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university of zakho biology deparment

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:     (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     (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:   (3) The Maclaurin series of Eqs . (3) has the following series:   (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:   (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     (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     (7) so that coefficients of are zero, that is,     (8)

16. Taking we have     (9) Now we consider yielding   (10)   Clearly the coefficients of are zero, so that we can write     (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.  

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,    

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 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: Maclaurin 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. Padé 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: Maclaurin 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. Padé 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: Maclaurin 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. Padé 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: Maclaurin approximation is generally stable within its radius of convergence but may become unstable outside this range, leading to inaccurate results or numerical errors. Padé 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.

43. References P. Bürkner , I. Kröker , S. Oladyshkin , W. N.-J. of Computational, and undefined 2023, “A fully Bayesian sparse polynomial chaos expansion approach with joint priors on the coefficients and global selection of terms,” Elsevier, 2023, Accessed: Dec. 13, 2023. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0021999123003054 I. Andrianov and A. Shatrov , “ Padé Approximants, Their Properties, and Applications to Hydrodynamic Problems. Symmetry 2021, 13, 1869,” 2021, doi : 10.3390/sym13101869.

44. [3] A. K. Prajapati and R. Prasad, “Model Reduction Using the Balanced Truncation Method and the Padé Approximation Method,” IETE Technical Review, vol. 39, no. 2, pp. 257–269, 2022, doi : 10.1080/02564602.2020.1842257. [4] M. F. Tabassum, M. Saeed, A. Akgül , M. Farman, and N. A. Chaudhry, “Treatment of HIV/AIDS epidemic model with vertical transmission by using evolutionary Padé -approximation,” Chaos Solitons Fractals, vol. 134, p. 109686, May 2020, doi : 10.1016/J.CHAOS.2020.109686.

45. [5] K. S. Nisar, J. Ali, M. K. Mahmood, D. Ahmed, and S. Ali, “Hybrid evolutionary padé approximation approach for numerical treatment of nonlinear partial differential equations,” Alexandria Engineering Journal, vol. 60, no. 5, pp. 4411–4421, Oct. 2021, doi : 10.1016/J.AEJ.2021.03.030. [6] K. M. Kim, S. H. Choe, J. M. Ryu, and H. Choi, “Computation of Analytical Zoom Locus Using Padé Approximation,” Mathematics 2020, Vol. 8, Page 581, vol. 8, no. 4, p. 581, Apr. 2020, doi : 10.3390/MATH8040581.