# Convergence acceleration by the Ford-Sidi W(m) algorithm

## Computation of infinite integrals involving non-oscillating or oscillating functions (trigonometric or Bessel) multiplied by Bessel function of an arbitrary order

Although we deal with infinite integrals we will use ${\displaystyle {d}^{(m)}}$  transformation for infinite series [2] because these integrals are evaluated by accelerating the convergence of sequence of partial sums. Still using the ${\displaystyle {D}^{(m)}}$  transformation for infinite integrals [2] produces excellent results.The integrals are of the form :${\displaystyle \int _{0}^{\infty }c(x)w(x)dx}$  where c(x) is non-oscillating function or oscillating function (trigonometric or Bessel) and w(x) is Bessel  function of an arbitrary order.

These partial sums (as proposed by A. Sidi) are computed by integration between consecutive zeros of the Bessel function (see[6]) by a low order Gaussian quadrature (see[7]), although other numerical quadrature methods can be employed.

The Ford - Sidi ${\displaystyle {W}^{(m)}}$ algorithm routine is applied to these partial sums for convergence acceleration.

The ${\displaystyle {d}^{(m)}}$ transformation is implemented very efficiently by the Ford-Sidi ${\displaystyle {W}^{(m)}}$  algorithm [1] . Special case with m=1 (${\displaystyle {W}^{(1)}}$ algorithm) is the Sidi W algorithm [4].

This Ford-Sidi ${\displaystyle {W}^{(m)}}$  algorithm [1] is cheap mainly because set of sophisticated recursions that enable to compute the g function and the rest of the algorithm only up to the value m and not to the full length of LMAX+1 as needed by direct application of the Ford-Sidi algorithm.

We propose here  Ford-Sidi ${\displaystyle {W}^{(m)}}$ algorithm computed by direct application of the Ford-Sidi algorithm, even if it costs more

than the Ford-Sidi ${\displaystyle {W}^{(m)}}$ algorithm [1] because :

1. It is simple and short as it does not include those sophisticated recursions.

2. The values of the g function above m up to LMAX+1 are possible easily using the g(k)=t*g(k-m) (see 3.10 p. 1219 in [1]).

3. Most of the arrays dimensions are reduced to 1-D.

In computing the g function the simplest sampling of r(l)=l is chosen. It enables easily the efficiency of the code by saving previous values of the partial sums. Previous values of the g function can be also saved but it requests computing of two dimensional g instead of one in this algorithm. This sampling is also very useful for alternating series.

The proposed Ford-Sidi ${\displaystyle {W}^{(m)}}$  algorithm produces accurate results even with simplest sampling of  r(l)=l and small value of m=2.

This algorithm copes successfully with:

1. Integrals with non oscillating function multiplied by Bessel function of arbitrary order.
2. Weber Schafheitin discontinuous integrals with oscillating  functions as, for example, cos(x) and  cos(x)/x multiplied by Bessel function of order one  with short range of r=2 while most extrapolation algorithms (except Wynn's Epsilon) fail.
3. Integrals with oscillating function as Bessel multiplied by Bessel function of arbitrary order.
4. Integrals that involve Bessel function of very high order (see also [6] and the ${\displaystyle {\overline {D}}^{(m)}}$ transformation of Sidi [3]).
5. Integrals as (27)(28) that there integrands are not integrable at infinity and need to be defined in the sense of "Abel sums". (see[5]).

30 integrals with 6 different ranges of r=0.05, 1.0 , 2.0, 4.0, 10.0 and 100.0 are tested successfully.

The proposed Ford-Sidi ${\displaystyle {W}^{(m)}}$ algorithm recursive scheme is :

${\displaystyle FSA_{0}^{(n)}={\frac {S^{(n)}}{g_{1}^{n}}}\qquad n=0...Nmax}$

${\displaystyle FSI_{0}^{(n)}={\frac {1}{g_{1}^{(n)}}}\qquad n=0...Nmax}$

${\displaystyle G_{0,i}^{(n)}={\frac {g_{i}^{(n)}}{g_{1}^{(n)}}}\qquad n=0...Nmax\qquad i=2...Nmax+1}$

${\displaystyle FSA_{k+1}^{(n)}={\frac {FSA_{k}^{(n+1)}-FSA_{k}^{(n)}}{G_{k,k+2}^{(n+1)}-G_{k,k+2}^{(n)}}}\qquad n\geq 0\qquad k=0...Nmax-1}$

${\displaystyle FSI_{k+1}^{(n)}={\frac {FSI_{k}^{(n+1)}-FSI_{k}^{(n)}}{G_{k,k+2}^{(n+1)}-G_{k,k+2}^{(n)}}}\qquad n\geq 0\qquad k=0...Nmax-1}$

${\displaystyle G_{k+1,i}^{(n)}={\frac {G_{k,i}^{(n+1)}-G_{k,i}^{(n)}}{G_{k,k+2}^{(n+1)}-G_{k,k+2}^{(n)}}}\qquad n\geq 0\qquad k=0...Nmax-1\qquad i=k+3...Nmax+1}$

${\displaystyle APPROX={\frac {FSA_{N_{max}}^{(0)}}{FSI_{N_{max}}^{(0)}}}}$

Rewriting with mm=n-(k+1) :

${\displaystyle FSA_{0}^{(n)}={\frac {S^{(n)}}{g_{1}}}\qquad n=0...Nmax}$

${\displaystyle FSI_{0}^{(n)}={\frac {1}{g_{1}}}\qquad n=0...Nmax}$

${\displaystyle G_{i}^{(n)}={\frac {g_{i}}{g_{1}}}\qquad n=0...Nmax\qquad i=2...Nmax+1}$

${\displaystyle FSA_{k+1}^{(mm)}={\frac {FSA_{k}^{(mm+1)}-FSA_{k}^{(mm)}}{G_{k+2}^{(mm+1)}-G_{k+2}^{(mm)}}}\qquad n\geq 0\qquad k\geq 0}$

${\displaystyle FSI_{k+1}^{(mm)}={\frac {FSI_{k}^{(mm+1)}-FSI_{k}^{(mm)}}{G_{k+2}^{(mm+1)}-G_{k+2}^{(mm)}}}\qquad n\geq 0\qquad k\geq 0}$

${\displaystyle G_{i}^{(mm)}={\frac {G_{i}^{(mm+1)}-G_{i}^{(mm)}}{G_{k+2}^{(mm+1)}-G_{k+2}^{(mm)}}}\qquad n\geq 0\qquad k\geq 0\qquad i=k+3...Nmax+1}$

${\displaystyle APPROX={\frac {FSA_{N}^{(0)}}{FSI_{N}^{(0)}}}}$

Formulating the arrays in  1-D with mm=n-(k+1) :

${\displaystyle FSA^{(n)}={\frac {S^{(n)}}{g_{1}}}\qquad n=0...Nmax}$

${\displaystyle FSI^{(n)}={\frac {1}{g_{1}}}\qquad n=0...Nmax}$

${\displaystyle G_{i}^{(n)}={\frac {g_{i}}{g_{1}}}\qquad n=0...Nmax\qquad i=2...Nmax+1}$

${\displaystyle FSA^{(mm)}={\frac {FSA^{(mm+1)}-FSA^{(mm)}}{G_{k+2}^{(mm+1)}-G_{k+2}^{(mm)}}}\qquad n\geq 0\qquad k\geq 0}$

${\displaystyle FSI^{(mm)}={\frac {FSI^{(mm+1)}-FSI^{(mm)}}{G_{k+2}^{(mm+1)}-G_{k+2}^{(mm)}}}\qquad n\geq 0\qquad k\geq 0}$

${\displaystyle G_{i}^{(mm)}={\frac {G_{i}^{(mm+1)}-G_{i}^{(mm)}}{G_{k+2}^{(mm+1)}-G_{k+2}^{(mm)}}}\qquad n\geq 0\qquad k\geq 0\qquad i=k+3...Nmax+1}$

${\displaystyle APPROX={\frac {FSA^{(0)}}{FSI^{(0)}}}}$

with S=partial sum Nmax=number of partial sums

the simple and short semi formal code for the Ford - Sidi ${\displaystyle {W}^{(m)}}$ algorithm is :

 1 // MAIN PROGRAM
2
3 S = 0
4
5 //// calling FUNCTION WMALG that computes the Ford-Sidi W(m) algorithm for convergence acceleration.
6 // and also calling FUNCTION MLTAG that computes the g funcion and the partial sums
7
8
9
10
11 END PROGRAM
12
13 //Routine for computing Ford-Sidi W(m) algorithm
14 FUNCTION WMALG
15 // prevent zero denominator - not exists in [1].
16 IF (ABS(G1) >= 1D-77) THEN
17  FSA(N) = S / G(1)
18  FSI(N) = 1 / G(1)
19  FOR I=2 TO Nmax + 1 DO
20   FSG(I,N) = G(I) / G(1)
21  ENDFOR
22  ELSE
23  FSA(N) = S
24  FSI(N) = 1
25  FOR I=2 TO Nmax + 1 DO
26   FSG(I,N) = G(I)
27  ENDFOR
28  ENDIF
29  FOR K=0 TO N-1 DO
30   MM=N-(K+1)
31   D = FSG(K+2,MM+1) - FSG(K+2,MM)
32   FOR I=K+3 TO Nmax+1 DO
33    FSG(I,MM) = (FSG(I,MM+1) - FSG(I,MM)) / D
34   ENDFOR
35   FSA(MM) = (FSA(MM+1) - FSA(MM)) / D
36   FSI(MM) = (FSI(MM+1) - FSI(MM)) / D
37  ENDFOR
38 ENDFOR
39 // prevent zero denominator
40 IF (ABS(FSA(0)) >= 1D-77) THEN
41 APPROX = FSA(0) / FSI(0)
42 ELSE
43 APPROX = 1D-77
44 ENDIF
45
46 END FUNCTION


with AN = sequence element S = partial sum   the simple and short semi formal code for computing the partial sums and the g function is :

//routine for computing partial sums and g function
FUNCTION MLTAG
// making partial sums S
//the simplicity of this part of the code is enabled because the choice of the simplest sampling
//R(L)=L (see B.5 p. 1228 in [1]). It also enables efficiency of the code by saving previous values
//of the partial sums (S).
AN=
S =S+AN

// making g function up to M
FOR K=1 TO M DO
//computing the sequence elements AN(needed for the g function)

//efficiency of this part of the code can be also achived by saving previous results of the sequence elements
//AN that were computed already for the prtial sums S.
AN=  G(K)
ENDFOR

// forward difference
FOR I=2 TO M DO
FOR J=M TO I STEP=-1 DO
G(J)=G(J) - G(J-1)
ENDFOR
ENDFOR

T=1/(N +1)

// making g function from M to Nmax+1 (see 3.10 p. 1219 in [1]).
FOR K=1 TO Nmax+1 DO
IF K <= M THEN
G(K)=G(K) * ((N+1) ^ k)
ELSE
G(K)=T * G(K-M)
ENDIF
ENDFOR
ENDFOR

END FUNCTION


where ${\displaystyle \alpha ={\frac {(1+i)}{\sqrt {2}}}\qquad i={\sqrt {-1}}}$  for integrals with complex functions (1), (4) and (6)

## The 30 tested integrals, with exact values, are:

(1) ${\displaystyle \int _{0}^{\infty }ke^{-\alpha k^{2}}J_{0}(kr)dk={\frac {e^{-r^{2}/4\alpha }}{2\alpha }}}$   rapidly convergent - complex function


(2) ${\displaystyle \int _{0}^{\infty }e^{-k}J_{1}(kr)dk={\frac {{\sqrt {r^{2}+1}}-1}{r{\sqrt {r^{2}+1}}}}}$  rapidly convergent

(3) ${\displaystyle \int _{0}^{\infty }1\cdot J_{0}(kr)dk={\frac {1}{r}}}$  slowly convergent

(4)${\displaystyle \int _{0}^{\infty }{\frac {k}{\sqrt {k^{2}+\alpha ^{2}}}}J_{0}(kr)dk={\frac {e^{-\alpha r}}{r}}}$  slowly convergent - complex function

(5)${\displaystyle \int _{0}^{\alpha }kJ_{0}(kr)dk=0}$  algebraically divergent

(6)${\displaystyle \int _{0}^{\infty }k{\sqrt {k^{2}+\alpha ^{2}}}J_{0}(kr)dk=-{\frac {e^{-\alpha r}}{r^{3}}}(\alpha r+1)}$  algebraically divergent - complex function

(7)${\displaystyle \int _{0}^{\infty }\cos(k)J_{1}(kr)dk={\begin{cases}{\frac {{\sqrt {1-r^{2}}}-1}{r{\sqrt {1-r^{2}}}}}&{\text{if }}r<1\\{\frac {1}{r}}&{\text{if }}r\geq 1\end{cases}}}$  oscillatory kernel

(8) ${\displaystyle \int _{0}^{\infty }{\frac {\cos(k)}{k}}J_{1}(kr)dk={\begin{cases}{\frac {\sqrt {r^{2}-1}}{r}}&{\text{if }}r>1\\0&{\text{if }}r\leq 1\end{cases}}}$  oscillatory kernel

(9)${\displaystyle \int _{0}^{\infty }e^{-k}J_{2}(kr)dk={\frac {({{\sqrt {r^{2}+1}}-1})^{2}}{r^{2}{\sqrt {r^{2}+1}}}}}$  Bessel function order greater than 1

(10)${\displaystyle \int _{0}^{\infty }J_{1}(kr)\ J_{0}(ka)\ dk={\begin{cases}{\frac {1}{r}}&{\text{if }}r>a\\{\frac {1}{2a}}&{\text{if }}r=a\\0&{\text{if }}r  products of Bessel functions

(11) ${\displaystyle \int _{0}^{\infty }J_{4}(kr)J_{3}(ka)dk={\begin{cases}{\frac {a^{3}}{r^{4}}}&{\text{if }}r>a\\{\frac {1}{2a}}&{\text{if }}r=a\\0&{\text{if }}r  products of Bessel functions

(12)${\displaystyle \int _{0}^{\infty }J_{1}(kr)k^{2}e^{-c^{2}k^{2}}J_{0}(ka)dk=e^{-{\frac {a^{2}+r^{2}}{4c^{2}}}}\cdot {\frac {rI_{0}({\frac {ar}{2c^{2}}})-aI_{1}({\frac {ar}{2c^{2}}})}{4c^{4}}}}$  products of Bessel functions

(13)${\displaystyle \int _{0}^{\infty }J_{1}(kr)ke^{-c^{2}k^{2}}J_{0}(ka)dk=e^{-{\frac {a^{2}+r^{2}}{4c^{2}}}}\cdot {\frac {I_{0}({\frac {ar}{2c^{2}}})}{2c^{2}}}}$  products of Bessel functions

(14)${\displaystyle \int _{0}^{\infty }J_{1}(kr)ke^{-3k}dk={\frac {r}{(9+r^{2})^{1.5}}}}$  rapidly convergent

(15)${\displaystyle \int _{0}^{\infty }J_{0}(kr)k^{2}e^{-3k}dk={\frac {18-r^{2}}{(9+r^{2})^{2.5}}}}$  rapidly convergent

Integrals with r = 1.0

(16) ${\displaystyle \int _{0}^{\infty }k^{2}J_{0}(k)dk=-1}$  divergent oscillatory (see[5][7])

(17)${\displaystyle \int _{0}^{\infty }{\frac {1}{2}}\ln(1+k^{2})J_{1}(k)=0.421024438240708333}$  (see[7])

(18)${\displaystyle \int _{0}^{\infty }{\frac {k}{1+k^{2}}}J_{0}(k)dk=0.421024438240708333}$

(19)${\displaystyle \int _{0}^{\infty }{\frac {1-e^{-k}}{k\ln(1+{\sqrt {2}})}}J_{0}(k)=1}$

(20)${\displaystyle \int _{0}^{\infty }{\frac {k}{1+k^{2}}}J_{10}(k)dk=0.098970545308402}$  big order of Bessel function

(21)${\displaystyle \int _{0}^{\infty }{\frac {k}{1+k^{2}}}J_{100}(k)dk=0.0099989997000302}$  very big order of Bessel function

(22)${\displaystyle \int _{0}^{\infty }k^{2}J_{1}(k)[J_{0}(k)]^{2}dk={\frac {4}{3\pi {\sqrt {3}}}}}$  see[9] oscillatory kernel - product of Bessel functions

(23)${\displaystyle \int _{0}^{\infty }{\frac {1}{k}}J_{0}(k)J_{1}(k)dk={\frac {2}{\pi }}}$  oscillatory kernel - product of Bessel functions (see [3])

(24)${\displaystyle \int _{0}^{\infty }{\frac {1}{\sqrt {16+k^{2}}}}J_{0}(k)dk=I_{0}(2.0)K_{0}(2.0)}$  see [6]

(25)${\displaystyle \int _{0}^{\infty }{\frac {1}{\sqrt {16+k^{2}}}}J_{10}(k)dk=I_{5}(2.0)K_{5}(2.0)}$  big order of Bessel function (see [6])

(26)${\displaystyle \int _{0}^{\infty }{\frac {1}{\sqrt {16+k^{2}}}}J_{100}(k)dk=I_{50}(2.0)K_{50}(2.0)}$  very big order of Bessel function (see [6])

(27)${\displaystyle \int _{0}^{\infty }J_{0}(k)dk=1}$  (see[3])

(28)${\displaystyle \int _{0}^{\infty }k^{4}J_{0}(k)dk=9}$  divergent oscillatory (see[5][7])

(29)${\displaystyle \int _{0}^{\infty }J_{0}({\frac {k^{4}+2k^{2}+5}{k^{2}+4}}){\sqrt {k^{2}+9k+20}}\,\ dk=2.627160401844}$  very oscillatory (see[8])

(30)${\displaystyle \int _{0}^{\infty }J_{0}(2k){\frac {k{\sqrt {k^{2}+{\frac {1}{3}}}}(2k^{2}\cdot e^{-0.2{\sqrt {k^{2}+1}}}-(2k^{2}+1)e^{-0.2{\sqrt {k^{2}+{\frac {1}{3}}}}})}{(2k^{2}+1)^{2}-4k^{2}{\sqrt {k^{2}+{\frac {1}{3}}}}{\sqrt {k^{2}+1}}}}dk=0.02660899812797}$  (see[3])

## Computed results of the integrals compared with exact values - Excellent accuracy

IN=Integral No.  R=Radius(Range)  ACCUR=Requested Accuracy COMPR=Real Computed Result COMPI=Imaginary Computed Result EXACTR=Real Exact Result  EXACTI=Imaginary Exact Resu lt

IN    R      ACCUR         COMPR            COMPI         EXACTR                EXACTI
1   0.05   0.1E-09     0.3535533216    -0.3532409596    0.3535533216       -0.3532409596
2   0.05   0.1E-09    0.2495322244E-01  0.000000000     0.2495322244E-01   0.000000000
3   0.05   0.1E-09     20.00000000      0.000000000     20.00000000            0.000000000
4   0.05   0.1E-09     19.29318268     -0.6824013726    19.29318268            -0.6824013726
5   0.05   0.1E-09   -0.1558940106E-10   0.000000000    0.000000000           0.000000000
6   0.05   0.1E-09    -7999.770489       9.764355802    -7999.770489          9.764355802
7   0.05   0.1E-09   -0.2504697287E-01   0.000000000    -0.2504697287E-01   0.000000000
8   0.05   0.1E-09    0.1681838313E-14   0.000000000     0.000000000            0.000000000
9   0.05   0.1E-09    0.6234411536E-03   0.000000000     0.6234411536E-03    0.000000000
10   0.05   0.1E-09   -0.2336820089E-13   0.000000000     0.000000000           0.000000000
11   0.05   0.1E-09   -0.1966156371E-14   0.000000000     0.000000000           0.000000000
12   0.05   0.1E-09   -0.1925418594E-26   0.000000000     0.9196825005E-10   0.000000000
13   0.05   0.1E-09    0.3861415026E-04   0.000000000     0.3861415026E-04   0.000000000
14   0.05  0.1E-09  0.1851080515E-02  0.000000000   0.1851080515E-02  0.000000000
15   0.05  0.1E-09  0.7401237782E-01 000000000      0.7401237782E-01  0.000000000

1   2.00   0.1E-09     0.2457791604     -0.1928180249E-01   0.2457791604     -0.1928180249E-01
2   2.00   0.1E-09     0.2763932023       0.000000000       0.2763932023      0.000000000
3   2.00   0.1E-09     0.5000000000       0.000000000       0.5000000000      0.000000000
4   2.00   0.1E-09    0.1895626091E-01   -0.1200712156      0.1895626091E-01  -0.1200712156
5   2.00   0.1E-09     0.9174218254E-13   0.000000000       0.000000000       0.000000000
6   2.00   0.1E-09   -0.5389270093E-01   0.6576733896E-01  -0.5389270093E-01  0.6576733896E-01
7   2.00   0.1E-09     0.5000000000       0.000000000       0.5000000000      0.000000000
8   2.00   0.1E-09     0.8660254038       0.000000000       0.8660254038      0.000000000
9   2.00   0.1E-09     0.1708203932       0.000000000       0.1708203932      0.000000000
10   2.00   0.1E-09     0.2500000000       0.000000000       0.2500000000      0.000000000
11   2.00   0.1E-09     0.6250000000E-01   0.000000000       0.6250000000E-01  0.000000000
12   2.00   0.1E-09     0.5731056602E-05   0.000000000       0.5731052806E-05  0.000000000
13   2.00   0.1E-09     0.4696496560E-04   0.000000000       0.4696496450E-04  0.000000000
14   2.00  0.1E-09  0.4266924586E-01  0.000000000   0.4266924586E-01  0.000000000
15   2.00  0.1E-09  0.2297574777E-01  0.000000000   0.2297574777E-01  0.000000000

1   4.00   0.1E-09    -0.1344288395E-01  0.2631845721E-01  -0.1344288395E-01  0.2631845721E-01
2   4.00   0.1E-09     0.1893660937       0.000000000       0.1893660937       0.000000000
3   4.00   0.1E-09     0.2500000000       0.000000000       0.2500000000       0.000000000
4   4.00   0.1E-09    -0.1405775698E-01  -0.4552202582E-02 -0.1405775698E-01  -0.4552202582E-02
5   4.00   0.1E-09     0.2293554563E-13   0.000000000       0.000000000        0.000000000
6   4.00   0.1E-09     0.2558970306E-02   0.3574319813E-02  0.2558970306E-02   0.3574319813E-02
7   4.00   0.1E-09     0.2500000000       0.000000000       0.2500000000       0.000000000
8   4.00   0.1E-09     0.9682458366       0.000000000       0.9682458366       0.000000000
9   4.00   0.1E-09     0.1478525782       0.000000000       0.1478525782       0.000000000
10   4.00   0.1E-09     0.2500000000       0.000000000       0.2500000000       0.000000000
11   4.00   0.1E-09     0.3906250000E-02   0.000000000       0.3906250000E-02   0.000000000
12   4.00   0.1E-09     0.4230955773E-04   0.000000000       0.4230954559E-04   0.000000000
13   4.00   0.1E-09     0.7499947354E-04   0.000000000       0.7499947370E-04   0.000000000
14  4.00  0.1E-09  0.3200000000E-01  0.000000000   0.3200000000E-01  0.000000000
15  4.00  0.1E-09  0.6400000000E-03  0.000000000   0.6400000000E-03  0.000000000

1   10.00  0.1E-09   -0.3962101630E-08  -0.9735953642E-08  -0.3962101582E-08  -0.9735953664E-08
2   10.00  0.1E-09    0.9004962810E-01   0.000000000        0.9004962810E-01   0.000000000
3   10.00  0.1E-09    0.1000000000       0.000000000        0.1000000000       0.000000000
4   10.00  0.1E-09    0.5990701076E-04  -0.6020541162E-04   0.5990701076E-04  -0.6020541162E-04
5   10.00  0.1E-09    0.3664870892E-14   0.000000000        0.000000000        0.000000000
6   10.00  0.1E-09   -0.9092300944E-05   0.6231542441E-06  -0.9092300945E-05   0.6231542437E-06
7   10.00  0.1E-09    0.1000000000       0.000000000        0.1000000000       0.000000000
8   10.00  0.1E-09    0.9949874371       0.000000000        0.9949874371       0.000000000
9   10.00  0.1E-09    0.8149379340E-01   0.000000000        0.8149379340E-01   0.000000000
10   10.00  0.1E-09    0.1000000000       0.000000000        0.1000000000       0.000000000
11   10.00  0.1E-09    0.9999999998E-04   0.000000000        0.1000000000E-03   0.000000000
12   10.00  0.1E-09    0.3729561332E-03   0.000000000        0.3729561345E-03   0.000000000
13   10.00  0.1E-09    0.3869011865E-03   0.000000000        0.3869011875E-03   0.000000000
14  10.00  0.1E-09   0.8787397112E-02  0.000000000  0.8787397112E-02   0.000000000
15  10.00  0.1E-09  -0.6610702415E-03  0.000000000  -0.6610702415E-03  0.000000000

1   100.00  0.1E-09   -0.2222804096E-13  0.8523120307E-13   0.000000000        0.000000000
2   100.00  0.1E-09   0.9900005000E-02   0.000000000        0.9900005000E-02   0.000000000
3   100.00  0.1E-09   0.1000000000E-01   0.000000000        0.1000000000E-01   0.000000000
4   100.00  0.1E-09   0.2980559246E-12  -0.3626946695E-12  -0.4851871203E-34  -0.1952579141E-32
5   100.00  0.1E-09   0.1216602506E-15   0.000000000        0.000000000        0.000000000
6   100.00  0.1E-09   -0.7847883994E-17 -0.3434899908E-17  -0.1345888854E-34   0.1434515653E-34
7   100.00  0.1E-09   0.1000000000E-01   0.000000000        0.1000000000E-01   0.000000000
8   100.00  0.1E-09   0.9999499987       0.000000000        0.9999499987       0.000000000
9   100.00  0.1E-09   0.9801499937E-02   0.000000000        0.9801499938E-02   0.000000000
10   100.00  0.1E-09   0.1000000000E-01   0.000000000        0.1000000000E-01   0.000000000
11   100.00  0.1E-09   0.9999978411E-08   0.000000000        0.1000000000E-07   0.000000000
12   100.00  0.1E-09   0.6601490111E-13   0.000000000        0.2152044132E-37   0.000000000
13   100.00  0.1E-09  -0.4478087354E-14   0.000000000        0.4211988247E-27   0.000000000
14  100.00  0.1E-09  0.9986515138E-04  0.000000000   0.9986515172E-04  0.000000000
15  100.00  0.1E-09  -0.9959575539E-06  0.000000000 -0.9959575826E-06  0.000000000

16    1.00   0.1E-09    -1.000000000      0.000000000        -1.000000000        0.000000000
17    1.00   0.1E-09    0.4210244382      0.000000000         0.4210244382       0.000000000
18    1.00   0.1E-09    0.4210244382      0.000000000         0.4210244382       0.000000000
19    1.00   0.1E-09     1.000000000      0.000000000         1.000000000        0.000000000
20    1.00   0.1E-09    0.9897054531E-01  0.000000000         0.9897054531E-01   0.000000000
21    1.00   0.1E-09    0.9998998075E-02  0.000000000         0.9998999700E-02   0.000000000
22    1.00   0.1E-09    0.6366197724      0.000000000         0.6366197724       0.000000000
23    1.00   0.1E-09    0.2450350646      0.000000000         0.2450350646       0.000000000
24    1.00    0.1E-09    0.2596307983        0.000000000      0.2596308154        0.000000000
25    1.00    0.1E-09    0.9266646414E-01    0.000000000      0.9266646571E-01    0.000000000
26    1.00    0.1E-09    0.9992004781E-02    0.000000000      0.9607746870E-02    0.000000000
27    1.00    0.1E-09    1.0000000000    0.000000000      1.0000000000    0.000000000
28    1.00    0.1E-09    9.0000000000    0.000000000      9.0000000000    0.000000000
29    1.00    0.1E-09    2.637633938    0.000000000      2.627160401844    0.000000000
30    1.00    0.1E-09  0.2630108981E-01 0.000000000      0.2660899796E-01 0.000000000



## References

[1] W.F. Ford and A. Sidi. An algorithm for a generalization of the Richardson extrapolation process. SIAM J. Numer. Anal., 24:1212–1232, 1987.

[2] D. Levin and A. Sidi. Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series. Appl. Math. Comp., 9:175–215, 1981. Originally appeared as a Tel Aviv University preprint in 1975.

[3] A. Sidi. Extrapolation methods for oscillatory infinite integrals. J. Inst. Maths. Applics., 26:1–20, 1980.

[4] A. Sidi. An algorithm for a special case of a generalization of the Richardson extrapolation process. Numer. Math., 38:299–307, 1982.

[5] A. Sidi. Extrapolation methods for divergent oscillatory infinite integrals that are defined in the sense of summability. J. Comp. Appl. Math., 17:105–114, 1987.

[6] A. Sidi. Computation of infinite integrals involving Bessel functions of arbitrary order by the D¯-transformation. J. Comp. Appl. Math., 78:125–130, 1997.

[7] A. Sidi. A User-Friendly extrapolation method for oscillatory infinite integrals. Math. Comp., 51,249–266, 1988.

[8] A. Sidi. The numerical evaluation of very oscillatory infinite integrals by exrtapolation. Math. Comp., 38, 517-529, 1982.
]9] A. Sidi. A User Friendly extrapolation method for computing infinite integrals of products oscillatory functions. IMA J. Numer. Anal., 32:602-631, 2012.