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A data fitting approach to series convergence acceleration
A Data
Fitting
K. J. Bunch,
Approach
to Series
Convergence
W. N. Cain, and R. W. Grow
Microwave Device and Physical Electronics Department of Electrical Engineering University of Utah Salt Lake City, Utah 84112
Transmitted
Acceleration
Laboratory
by Melvin R. Scott
ABSTRACT A method is shown to accelerate the convergence of a partial series. It is based on a generalized form of Kummer’s comparison method in which a set of functions with unique properties are fitted to the summand values. This process subtracts off the tail of an infinite sum and accelerates the convergence of the remaining partial sum.
DISCUSSION Frequently, problems of a partial series
in science
and engineering
S,=
5
have solutions
in terms
a,.
(1)
n=l
Increasing the number of series terms (N) brings the partial sum S, closer to the exact solution. In many cases, each element of the summation may be so time-consuming to calculate that almost any method to accelerate series convergence is advantageous computationally. This paper presents a generalized data fitting approach to convergence acceleration based on Kummer’s comparison method [I]. Assume that the series elements asymptotically approach a function with a known sum, i.e.,
a, - W(n) > where
B is a constant.
Denote
n large,
the infinite sum by y: y=
ef(n). n=l
APPLIED MATHEMATICS AND COMPUTATION 42~189-195 (1991) 0 Elsevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas, New York, NY 10010
189
0096-3003/91/$03.50
190
K. J. BUNCH, W. N. CAIN, AND R. W. GROW
It is then true that the transformed
series
converges faster than the original series. This method “subtracts out” the tail of the series, and it can be generalized as follows: Assume that the series elements can be expressed as a sum of functions, a, -kf,W+W-a(n)+ where
B,, B,, . . . , B,, are constants
.** +R,f,(nL
(5)
and
m
c fk(nn)= Yk.
(6)
n=l
The transformed
series is then given by a, -
5 Bkfk(n) + fI:Bkyk.
I
k=l
(7)
k=l
The set of functions fk(n) can be chosen to best fit a wide range of series with known asymptotic forms. For example, consider the functions npk, so that
The infinite
sum of nPk is given by [2]
5 n_k=l(k),
k>l,
n=l
(8b)
where 5 is the Riemann zeta function [3, 41. The problem of choosing the constants B, in Equation (7) can be viewed as fitting the series element “data points” a,, a2,. . , , aN with the “fitting functions” nek, R,
R,
R,
l3
l4
B,
B,
~+-+-+.*.-i-~=a,
B,
2+23+25+...+2p=a2
RP
BP (9)
Series Contjergence
191
Acceleration
or in matrix form,
(10)
MBZA. The set of coefficients B can be chosen to best solve Equation least-squares sense through the normal equations [5]
(10) in a
M+MB = M+A,
(11)
where Mt is the transpose of the matrix M. As an example of this “zeta transform” method, the following series were evaluated using several orders [p in Equation (S)] of the transformation: 1 0.39493407
= 5 n=l
(12)
(n+l)’ 1
0.233700551=
-0.69314718
F n=l
(13)
(2n+1)“’
m (-1)” = c n-2.
(14)
n=l
Table 1 compares the results of the zeta transform with those of the partial series. The first two series [Equations (12) and (13)] are monotonically decreasing, and the functions trek accurately represent the asymptotic forms better results of the series terms. The zeta transform produces consistently for these two series as the order increases. Unfortunately, the results are poor with the third series [Equation (14)], in which the summand alternates in sign. Here the fitting functions n -k do not fit the oscillatory nature of the series terms. Another type of function set is designed to fit a general set of functions over an infinite interval. The summation index is defined by the infinite interval, and it is convenient to have a set of functions that is complete over this interval. The Laguerre polynomials L,(r) [6] are such a set of functions. They are orthogonal with respect to the weighting function e --*: m
/0 For the purpose given the form
e-xL,,l(x)L,(x)dx=
of series evaluation,
{
y>
,
the functions
fk( n) = e-““Lk(
n).
m+nT m=n. fk(n)
in Equation
(15) (7) are
(16)
K. J. BUNCH, W. N. CAIN, AND R. W. GROW
192
TABLE 1 THE
SERIES
VALUES
OF THE
VERSUS
ZETA
THAT
TRANSFORM
OF THE
PARTIAL
FOR VAR,O”S
SUMS
SUM
Zeta transform value N
Partial sum
Order 1
2
3
4
5
5 10 15 m
IZ(N+2)-" 0.26179705 0.28424239 0.33537904 0.37199770 0.387325043 0.39261211 0.31497664 0.32682776 0.35602767 0.37991337 0.39040989 0.39376305 0.33780674 0.34584543 0.36610192 0.38051053 0.39158128 0.39411720 0.39493407
5 10 15 m
E(2N+l)-" 0.19212942 0.21302190 0.22922948 0.23310440 0.23363554 0.23369420 0.21098888 0.22196931 0.23097516 0.23333808 0.23366585 0.23369774 0.21808063 0.22552337 0.23173867 0.23343244 0.23367544 0.23369867 0.233700551
5 10 15 co
E:(-l)NN-" -0.83861111 -1.44763234-0.55775575-1.33408753 0.07647335-7.02483377 -0.81796218 -0.90129966-0.66474344-1.04166132-0.47528457 -1.37369308 -0.82454176 -0.88097899-0.71879609-0.98073866-0.60861722 -1.14434474 -0.82246703
Let the infinite sum be defined by
5 eMRnLk(n),
S,(a) =
(17)
?%=I
so that
(18) Using the recursion L,+l(
relation for Laguerre x)
=
polynomials [6],
(2n+1-x)L(x) n+l
- cl-,(x)
(19)
with
&l(r) = 1,
(20)
&(x)=-x+1,
(21)
193
Series Convergence Acceleration the infinite sum of Equation relation, namely,
5 #%+A”)=LlWl,=,,,
n=l
(17) can be shown to satisfy a similar recursion
=[
2k+‘~y”Sdu)KSk-lb)]~~l,2 (22)
with
(23)
s,=
2 (-n+1)e--5.
(24)
It=1
Evaluation
of the first 11 terms for
u = i
gives
S,( +) = 1.54149408253679, S,(i)
= -2.37620400649596,
s,( +) = 1.70405537039732, s,(f)
= - 2.21903578509954,
S,( +) = 1.85329366461454, S,(i)
= -2.08007711777906,
S,( +) = 1.97986526287910, S,( +) = - 1.96770868953962, S,(f)
= 2.07654761691417,
S,( +) = - 1.88782852326230, S,,( ;) = 2.13890167836134. Table 2 compares the results of the Laguerre transform with that of the partial series for Equations (12)-(14). The Laguerre transform gives somewhat poorer results than the zeta transform for the two monotonically decreasing series (12) and (13). It does, however, give somewhat better results for the series with alternating terms. This result is as expected, since
194
K. J. BUNCH, W. N. CAIN, AND R. W. GROW TABLE 2 THE
SERIES VERSUS
VALUES THAT
OF THE
OF THE
LAGUERRE
PARTIAL
SUM
TRANSFORM AND
THE
FOR WYNN
VARIOUS
SUMS
TRANSFORM
Laguerre transform value N
Partial sum
Order 0
1
2
3
9
5 10 15 m
Yz(‘v+2)-' 0.26179705 0.28675517 0.29630709 0.39443407 0.39493407 0.31497664 0.32354334 0.32543606 0.33307944 0.33838482 0.31216022 0.33780674 0.40526242 0.34172890 0.34519253 0.34863366 0.36303778 0.39493407
5 10 15 cc
13(2iv+1)-' 0.19212942 0.25012532 0.19619916 0.23386462 0.15994839 0.21098888 0.31343990 0.21401399 0.21750482 0.21688449-0.43537972 0.21808063 0.44641453 0.21925292 0.22033849 0.22123105 0.21537475 0.233700551 C(- 1).yj-"
5 10 15 cc
-0.83861111 -0.71559748-0.75541471-1.65545735 0.82995414 -0.81796218 -2.7172422 -0.80947575-0.81276333-0.720891651714.7478 -0.82454176 -1.2275302 -0.82564392-0.82898380-0.83470871-51.512365 -0.82246703
the Laguerre transform provides in general a better fit to a series than does the zeta transform. However, the latter is the transform of choices for series whose summands vary asymptotically as nPk. It’s not surprising that the data fitting approach to series convergence acceleration works best when the fitting functions closely approximate the asymptotic forms for the summand (as the above results indicate). The Laguerre transform method is probably best for general series, whereas specific series with known asymptotic forms are best fitted with other function sets. This method is not confined to the fitting functions illustrated in this paper, and other mnctions may be more suited to a particular application.
CONCLUSIONS This paper has presented a method to accelerate series convergence. The method fits the summand values with function sets having known infinite sums, and it is a generalized form of Kummer’s comparison method. The best
Series
Convergence
195
Acceleration
results are obtained when the fitting functions best approximate the asymptotic value of the summand. This method allows the fitting functions to be tailored to a series when the summands have known asymptotic
values.
REFERENCES 1
R. W. Hamming,
Numerical
Methods for Scientists and Engineers,
2nd ed., Dover,
New York, 1973, pp. 195-196. 2
A. D. Wheelon,
Tables of Summable
tions, Holden-Day, 3
L. C. Andrews, Macmillan,
4
San Francisco, Special
Functions
Series
and Integrals
1968, Chapter for
Engineers
Involving
Bessel
Func-
2. and Applied
Mathematicians,
New York, 1985, pp. 83-91.
M. Abramowitz
and I. A. Stegun,
Handbook
of Mathematical
Functions,
Dover,
New York, 1972, p. 807. 5
C. L. Lawson Englewood
6
Reference
and R. J. Hanson,
Cliffs, N.J., 1974. [3], pp. 176-182.
Solving
Least Squares
Problems,
Prentice-Hall,
Fitting
K. J. Bunch,
Approach
to Series
Convergence
W. N. Cain, and R. W. Grow
Microwave Device and Physical Electronics Department of Electrical Engineering University of Utah Salt Lake City, Utah 84112
Transmitted
Acceleration
Laboratory
by Melvin R. Scott
ABSTRACT A method is shown to accelerate the convergence of a partial series. It is based on a generalized form of Kummer’s comparison method in which a set of functions with unique properties are fitted to the summand values. This process subtracts off the tail of an infinite sum and accelerates the convergence of the remaining partial sum.
DISCUSSION Frequently, problems of a partial series
in science
and engineering
S,=
5
have solutions
in terms
a,.
(1)
n=l
Increasing the number of series terms (N) brings the partial sum S, closer to the exact solution. In many cases, each element of the summation may be so time-consuming to calculate that almost any method to accelerate series convergence is advantageous computationally. This paper presents a generalized data fitting approach to convergence acceleration based on Kummer’s comparison method [I]. Assume that the series elements asymptotically approach a function with a known sum, i.e.,
a, - W(n) > where
B is a constant.
Denote
n large,
the infinite sum by y: y=
ef(n). n=l
APPLIED MATHEMATICS AND COMPUTATION 42~189-195 (1991) 0 Elsevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas, New York, NY 10010
189
0096-3003/91/$03.50
190
K. J. BUNCH, W. N. CAIN, AND R. W. GROW
It is then true that the transformed
series
converges faster than the original series. This method “subtracts out” the tail of the series, and it can be generalized as follows: Assume that the series elements can be expressed as a sum of functions, a, -kf,W+W-a(n)+ where
B,, B,, . . . , B,, are constants
.** +R,f,(nL
(5)
and
m
c fk(nn)= Yk.
(6)
n=l
The transformed
series is then given by a, -
5 Bkfk(n) + fI:Bkyk.
I
k=l
(7)
k=l
The set of functions fk(n) can be chosen to best fit a wide range of series with known asymptotic forms. For example, consider the functions npk, so that
The infinite
sum of nPk is given by [2]
5 n_k=l(k),
k>l,
n=l
(8b)
where 5 is the Riemann zeta function [3, 41. The problem of choosing the constants B, in Equation (7) can be viewed as fitting the series element “data points” a,, a2,. . , , aN with the “fitting functions” nek, R,
R,
R,
l3
l4
B,
B,
~+-+-+.*.-i-~=a,
B,
2+23+25+...+2p=a2
RP
BP (9)
Series Contjergence
191
Acceleration
or in matrix form,
(10)
MBZA. The set of coefficients B can be chosen to best solve Equation least-squares sense through the normal equations [5]
(10) in a
M+MB = M+A,
(11)
where Mt is the transpose of the matrix M. As an example of this “zeta transform” method, the following series were evaluated using several orders [p in Equation (S)] of the transformation: 1 0.39493407
= 5 n=l
(12)
(n+l)’ 1
0.233700551=
-0.69314718
F n=l
(13)
(2n+1)“’
m (-1)” = c n-2.
(14)
n=l
Table 1 compares the results of the zeta transform with those of the partial series. The first two series [Equations (12) and (13)] are monotonically decreasing, and the functions trek accurately represent the asymptotic forms better results of the series terms. The zeta transform produces consistently for these two series as the order increases. Unfortunately, the results are poor with the third series [Equation (14)], in which the summand alternates in sign. Here the fitting functions n -k do not fit the oscillatory nature of the series terms. Another type of function set is designed to fit a general set of functions over an infinite interval. The summation index is defined by the infinite interval, and it is convenient to have a set of functions that is complete over this interval. The Laguerre polynomials L,(r) [6] are such a set of functions. They are orthogonal with respect to the weighting function e --*: m
/0 For the purpose given the form
e-xL,,l(x)L,(x)dx=
of series evaluation,
{
y>
,
the functions
fk( n) = e-““Lk(
n).
m+nT m=n. fk(n)
in Equation
(15) (7) are
(16)
K. J. BUNCH, W. N. CAIN, AND R. W. GROW
192
TABLE 1 THE
SERIES
VALUES
OF THE
VERSUS
ZETA
THAT
TRANSFORM
OF THE
PARTIAL
FOR VAR,O”S
SUMS
SUM
Zeta transform value N
Partial sum
Order 1
2
3
4
5
5 10 15 m
IZ(N+2)-" 0.26179705 0.28424239 0.33537904 0.37199770 0.387325043 0.39261211 0.31497664 0.32682776 0.35602767 0.37991337 0.39040989 0.39376305 0.33780674 0.34584543 0.36610192 0.38051053 0.39158128 0.39411720 0.39493407
5 10 15 m
E(2N+l)-" 0.19212942 0.21302190 0.22922948 0.23310440 0.23363554 0.23369420 0.21098888 0.22196931 0.23097516 0.23333808 0.23366585 0.23369774 0.21808063 0.22552337 0.23173867 0.23343244 0.23367544 0.23369867 0.233700551
5 10 15 co
E:(-l)NN-" -0.83861111 -1.44763234-0.55775575-1.33408753 0.07647335-7.02483377 -0.81796218 -0.90129966-0.66474344-1.04166132-0.47528457 -1.37369308 -0.82454176 -0.88097899-0.71879609-0.98073866-0.60861722 -1.14434474 -0.82246703
Let the infinite sum be defined by
5 eMRnLk(n),
S,(a) =
(17)
?%=I
so that
(18) Using the recursion L,+l(
relation for Laguerre x)
=
polynomials [6],
(2n+1-x)L(x) n+l
- cl-,(x)
(19)
with
&l(r) = 1,
(20)
&(x)=-x+1,
(21)
193
Series Convergence Acceleration the infinite sum of Equation relation, namely,
5 #%+A”)=LlWl,=,,,
n=l
(17) can be shown to satisfy a similar recursion
=[
2k+‘~y”Sdu)KSk-lb)]~~l,2 (22)
with
(23)
s,=
2 (-n+1)e--5.
(24)
It=1
Evaluation
of the first 11 terms for
u = i
gives
S,( +) = 1.54149408253679, S,(i)
= -2.37620400649596,
s,( +) = 1.70405537039732, s,(f)
= - 2.21903578509954,
S,( +) = 1.85329366461454, S,(i)
= -2.08007711777906,
S,( +) = 1.97986526287910, S,( +) = - 1.96770868953962, S,(f)
= 2.07654761691417,
S,( +) = - 1.88782852326230, S,,( ;) = 2.13890167836134. Table 2 compares the results of the Laguerre transform with that of the partial series for Equations (12)-(14). The Laguerre transform gives somewhat poorer results than the zeta transform for the two monotonically decreasing series (12) and (13). It does, however, give somewhat better results for the series with alternating terms. This result is as expected, since
194
K. J. BUNCH, W. N. CAIN, AND R. W. GROW TABLE 2 THE
SERIES VERSUS
VALUES THAT
OF THE
OF THE
LAGUERRE
PARTIAL
SUM
TRANSFORM AND
THE
FOR WYNN
VARIOUS
SUMS
TRANSFORM
Laguerre transform value N
Partial sum
Order 0
1
2
3
9
5 10 15 m
Yz(‘v+2)-' 0.26179705 0.28675517 0.29630709 0.39443407 0.39493407 0.31497664 0.32354334 0.32543606 0.33307944 0.33838482 0.31216022 0.33780674 0.40526242 0.34172890 0.34519253 0.34863366 0.36303778 0.39493407
5 10 15 cc
13(2iv+1)-' 0.19212942 0.25012532 0.19619916 0.23386462 0.15994839 0.21098888 0.31343990 0.21401399 0.21750482 0.21688449-0.43537972 0.21808063 0.44641453 0.21925292 0.22033849 0.22123105 0.21537475 0.233700551 C(- 1).yj-"
5 10 15 cc
-0.83861111 -0.71559748-0.75541471-1.65545735 0.82995414 -0.81796218 -2.7172422 -0.80947575-0.81276333-0.720891651714.7478 -0.82454176 -1.2275302 -0.82564392-0.82898380-0.83470871-51.512365 -0.82246703
the Laguerre transform provides in general a better fit to a series than does the zeta transform. However, the latter is the transform of choices for series whose summands vary asymptotically as nPk. It’s not surprising that the data fitting approach to series convergence acceleration works best when the fitting functions closely approximate the asymptotic forms for the summand (as the above results indicate). The Laguerre transform method is probably best for general series, whereas specific series with known asymptotic forms are best fitted with other function sets. This method is not confined to the fitting functions illustrated in this paper, and other mnctions may be more suited to a particular application.
CONCLUSIONS This paper has presented a method to accelerate series convergence. The method fits the summand values with function sets having known infinite sums, and it is a generalized form of Kummer’s comparison method. The best
Series
Convergence
195
Acceleration
results are obtained when the fitting functions best approximate the asymptotic value of the summand. This method allows the fitting functions to be tailored to a series when the summands have known asymptotic
values.
REFERENCES 1
R. W. Hamming,
Numerical
Methods for Scientists and Engineers,
2nd ed., Dover,
New York, 1973, pp. 195-196. 2
A. D. Wheelon,
Tables of Summable
tions, Holden-Day, 3
L. C. Andrews, Macmillan,
4
San Francisco, Special
Functions
Series
and Integrals
1968, Chapter for
Engineers
Involving
Bessel
Func-
2. and Applied
Mathematicians,
New York, 1985, pp. 83-91.
M. Abramowitz
and I. A. Stegun,
Handbook
of Mathematical
Functions,
Dover,
New York, 1972, p. 807. 5
C. L. Lawson Englewood
6
Reference
and R. J. Hanson,
Cliffs, N.J., 1974. [3], pp. 176-182.
Solving
Least Squares
Problems,
Prentice-Hall,