This research proposal investigates the Padé approximation technique for function approximation, comparing its accuracy against the Maclaurin series. The study will evaluate both methods' performance through numerical results and discussion, emphasizing error margins and approximation efficiency. The conclusions highlight the strengths and weaknesses of each method, ultimately guiding future applications in mathematical and scientific fields.

1. Using the Pade Technique to
Approximate the Function
Prepared by: Supervisor:
Ammar H. Abdulsalam M. Ahmed J. Sabali
Amer Z. Kreet

2. Abstract
Chapter One
Introduction and Mathematical Functions
1.1.Preliminaries
1.2 Mathematical Functions
1.3 Literature review
Chapter Two
Basic Idea
2.1 Pade Technique
Chapter Three
3.1 Numerical Results
3.2 Discussion
3.3 Conclusion
References:
Outline

3. Abstract
This research proposal aims to conduct an in-depth study of the Padé
approximation technique, a powerful tool for solving complex mathematical
problems, and compare it with the Maclaurin series in terms of absolute errors.
The Padé approximation, a method of approximating functions by rational
functions, will be meticulously examined and its effectiveness evaluated. The
Maclaurin series, a representation of a function as an infinite sum of terms, will
serve as a benchmark for comparison.

4. The primary objective is to discern which method provides a more accurate
solution with fewer absolute errors. All calculations, including the generation of
series, tabulation of data, and creation of plots, will be performed using
Mathematica software, ensuring precision and reproducibility. This study will
contribute to the existing body of knowledge by providing a comprehensive
comparison of these two techniques and their error margins, thereby guiding
future research and applications in various mathematical and scientific fields.

5. Chapter One

6. Introduction and Mathematical Functions
1.1. Preliminaries
The Padé method, named after the French mathematician Henri Padé, is a powerful
mathematical technique used to approximate functions, particularly rational
functions, by constructing rational approximants. The method originated in the late
19th and early 20th centuries and has since found widespread applications in various
fields, including physics, engineering, and computational mathematics.

7. Henri Padé introduced his method in a series of papers published between 1880
and 1892. The motivation behind the development of the Padé method was to
find rational approximations that could accurately represent functions, especially
when polynomial approximations were insufficient. The fundamental idea behind
the Padé method is to represent a given function by a rational function of the
form:
𝑅 ( 𝑥 )=
𝑝 ( 𝑥 )
𝑞 ( 𝑥 ) (1)
where and are the polynomials, whose degrees add to .

8. Given a function expanded in a Maclaurin series , we can use the coefficients of the
series to represent the function by a ratio of two polynomials
𝑎0+ 𝑎1 𝑥+⋯ +𝑎𝐿 𝑥𝐿
𝑏0 +𝑏1 𝑥+ ⋯+𝑏𝑀 𝑥
𝑀
(2)
The goal is to choose these coefficients such that the rational function provides the best
possible approximation to over a specified interval or set of points.
The Padé method achieves this by matching the Taylor series expansions of and up to
a certain order. By equating the coefficients of corresponding terms in the series
expansions, a system of linear equations is formed, from which the coefficients of the
rational function can be solved.

9. One of the key advantages of the Padé method is its ability to provide highly
accurate approximations with relatively simple rational functions. This makes it
particularly useful in situations where more traditional approximation methods,
such as polynomial interpolation, may lead to poor results or require higher
degrees of approximation.
Over the years, the Padé method has been further developed and refined, with
extensions to handle various types of functions and improve computational
efficiency. It has been applied in diverse areas of science and engineering,
including signal processing, numerical analysis, quantum mechanics, and control
theory, among others.

10. 1.2 Mathematical Functions
We consider the following function:
𝑓 ( 𝑥 ) =𝑒
𝑥
(3)
The Maclaurin series of Eqs. (3) has the following series:
𝑓 (𝑥)¿𝑒
𝑥
=∑
𝑛=0
∞
𝑥𝑛
𝑛!
=1+𝑥+
𝑥2
2!
+
𝑥3
3!
+
𝑥4
4!
+… (4)

11. The functions in Eqs. (3) can be sketched as:
Fig.1. The plot of when

12. Chapter Two

13. 2.1 Pade Technique
There are a sufficient number of polynomials to approximate any continuous
function on a closed interval. But using polynomials for approximation, there is
a disadvantage because of oscillatory behavior of the polynomials, causing
error bounds in polynomial approximation to significantly exceed the average
approximation error [1].
We now discuss a technique that disseminates the approximation error more
evenly over the interval of approximation. The procedure is to seek a rational
function for the approximation.

14. A rational function r(x) of degree n has the following form:
𝑟 ( 𝑥 ) =
𝑝 ( 𝑥 )
𝑞 ( 𝑥 )
(5)
where and are the polynomials, whose degrees add to .
Given a function expanded in a Maclaurin series , we can use the coefficients of
the series to represent the function by a ratio of two polynomials
𝑎0+𝑎1 𝑥+⋯ +𝑎𝐿 𝑥𝐿
𝑏0+𝑏1 𝑥+⋯+𝑏𝑀 𝑥
𝑀 (6)

15. denoted by and called the Padé approximant. The basic idea is to match the series
coefficients as far as possible. Even though the series has a finite region of
convergence, we can obtain the limit of the function as if .
Note that if we wish to obtain we will have
(𝑎0+𝑎1 𝑥)=(𝑏0+𝑏1 𝑥)(𝑐0+𝑐1 𝑥+𝑐2 𝑥2
+⋯) (7)
so that coefficients of are zero, that is,
𝑏1 𝑐1+ 𝑏0 𝑐2=0 (8)

16. Taking we have
𝑏1 𝑐1+ 𝑐2 =0 (9)
Now we consider yielding
(10)
(𝑎0+𝑎1𝑥+𝑎2𝑥
2
)=(𝑏0+𝑏1𝑥+𝑏2𝑥
2
)(𝑐0+𝑐1𝑥+𝑐2𝑥
2
+𝑐3𝑥
3
+⋯)
Clearly the coefficients of are zero, so that we can write
𝑏2 𝑐1+𝑏1 𝑐2 +𝑏0 𝑐3 =0 (11)

17. In general, we note that there are independent coefficients in the numerator
and coefficients in the denominator. To make the system determinable, it is
customary to let . We then have independent coefficients in the denominator,
and thus we have independent coefficients in total. Now the approximant
can fit the power series through orders with an error of .

18. For example, let us consider
We have
Consequently,

19. Equating coefficients of successively to zero, we obtain
Now setting , we have linear equations for the coefficients in the denominator.

20. [¿
¿𝑐𝐿− 𝑀+1 ¿𝑐𝐿− 𝑀+2 ⋅ ⋅ ⋅ ¿𝑐𝐿
𝑐𝐿− 𝑀+ 2 𝑐𝐿− 𝑀+ 3 ⋅ ⋅ ⋅ 𝑐𝐿+1
⋅ . . . . .
⋅ . . . . .
⋅ . . . . .
𝑐𝐿 𝑐𝐿+1 ⋅ ⋅ ⋅ 𝑐𝐿 −𝑀 −1
¿
][¿
¿
¿𝑏𝑀
𝑏𝑀− 1
⋅
⋅
⋅
𝑏1
¿
]=
[¿
¿
¿𝑐𝐿+1
𝑐𝐿+2
⋅
⋅
⋅
𝑐𝐿+ 𝑀
¿
]

21. Solving the above system of equations, we determine the coefficients for .
Since we know the coefficients , we can equate the coefficients of to obtain
the remaining coefficients . Thus

22. Thus, the numerator and denominator of the Padé approximant are all
determined and we have agreement with the original series through order .
The same or higher order accuracy can be achieved by using lower order
polynomials in than direct polynomial approximation. For higher orders
approximants, one can use symbolic programs.
For , we have

23. When the function is such that its limiting value is finite as , polynomial
approximations give very poor results. On the other hand, the Padé approximations
give much better results [2].

24. Chapter Three

25. 3.1 Numerical Results
Example: Find the Pade approximation for the function with numerator and
denominator each of polynomial of degree three. Find the maximum error in the
interval .
Solution: Suppose and let is a rational function which is an approximation to in which
the number of constants are In the present problem, we are given and therefore, or .
Thus, we from Maclaurin series for as
(12)

26. Then,
(13)

27. Equating the coefficients of to in the numerator of Eq. (13) to zero, we have
(14)

28. Solving the last three equations of (3) we get
Using these values, the first four equations of (14) yield
Hence, the required Pade approximation to is
(15)

29. The following table gives the comparison of errors in both Pade and Maclaurin
approximations to
True value
of
Pade
approximation
Error Maclaurin
approximation
Error
0.2
0.4
0.6
0.8
1.0
1.221403
1.491825
1.822119
2.225541
2.718282
1.22140
1.49182
1.82212
2.22555
2.71831
Table 3.1 Comparison between Maclaurin and Pade approximation for
different values of

30. From the above table, it is clear that enough terms are available even in
Maclaurin series to get five decimal accuracy when takes values from through .
However, when , which is the limit for convergence of the Maclaurin series, the
magnitude of the error is more when compared with Pade approximation. In
this example, the maximum error due to Pade approximation is and Pade
approximation is more accurate.

31. Figure 3.1: Comparison between Exponential function and Maclurin
approximation.

32. Figure 3.2. Comparison between Exponential function and Padé approximation.

33. Figure 3.3. Comparison between Exponential function, Maclurin approximation,
and Pade approximation.

34. 3.2 Discussion
We used a technique called Pade approximation to evaluated continuous function,
applying the technique on mathematical function (3). Thus, the discussion
between Maclaurin approximation and Padé approximation revolves around their
respective strengths and weaknesses in terms of precision, speed, accuracy,
divergence, convergence, and stability. The results were as follows:

35. Precision:
oMaclaurin approximation is based on Taylor series expansions around the point ,
providing local polynomial approximations. It is highly precise near the expansion
point but may suffer from poor accuracy away from it, especially for functions with
complex behavior.
oPadé approximation constructs rational functions that can offer more accurate
approximations over a wider range of the function's domain compared to Maclaurin
series. It achieves this by incorporating both numerator and denominator polynomials,
allowing for better representation of function behavior beyond the expansion point.

36. Speed:
oMaclaurin approximation involves straightforward polynomial calculations, making
it computationally efficient and relatively fast for obtaining local approximations.
However, for functions with significant nonlinearity, higher-order terms may be
required, potentially increasing computational complexity.
oPadé approximation, while conceptually more involved due to solving linear systems
for rational function coefficients, can still be implemented efficiently. It may require
more computational resources compared to Maclaurin approximation but often
provides superior accuracy, justifying the slight increase in computational cost.

37. Convergence:
oMaclaurin approximation converges within its radius of convergence, which
depends on the behavior of the function and the order of the approximation.
It may diverge outside this radius or fail to accurately represent the function.
oPadé approximation, when properly constructed, can converge rapidly and
reliably over a broader domain compared to Maclaurin approximation. Its
rational form allows for better convergence properties, making it more suitable
for approximating functions with diverse behavior.

38. Stability:
oMaclaurin approximation is generally stable within its radius of convergence
but may become unstable outside this range, leading to inaccurate results or
numerical errors.
oPadé approximation exhibits improved stability compared to Maclaurin
approximation, particularly for functions with more complex behavior or
singularities. Its rational form provides better stability properties, allowing for
more reliable approximations over a broader domain.

39. In summary, while Maclaurin approximation offers simplicity and
computational efficiency for local approximations, Padé approximation provides
higher accuracy, broader convergence, and improved stability, especially for
functions with complex behavior or non-negligible higher-order derivatives.
The choice between the two methods depends on the specific requirements of
the problem at hand, balancing computational resources with the desired level
of accuracy and reliability.

40. 3.3 Conclusion
In conclusion, when comparing Maclaurin approximation and Padé approximation, it
becomes evident that each method possesses distinct advantages and limitations across
various aspects of precision, speed, accuracy, convergence, divergence, and stability.
Maclaurin approximation, rooted in Taylor series expansions around the origin,
provides local polynomial approximations that are computationally efficient and fast to
compute. However, its precision diminishes as one moves away from the expansion
point, limiting its accuracy for functions with non-trivial behavior. Additionally,
Maclaurin approximations may suffer from divergence outside their radius of
convergence, making them less reliable for broader ranges of function domains.

41. On the other hand, Padé approximation offers a compelling alternative by
constructing rational functions that provide more accurate approximations over wider
function domains. Despite its slightly higher computational complexity, Padé
approximation exhibits superior accuracy, convergence, and stability compared to
Maclaurin approximation. By incorporating both numerator and denominator
polynomials, Padé approximants can better capture intricate function features, making
them particularly effective for functions with complex behavior or non-negligible
higher-order derivatives.

42. Ultimately, the choice between Maclaurin approximation and Padé
approximation depends on the specific requirements of the problem at hand.
While Maclaurin approximation offers simplicity and computational efficiency
for local approximations, Padé approximation provides higher accuracy, broader
convergence, and improved stability, especially for functions with intricate
behavior or non-trivial derivatives. Researchers and practitioners must carefully
weigh these factors to select the most suitable approximation method for their
particular application, ensuring a balance between computational resources and
the desired level of accuracy and reliability.

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