)||f(x)|

6 (4 + ∆)Mn − 1M− 1

u(s)

M = max{M1, . . . ,Mn}

53 / 63

Project

Book

Maple


Example:


2x2x2 + 1 +

∞Km=1

am(x)

1

∆ =16

1 − 16u(s),

n = 13,

M = M13 6 3.84× 10−2

(4 +

161 − 8β−s+1

)Mn − 1M− 1

β−s+1

26

12β−39 ⇒ s > 42

54 / 63

Project

Book

Maple

C++Type S.IType S.IISeriesType F.IType F.IIFractions. CombineWeb

Combinations

×,÷ of previous types:

relative error of the form∣∣∣∣∣∏hi=1(1 + δi)∏h+ki=h+1(1 + δi)

∣∣∣∣∣is bounded by 1 + ε if for 1 6 i 6 h+ k

|δi| 6diε

1 + ε,

h+k∑i=1

di = 1

55 / 63

Project

Book

Maple


Example:

J5/2(x) = J1/2(x)J3/2(x)

J1/2(x)

J5/2(x)

J3/2(x)

J1/2(x) =

√2πx

sin(x)

Jν+1(x)

Jν(x)=x/(2ν+ 2)

1 +

∞Km=2

(ix)2/(4(ν+m− 1)(ν+m))

1,

x ∈ R, ν > 0

am(x)↗ 0

56 / 63

Project

Book

Maple


special function series continued fractionγ(a, x) a > 0, x 6= 0Γ(a, x) a ∈ R, x > 0erf(x) |x| 6 1 identity via erfc(x)erfc(x) identity via erf(x) |x| > 1

dawson(x) |x| 6 1 |x| > 1Fresnel S(x) x ∈ RFresnel C(x) x ∈ REn(x), n > 0 n ∈ N, x > 02F1(a,n; c; x) a ∈ R,n ∈ Z,

c ∈ R \ Z−0 , x < 1

1F1(n; c; x) n ∈ Z,c ∈ R \ Z−

0 , x ∈ RIn(x) n = 0, x ∈ R n ∈ N, x ∈ RJn(x) n = 0, x ∈ R n ∈ N, x ∈ R

In+1/2(x) n = 0, x ∈ R n ∈ N, x ∈ RJn+1/2(x) n = 0, x ∈ R n ∈ N, x ∈ R

57 / 63

Project

Book

Maple

C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombine. Web

Web interface

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions

Error function and related integrals » erfc » evaluate

Representation

Input

parameters none

base 10^k10^k k= (base 2^k: 1 ≤ k ≤ 24; base 10^k: 1 ≤ k ≤ 7)


verbose yes no

x

Continue


Handbook


Software



1

40

6.5

58 / 63

Project

Book

Maple



1 of 1 09/12/2008 04:38 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions

Error function and related integrals » erfc » evaluate

Representation

Input

parameters none



verbose yes no

x

Output

value 3.842148327120647469875804543768776621449e-20

absolute error bound 6.655e-61 (CF)

relative error bound 1.732e-41 (CF)

tail estimate -3.691114343068670e-02

working precision 42 (CF)

approximant 13

Continue


Handbook


Software



1

40

6.5

59 / 63

Project

Book

Maple



1 of 1 09/12/2008 03:16 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Continue


Handbook


Software



42

13

6.5

-3.691114343068670e-02

60 / 63

Project

Book

Maple



1 of 1 09/12/2008 03:17 PM

Function categories



functions


integrals

erf

dawson

erfc

Fresnel C

Fresnel S


related functions



functions

Bessel functions


Representation

Input

rhs J-fraction

parameters none

base 1010



z



Output

approximant 3.84214832712064746987580454376877662144924e-20



tail estimate -3.691114343068670e-02

Continue


Handbook


Software



42

13

6.5

-3.691114343068670e-02

61 / 63

Project

Book

Maple


Function categories



functions


integrals


related functions



functions

Bessel functions

J(nu, z)

J(n, z)

J(nu+1, z) / J(nu, z)

j(n, z)

j(n+1, z) / j(n, z)

I(nu, z)

I(n, z)

I(nu+1, z) / I(nu, z)

i(n, z)

i(n+1, z) / i(n, z)

Bessel functions » I(n, z) » evaluate

Representation

Input

parameters n



verbose yes no

x

n

Continue


Handbook


Software



1

50

4.5

4

62 / 63

Project

Book

Maple


!"#$%&'()%!*!$)++)&%,-./0'1.+%!2!34.5%67 8009:;;<<<=/,8>&'()=-?=?/=>);)(?&-?0);'.@)A=989

B%1,%B BC;BB;CDDE%DF:FF%GH

!"#$%&'#($)%*+',&*-

!"#$#%&'()*+,%-&./%0

1'$$'*+,%-&./%*'%2*(#"'&#2

+,%-&./%0

!((/(*+,%-&./%*'%2*(#"'&#2

.%&#3('"0

!45/%#%&.'"*.%&#3('"0*'%2

(#"'&#2*+,%-&./%0

6)5#(3#/$#&(.-*+,%-&./%0

7/%+",#%&*8)5#(3#/$#&(.-

+,%-&./%0

9#00#"*+,%-&./%0

:;%,<*=>

:;%<*=>

:;%,?@<*=>*A*:;%,<*=>

B;%<*=>

B;%?@<*=>*A*B;%<*=>

C;%,<*=>

C;%<*=>

C;%,?@<*=>*A*C;%,<*=>

.;%<*=>

.;%?@<*=>*A*.;%<*=>

.*--*/(0"#$%&'#-(1(23#4(56(1(*7)/")%*(

D#5(#0#%&'&./%

C%5,&

5'('$#&#(0 #

E'0#* !"#$!"#$ (89( (3:)-*(;<8=(>(?(8(?(;@A(:)-*(>B<8=(>(?(8(?(C6

2.3.&0* (3D(?(E&+&%-(?(FFF6

F#(E/0#* G*-( (#'(

4

%

G,&5,&

F'",# ;HCI@C;;;CJJFIBICKDDF@CB@DBJ>KID@I>>@JIBIDB;DJ>;JDC

'E0/",&#*#((/(*E/,%2 >H>BJ*LD>4(>HB>C*LDI4(;H>;K*LDI4(JHKD>*LDI(3M!64(>HBIJ*LDI(3NO6

(#"'&.F#*#((/(*E/,%2 >H;DJ*LD>4(>H@C>*LDI4(IHK>K*LDI4(>H@KJ*LD;(3M!64(DHFI>*LDD(3NO6

&'."*#0&.$'&#CHCIIBJ;F@@FIFKD@*LBI4(CH>JIKJ>CBKJIBD@K*LBI4(

CH>JIKJ>CBKJIBD@K*LBI4(CH>JIKJ>CBKJIBD@K*LBI

H/(I.%3*5(#-.0./% DC4(DF4(DK4(DC(3M!64(DJ(3NO6

'55(/4.$'%& ;D4(;D4(;@4(;I(3M!64(II(3NO6

%&'()'*+

NP*$&)/(0"#$%&'#-(=$'#%&#"*E(0,)$%&'#()#E(-*,&*-(,*P,*-*#%)%&'#-

Q)#E:''8

RH(M"G%SH(.H(T'#*-UH(V*%*,-*#.H(U*,E'#8QH(S))E*/)#E

N'0%W),*

!H(.)$8*/X)"WNH(.*$"W*YH(M'/Z)#RH(M"G%[H(\'$]

^(M'PG,&+_%(;BBK(J%.F#(0.&#.&*K%&H#(5#% $'#%)$%`$0_:/&7*H")H)$H:*

!

,"

-.,

-

63 / 63

convergence acceleration and continued fraction ...luminy09/slides/cuyt.pdf · of continued...

Documents