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
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Book
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Motivation
Related to large-scale projects:
I dlmf.nist.gov
“the NBS handbook rewritten with regard to the needsof today”
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Book
Maple
C++
3 / 63
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Book
Maple
C++
Motivation
Related to large-scale projects:
I dlmf.nist.gov
“the NBS handbook rewritten with regard to the needsof today”
I functions.wolfram.com
“the world´ s largest collection of formulas and graphicsavailable on the web”
4 / 63
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Book
Maple
C++
5 / 63
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Book
Maple
C++
Motivation
Related to large-scale projects:
I dlmf.nist.gov
“the NBS handbook rewritten with regard to the needsof today”
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
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Book
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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)
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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!
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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) + · · ·
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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
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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
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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
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Book. Part IPart IIPart IIIWeb
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Part I: Basic theory
I BasicsI Continued fraction representation of functionsI Convergence criteriaI Padé approximantsI Moment theory and orthogonal polynomials
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Part II: Numerics
I Continued fraction constructionI Truncation error boundsI Continued fraction evaluation
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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
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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).
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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
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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
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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
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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).
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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).
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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)
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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) – – –
– – –
– – –
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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
. . .
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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
. . .
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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
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BookPart IPart IIPart III. Web
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
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
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]
10
.1,.15,.2,.25,.3,.35,.4,.45
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BookPart IPart IIPart III. Web
Maple
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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
Input
parameters none
base 1010
approximant (1 ≤ approximant ≤ 999)
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
Special functions :continued fraction and series representations
Handbook
A. CuytW. B. JonesV. PetersenB. VerdonkH. Waadeland
Software
F. BackeljauwS. BecuweM. ColmanA. CuytT. Docx
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5,10,15,20,25
.25
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Maple library
symbolic:
f(z) = c0 + c1z+ c2z2 + . . .
f(z) = b0(z) +a1(z)
b1(z) +a2(z)
b2(z) +a3(z)
b3(z) + · · ·
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Web interface
Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
erf
dawson
erfc
Fresnel C
Fresnel S
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
functions
Bessel functions
Error function and related integrals » erfc » approximate
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]
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CFHB live | Error function and related integrals » erfc http://www.cfhblive.ua.ac.be/evaluate/index.php
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Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
erf
dawson
erfc
Fresnel C
Fresnel S
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
functions
Bessel functions
Error function and related integrals » erfc » approximate
Representation
Input
rhs J-fraction
parameters none
base 1010
digits (5 ≤ digits ≤ 999)
approximant (1 ≤ approximant ≤ 999)
z
tail estimatenone standard improved
user defined (simregular form)
Continue
Special functions :continued fraction and series representations
Handbook
A. CuytW. B. JonesV. PetersenB. VerdonkH. Waadeland
Software
F. BackeljauwS. BecuweM. ColmanA. CuytT. Docx
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CFHB live | Error function and related integrals » erfc http://www.cfhblive.ua.ac.be/evaluate/index.php
1 of 1 09/12/2008 03:12 PM
Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
erf
dawson
erfc
Fresnel C
Fresnel S
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
functions
Bessel functions
Error function and related integrals » erfc » approximate
Representation
Input
rhs J-fraction
parameters none
base 1010
digits (5 ≤ digits ≤ 999)
approximant (1 ≤ approximant ≤ 999)
z
tail estimatenone standard improved
user defined (simregular form)
Output
approximant 3.84214832712064746987580452585280852276469e-20
absolute error 1.792e-46
relative error 4.663e-27
tail estimate 0
Continue
Special functions :continued fraction and series representations
Handbook
A. CuytW. B. JonesV. PetersenB. VerdonkH. Waadeland
Software
F. BackeljauwS. BecuweM. ColmanA. CuytT. Docx
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CFHB live | Error function and related integrals » erfc http://www.cfhblive.ua.ac.be/evaluate/index.php
1 of 1 09/12/2008 03:13 PM
Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
erf
dawson
erfc
Fresnel C
Fresnel S
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
functions
Bessel functions
Error function and related integrals » erfc » approximate
Representation
Input
rhs J-fraction
parameters none
base 1010
digits (5 ≤ digits ≤ 999)
approximant (1 ≤ approximant ≤ 999)
z
tail estimatenone standard improved
user defined (simregular form)
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]
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CFHB live | Error function and related integrals » erfc http://www.cfhblive.ua.ac.be/evaluate/index.php
1 of 1 09/12/2008 03:14 PM
Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
erf
dawson
erfc
Fresnel C
Fresnel S
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
functions
Bessel functions
Error function and related integrals » erfc » approximate
Representation
Input
rhs J-fraction
parameters none
base 1010
digits (5 ≤ digits ≤ 999)
approximant (1 ≤ approximant ≤ 999)
z
tail estimatenone standard improved
user defined (simregular form)
Output
approximant 3.84214832712064746987580500914386200233213e-20
absolute error 4.654e-45
relative error 1.211e-25
tail estimate -6.67500000000000000000000000000000000000000e+01
Continue
Special functions :continued fraction and series representations
Handbook
A. CuytW. B. JonesV. PetersenB. VerdonkH. Waadeland
Software
F. BackeljauwS. BecuweM. ColmanA. CuytT. Docx
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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]
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. 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)
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. C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombineWeb
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 ?
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. C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombineWeb
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
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. C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombineWeb
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
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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
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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
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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)|
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C++Type S.IType S.II. SeriesType F.IType F.IIFractionsCombineWeb
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
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C++Type S.IType S.II. SeriesType F.IType F.IIFractionsCombineWeb
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)
)
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C++Type S.IType S.II. SeriesType F.IType F.IIFractionsCombineWeb
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
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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 + · · ·
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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
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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
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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
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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
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C++Type S.IType S.IISeriesType F.IType F.II. FractionsCombineWeb
Example:
exp(x2)√π erfc(x) =
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
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C++Type S.IType S.IISeriesType F.IType F.II. FractionsCombineWeb
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}
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C++Type S.IType S.IISeriesType F.IType F.II. FractionsCombineWeb
Example:
exp(x2)√π erfc(x) =
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
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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
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C++Type S.IType S.IISeriesType F.IType F.IIFractions. CombineWeb
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
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C++Type S.IType S.IISeriesType F.IType F.IIFractions. CombineWeb
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
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C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombine. Web
Web interface
Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
erf
dawson
erfc
Fresnel C
Fresnel S
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
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)
digits (5 ≤ digits ≤ 999)
verbose yes no
x
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]
1
40
6.5
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CFHB live | Error function and related integrals » erfc http://www.cfhblive.ua.ac.be/evaluate/index.php
1 of 1 09/12/2008 04:38 PM
Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
erf
dawson
erfc
Fresnel C
Fresnel S
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
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)
digits (5 ≤ digits ≤ 999)
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
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]
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CFHB live | Error function and related integrals » erfc http://www.cfhblive.ua.ac.be/evaluate/index.php
1 of 1 09/12/2008 03:16 PM
Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
erf
dawson
erfc
Fresnel C
Fresnel S
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
functions
Bessel functions
Error function and related integrals » erfc » approximate
Representation
Input
rhs J-fraction
parameters none
base 1010
digits (5 ≤ digits ≤ 999)
approximant (1 ≤ approximant ≤ 999)
z
tail estimatenone standard improved
user defined (simregular form)
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]
42
13
6.5
-3.691114343068670e-02
60 / 63
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Book
Maple
C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombine. Web
CFHB live | Error function and related integrals » erfc http://www.cfhblive.ua.ac.be/evaluate/index.php
1 of 1 09/12/2008 03:17 PM
Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
erf
dawson
erfc
Fresnel C
Fresnel S
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
functions
Bessel functions
Error function and related integrals » erfc » approximate
Representation
Input
rhs J-fraction
parameters none
base 1010
digits (5 ≤ digits ≤ 999)
approximant (1 ≤ approximant ≤ 999)
z
tail estimatenone standard improved
user defined (simregular form)
Output
approximant 3.84214832712064746987580454376877662144924e-20
absolute error 4.000e-61
relative error 1.041e-41
tail estimate -3.691114343068670e-02
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]
42
13
6.5
-3.691114343068670e-02
61 / 63
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C++Type S.IType S.IISeriesType F.IType F.IIFractionsCombine. Web
Function categories
Elementary functions
Gamma function and related
functions
Error function and related
integrals
Exponential integrals and
related functions
Hypergeometric functions
Confluent hypergeometric
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
base 10^k10^k k= (base 2^k: 1 ≤ k ≤ 24; base 10^k: 1 ≤ k ≤ 7)
digits (5 ≤ digits ≤ 999)
verbose yes no
x
n
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]
1
50
4.5
4
62 / 63
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!"#$%&'()%!*!$)++)&%,-./0'1.+%!2!34.5%67 8009:;;<<<=/,8>&'()=-?=?/=>);)(?&-?0);'.@)A=989
B%1,%B BC;BB;CDDE%DF:FF%GH
!"#$%&'#($)%*+',&*-
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1'$$'*+,%-&./%*'%2*(#"'
+,%-&./%0
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.%('"0
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+,%-&./%0
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'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
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CH>JIKJ>CBKJIBD@K*LBI4(CH>JIKJ>CBKJIBD@K*LBI
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%&'()'*+
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:*
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