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A difference equation model of partial series
A Difference Equation Model of Partial Series K. J. Bunch and R. W. Grow Microwave Device and Physical Electronics Laboratory Department of Electrical Engineering University of Utah Salt Lake City, Utah 84112
Transmitted
by Melvin Scott
ABSTRACT A model for partial series is given in terms of difference equations: This model allows the techniques of digital signal processing to be applied to series for the purpose of series evaluation, filtering, and extrapolation.
1.
DISCUSSION
This paper presents an alternative manner of viewing partial series that allows the application of linear systems theory and digital signal processing techniques to the handling of partial series. Some progress along these lines has already been made. For example, consider the partial sum defined by
SN
(1)
= n=l
Shanks [I] assumes the behavior transients of the form:
sN
=
of this series to be composed
B +
&,e-“kN.
of a sum of p
(2)
k=O APPLJED MATHEMATICS
AND COMPLJZATION
0 Elsevier Science Inc., 1994 655 Avenue of the Americas, New York, NY 10010
6 11301-305 (1994)
301 0096-3003/94/$7.00
K. J. BUNCH AND R. W. GROW
302
The exponents CY~can be complex to model the oscillatory, divergent, and convergent nature of the partial sum as N increases. The limit of the infinite sumrnation is B. From this model, Shanks [l] derives a powerful extrapolation method to obtain this limit from a finite number of partial series terms. Wheelon [2] assumes that a, in (1) is a continuous function of TZ,which is the Laplace transform of another function,
an =
r
f(t)e-%.
(3)
0
By substituting
for arr in (l), the series is converted to an integral:
In this manner, a wide class of series can be evaluated using table look-up methods [3]. Bunch et al. [4] have extended this treatment by using z-transforms [5] that were developed for discrete processes. Although the models developed by Shanks [ 11 and Wheelon [2] are powerful, they both consider the partial summation SN to be a continuous function that is only evaluated at discrete values of N. The authors propose viewing the partial series as describing a discrete waveform that settles to a limiting value as N approaches infinity. The partial series can then be modeled by a linear difference equation of the form:
&c#)sN-~ =f(N).
(5)
k=O The solution to this difference equation will be of the form:
SN
=
$,(N) + s,(N).
where p denotes the particular solution and c denotes the complementary [5]. As an example, consider the series,
(f-5) solution
A DifSerence Equation Model
303 N
SN
=
c eMat.
(7)
t=1
This series is geometric whose sum is [6] e-a(N--l)
e --a sN=--
e-a
1 -
1 - ema
*
(8)
Thus, the series defined by (7) is an ideal first-order transient series, as defined by (2). The remainder defined by
RN =
so0
-
SN
e-a(N+l) =
’
1-e-a
(9)
satisfies the first-order difference equation,
RN - emaNRN_l
= 0.
Thus, the remainder of a first-order transient series satisfies a first-order linear homogeneous difference equation. It is easy to show that higher-order transient series satisfy similar difference equations. Consider the @h-order linear homogeneous difference equation,
P
c
ak
RN-k = 0.
(11)
k=O Assume the solution is of the form: RN =
CZ-~,
so that (11) becomes an algebraic equation of the form
(12)
K. J. BUNCH AND R. W. GROW
304
f:
akz k -- 0.
(13)
k=O
Assume the roots of this equation are distinct and labeled qk, k = 1, . . . , p so that (13) can be factored into the form:
=o.
f&-qk)
(14)
k=l
The solution to the difference equation (14) is of the form [5]: P
RN
=
c c
%!k
-N
k=l P
=
eke
-akN
,
(15)
k=l
where
Thus, the difference equation, (11) accurately models the remainder of a transient series defined by Shanks (2). The difference equation model of partial series allows one to pull in the techniques developed for discrete processes [7] in the analysis of partial series. Consider the z-transform of (2) and [5]:
(17)
Thus, the distribution of poles in the complex plane indicates the transient nature of the series. This distribution can be used to design extrapolation methods to filter out the series transients, leaving the steady-state solution of the series, i.e., the desired series limit [5]. Another approach to extrapolation may be to use a discrete Fourier transform
A DifSerenceEquationModel
305
[7] on the partial summation, use a filter in the frequency domain, and then use the inverse transform to obtain an approximation to the series limit. Each filter can be fine tuned to the set of summations to be evaluated.
2.
CONCLUSIONS
It has been suggested to model partial summations as satisfying a difference equation. This viewpoint considers the limit of a series as the steady-state solution of this equation. The techniques of linear systems theory and digital signal processing are then available to filter out this steady-state solution. REFERENCES 1
D. Shanks, Non-linear transformations
of divergent and slowly convergent sequences,
J. Math. Phys. 34:1-42 (1955). 2 3 4 5 6 7
A. D. Wheelon, On the summation of infinite series in closed form, 1. AppZ. Phys. 23:113-118 (1954). A, D. Wheelon, Tables of Summable Series and Integrals Involving Bessel Functions, Holden-Day, San Francisco, California, 1968. K. J. Bunch, W. N. Cain, and R. W. Grow, The z-transform method of evaluating partial summations in closed form, J. Phys. A: Math. Gen. 23: 1990. J. M. Smith, Mathematical Modeling and Digital Simulationfor Engineers and Scientists, 2nd Ed., John Wiley and Sons, New York, 1987. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Orlando, Florida, 1980, p. 1. A. U. Oppenheim and R. W. Schafer, Digital Signal Processing, Prentice-Hall, Englewood Cliffs, New Jersey, 1975.
Transmitted
by Melvin Scott
ABSTRACT A model for partial series is given in terms of difference equations: This model allows the techniques of digital signal processing to be applied to series for the purpose of series evaluation, filtering, and extrapolation.
1.
DISCUSSION
This paper presents an alternative manner of viewing partial series that allows the application of linear systems theory and digital signal processing techniques to the handling of partial series. Some progress along these lines has already been made. For example, consider the partial sum defined by
SN
(1)
= n=l
Shanks [I] assumes the behavior transients of the form:
sN
=
of this series to be composed
B +
&,e-“kN.
of a sum of p
(2)
k=O APPLJED MATHEMATICS
AND COMPLJZATION
0 Elsevier Science Inc., 1994 655 Avenue of the Americas, New York, NY 10010
6 11301-305 (1994)
301 0096-3003/94/$7.00
K. J. BUNCH AND R. W. GROW
302
The exponents CY~can be complex to model the oscillatory, divergent, and convergent nature of the partial sum as N increases. The limit of the infinite sumrnation is B. From this model, Shanks [l] derives a powerful extrapolation method to obtain this limit from a finite number of partial series terms. Wheelon [2] assumes that a, in (1) is a continuous function of TZ,which is the Laplace transform of another function,
an =
r
f(t)e-%.
(3)
0
By substituting
for arr in (l), the series is converted to an integral:
In this manner, a wide class of series can be evaluated using table look-up methods [3]. Bunch et al. [4] have extended this treatment by using z-transforms [5] that were developed for discrete processes. Although the models developed by Shanks [ 11 and Wheelon [2] are powerful, they both consider the partial summation SN to be a continuous function that is only evaluated at discrete values of N. The authors propose viewing the partial series as describing a discrete waveform that settles to a limiting value as N approaches infinity. The partial series can then be modeled by a linear difference equation of the form:
&c#)sN-~ =f(N).
(5)
k=O The solution to this difference equation will be of the form:
SN
=
$,(N) + s,(N).
where p denotes the particular solution and c denotes the complementary [5]. As an example, consider the series,
(f-5) solution
A DifSerence Equation Model
303 N
SN
=
c eMat.
(7)
t=1
This series is geometric whose sum is [6] e-a(N--l)
e --a sN=--
e-a
1 -
1 - ema
*
(8)
Thus, the series defined by (7) is an ideal first-order transient series, as defined by (2). The remainder defined by
RN =
so0
-
SN
e-a(N+l) =
’
1-e-a
(9)
satisfies the first-order difference equation,
RN - emaNRN_l
= 0.
Thus, the remainder of a first-order transient series satisfies a first-order linear homogeneous difference equation. It is easy to show that higher-order transient series satisfy similar difference equations. Consider the @h-order linear homogeneous difference equation,
P
c
ak
RN-k = 0.
(11)
k=O Assume the solution is of the form: RN =
CZ-~,
so that (11) becomes an algebraic equation of the form
(12)
K. J. BUNCH AND R. W. GROW
304
f:
akz k -- 0.
(13)
k=O
Assume the roots of this equation are distinct and labeled qk, k = 1, . . . , p so that (13) can be factored into the form:
=o.
f&-qk)
(14)
k=l
The solution to the difference equation (14) is of the form [5]: P
RN
=
c c
%!k
-N
k=l P
=
eke
-akN
,
(15)
k=l
where
Thus, the difference equation, (11) accurately models the remainder of a transient series defined by Shanks (2). The difference equation model of partial series allows one to pull in the techniques developed for discrete processes [7] in the analysis of partial series. Consider the z-transform of (2) and [5]:
(17)
Thus, the distribution of poles in the complex plane indicates the transient nature of the series. This distribution can be used to design extrapolation methods to filter out the series transients, leaving the steady-state solution of the series, i.e., the desired series limit [5]. Another approach to extrapolation may be to use a discrete Fourier transform
A DifSerenceEquationModel
305
[7] on the partial summation, use a filter in the frequency domain, and then use the inverse transform to obtain an approximation to the series limit. Each filter can be fine tuned to the set of summations to be evaluated.
2.
CONCLUSIONS
It has been suggested to model partial summations as satisfying a difference equation. This viewpoint considers the limit of a series as the steady-state solution of this equation. The techniques of linear systems theory and digital signal processing are then available to filter out this steady-state solution. REFERENCES 1
D. Shanks, Non-linear transformations
of divergent and slowly convergent sequences,
J. Math. Phys. 34:1-42 (1955). 2 3 4 5 6 7
A. D. Wheelon, On the summation of infinite series in closed form, 1. AppZ. Phys. 23:113-118 (1954). A, D. Wheelon, Tables of Summable Series and Integrals Involving Bessel Functions, Holden-Day, San Francisco, California, 1968. K. J. Bunch, W. N. Cain, and R. W. Grow, The z-transform method of evaluating partial summations in closed form, J. Phys. A: Math. Gen. 23: 1990. J. M. Smith, Mathematical Modeling and Digital Simulationfor Engineers and Scientists, 2nd Ed., John Wiley and Sons, New York, 1987. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Orlando, Florida, 1980, p. 1. A. U. Oppenheim and R. W. Schafer, Digital Signal Processing, Prentice-Hall, Englewood Cliffs, New Jersey, 1975.