The Fourth Version Of Richardson s Extrapolation Spreadsheet Calculator Using Vba Programming For Numerical Differentiations

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1 Available online at GlobalIlluminators FULL PAPER PROCEEDING Multidisciplinary Studies Full Paper Proceeding GTAR-2014, Vol. 1, ISBN: GTAR-14 The Fourth Version Of Richardson s Extrapolation Spreadsheet Calculator Using Vba Programming For Numerical Differentiations Kim Gaik Taya 1*, Tau Han Cheongb 2, Sie Long Kekc 3 and Rosmila Abdul-Kahard 4 1Faculty of Electrical & Electronic Engineering, 2Department of Mathematics & Statistics,3Faculty of Science, Technology & Human Development, Universiti Tun Hussein Onn Malaysia, 4Faculty of Education, Universiti Teknologi Mara Abstract In this paper, we have further upgraded the limitation of our third version of Richardson s extrapolation spreadsheet calculator for computing differentiations numerically. In the third version, users have to input function f(x) using Excel Command. However, in this latest Richardson s extrapolation spreadsheet calculator, users only have to type in function f(x) naturally via programming syntax or mathematical form instead of using Excel command. Using one customized written function in VBA programming, the function f(x) is later evaluated in Excel. The latest version is more user-friendly especially to users who are not familiar with Excel. We believe this is the ideal version The Authors. Published by Global Illuminators. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the Scientific & Review committee of GTAR Keywords Spreadsheet PACS: gb, H; Introduction In real-world problems, applications of the rate of change have been well-defined. For example, flow rate in a tank, productivity of an industrial process and marginal revenue of a manufacturing company. In fact, calculating the rate of change requires the knowledge of differentiation, which is an interesting topic in numerical analysis. Since the physical problems are complex, the exact differentiation is impossible to obtain. In such situations, an approximation of the definite differentiation is important to provide a practical solution. Richardson s extrapolation (Gaunt, 1927; Richardson, 1911), which is one of the most powerful approaches in numerical differentiation, improves the rate of convergence of a sequence in numerical differentiation. Implementation of Richardson s extrapolation via mathematical software or any computational tool leads to a desired definite differentiation. According to the works of (S. L. K. a. A.-K. K. G. Tay, 2012; S. L. K. a. R. A.-K. K. G. Tay, 2009, 2010; S. L. K. a. R. A.- K. K.G. Tay, 2009; Kek, 2008, 2009a, 2009b; S. L. & Tay S. L. Kek, K. G, 2008; S.L. & Tay *All correspondence related to this article should be directed to Kim Gaik Taya, Faculty of Electrical & Electronic Engineering, Universiti Teknologi Mara atay@uthm.edu.my 2015 The Authors. Published by Global Illuminators. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the Scientific & Review committee of GTAR-2015.

2 S. L. Kek, K. G, 2009), Excel spreadsheet shows the calculation ability for a wide range of numerical methods. In addition, the application of Excel spreadsheet in implementing Richardson s extrapolation for numerical differentiation has been developed in (S. L. K. K.G. Tay, R. Abdul-Kahar M.A. Azlan & M.F. Lee, 2013) (S. L. K. K.G. Tay, R. Abdul-Kahar, 2013, 2014) recently. However, the previous Richardson s extrapolation Excel spreadsheet calculator (S. L. K. K.G. Tay, R. Abdul-Kahar M.A. Azlan & M.F. Lee, 2013) is limited to level four in calculating numerical differentiation and the initial value D(0, 0) of approximation to the derivative of a function is required to be keyed in by users using 3-point central difference formula. The improved version of Richardson s extrapolation Excel spreadsheet calculator (S. L. K. K.G. Tay, R. Abdul-Kahar, 2013) overcomes the previous two limitations, in which users only need to enter x, and h instead of the initial value D(0, 0) and it is able to calculate up to level 10. The new improved version of Richardson s extrapolation Excel spreadsheet calculator (S. L. K. K.G. Tay, R. Abdul-Kahar, 2014) further improved the flexibility of the previous two versions up to any level using VBA programming. However, there is still one drawback in the third version, that is users need to enter the function using Excel command rather than programming syntax or mathematical form. Hence it will be not user-friendly for users who are not familiar with Excel. Therefore, in this work, we further improved this drawback using one customized written function in VBA programming. Richardson s Extrapolation There are two ways to improve derivative estimates when employing finite divided differences; decreasing the step size or using a higher-order formula that employs more points. A third approach, based on Richardson s extrapolation which uses two derivatives, estimates a more accurate approximation. (Chapra & Canale, 2006) For a given function of approximations to the derivative of at a specified value of can be computed for a chosen value of using Richardson s extrapolation method as follows: (Mathews & Fink, 2004). K 4 D( J, K 1) D( J 1, K 1) 2K 2 D( J, K) O( h ), J 1, 2,..., K 1, 2,... K 4 1 (1) f ( x hj) f ( x hj) D( J,0), J 0,1, 2,... 2hJ (2) h h h h0 h, h1, h2,..., hj J (3) The iteration process (1) is repeated until D( J, J) D( J, J 1) or D( J, J) D( J 1, J 1) for a specified value of. The value of D( J, J ) then approximates f ( x) at this level. Richardson s extrapolation table for K from 1 to 4 is shown in Table (1). 186

3 Table 1: Richardson s Extrapolation table for level values from 1 to 4. J h J K DJ (,0) DJ (,1) DJ (,2) DJ (,3) DJ (,4) 0 h 0 D (0,0) 1 h 1 D (1,0) D (1,1) 2 h 2 D (2,0) D (2,1) D (2, 2) 3 h 3 D (3,0) D (3,1) D (3, 2) D (3,3) 4 h 4 D (4,0) D (4,1) D (4, 2) D (4,3) D (4, 4) Numerical Example In this section, a numerical example to be solved by Richardson s extrapolation is provided. Question: The temperature distribution of a cup of hot tea is given by kt T( t) T ( T T ) e, s 0 s where Ts is surrounding temperature, T0 is initial temperature of the tea at and k is a using Richardson s extrapolation. Find the exact value and absolute error. Solution: then into f(t), then Hence, the cooling rate function is given by Richardson s extrapolation at with can be approximated using The numerical solution of this example is shown in Figure 1 in Section 4, where is approximately , while the absolute error is shown in cell G4 in Figure

4 The Fourth Version Of Richardson s Extrapolation Spreadsheet Calculator Figure 1 illustrates the fourth version of Richardson s extrapolation spreadsheet calculator. This spreadsheet calculator is able to compute Richardson s extrapolation value up to any level similar to the third version. However, in this version, users can input the function to be differentiated into cell C2 via programming syntax or mathematical form rather than Excel Command. As in the previous version, will be entered into cell D4 and a chosen point of x can be entered into cell C3. The function will be evaluated at the chosen point of x and its value will be shown in cell C4. The command to evaluate the function value is shown on top of the formula bar as given in Figure 1. The customized eval function is given in Appendix A. Users can select the desired accuracy of one decimal place, two decimal places, up to nine decimal places calculation from the drop down menu in cell J3. To obtain the changes on the Richardson s extrapolation table up to the desired accuracy, users need to click on the APPLY button and the values in the Richardson s extrapolation table will be calculated automatically. The APPLY button is associated with VBA programming calculating the values in the Richardson s extrapolation table based on the stopping criteria. The details of the programming can be found in Appendix A. The final answer is shown in cell E4. To calculate the absolute error, users need to enter the exact differentiated function into cell F4. The absolute error will then be calculated automatically in cell G4. Figure 1: Solution using fourth version of Richardson s extrapolation spreadsheet calculator. The advantage of this fourth version of Richardson s extrapolation spreadsheet calculator is that it is more user- friendly especially to those who are not familiar with Microsoft Excel. Conclusion A fourth version of Richardson s extrapolation spreadsheet calculator has been developed to approximate the derivative of the given function at a chosen point with a step size The VBA programming that calculates Richardson s extrapolation values is given in Appendix A and the layout of the spreadsheet calculator is shown in Figure 1. The spreadsheet calculator is very user-friendly. It provides an alternative tool for approximating the numerical differentiation by Richardson s extrapolation. It can be used as a marking scheme for educators and students who need full solutions. Moreover, it reduces calculation time and is hoped to improve students learning ability. 188

5 Acknowledgements This project is financially supported by UTHM MDR research grant scheme vote References Chapra, S. C., & Canale, R. P. (2006). Numerical methods for engineers (Vol. 2). McGraw-Hill. Gaunt, L. F. R. a. J. A. (1927). Philosophical Transactions of the Royal Society of London. K. G. Tay, S. L. K. a. A.-K. (2012). Spreadsheets in Education K. G. Tay, S. L. K. a. R. A.-K. (2009). National Seminar on Science and Technology 2009 K. G. Tay, S. L. K. a. R. A.-K. (2010). Proceeding of the National Symposium on Application of Science Mathematics 2010and 18th Mathematical Science National Symposium. K.G. Tay, S. L. K., R. Abdul-Kahar. (2013). Proceeding of the 21st National Symposium on Mathematical Science (SKSM 2013): Gurney Resort Penang. K.G. Tay, S. L. K., R. Abdul-Kahar. (2014). Proceeding of the International Conference on Mathematics, Engineering and Industrial Applications (ICoMEIA 2014): Gurney Resort Penang. K.G. Tay, S. L. K., R. Abdul-Kahar M.A. Azlan & M.F. Lee. (2013). Spreadsheets in Education K.G. Tay, S. L. K. a. R. A.-K. (2009). Proceeding of the 3rd International Conference on Science and Mathematics Education. Kek, K. G. T. a. S. L. (2008). Proceeding of the National Symposium on Application of Science Mathematics Kek, K. G. T. a. S. L. (2009a). Proceeding of the 17th National Symposium on Mathematical Science. Kek, K. G. T. a. S. L. (2009b). Proceedings of the 4th International Conference on Research and Education in Mathematics. Mathews, J. H., & Fink, K. D. (2004). Numerical methods using MATLAB (Vol. 3): Prentice hall Upper Saddle River, NJ. Richardson, L. F. (1911). Philosophical Transactions of the Royal Society of London. S. L. Kek, S. L. T., K. G. (2008). Proceeding of the National Symposium on Application of Science Mathematics S. L. Kek, S. L. T., K. G. (2009). National Seminar on Science and Technology

6 Appendix A Sub richardson() ' ' richardson Macro ' ' Dim J As Integer Range("A8:N65536").Clear Range("A8:N65536").ShrinkToFit = True Columns("A:N").HorizontalAlignment = xlcenter Cells(2, 3).HorizontalAlignment = xlleft 'column i 'J=0 'Fill in the table of x-h, x, x+h and f(x-h), f(x) and f(x+h) J = 0 Cells(4, 12) = J Cells(4, 14) = (Cells(3, 3) - Cells(4, 4)) / 2 ^ J Cells(4, 15) = Cells(3, 3) Cells(4, 16) = (Cells(3, 3) + Cells(4, 4)) / 2 ^ J 'copy f(x) and paste into f(x-h), f(x) and f(x+h) Cells(4, 3).Select Selection.Copy Cells(5, 14).Select Cells(5, 15).Select Cells(5, 16).Select 'fill in label J Cells(8, 2) = "J" Cells(8, 2).Font.Bold = True Cells(8, 2).Borders.LineStyle = xlcontinuous Cells(8, 2).Interior.Color = RGB(255, 255, 0) 'fill in label h Cells(8, 3) = "h" Cells(8, 3).Font.Bold = True Cells(8, 3).Borders.LineStyle = xlcontinuous Cells(8, 3).Interior.Color = RGB(255, 100, 160) 'fill in K=0 Cells(8, 4) = 0 190

7 Cells(8, 4).Borders.LineStyle = xlcontinuous Cells(8, 4).Interior.Color = RGB(255, 255, 0) 'fill in K=1 Cells(8, 5) = 1 Cells(8, 5).Borders.LineStyle = xlcontinuous Cells(8, 5).Interior.Color = RGB(255, 255, 0) 'fill in row J=0 Cells(9, 2) = J Cells(9, 2).Borders.LineStyle = xlcontinuous Cells(9, 2).Interior.Color = RGB(255, 255, 0) Cells(9, 3) = Cells(4, 4) Cells(9, 3).Borders.LineStyle = xlcontinuous Cells(9, 3).Interior.Color = RGB(255, 100, 160) Cells(9, 4) = Round((Cells(5, 16) - Cells(5, 14)) / (2 * Cells(4, 4)), Cells(2, 19)) Cells(9, 4).Borders.LineStyle = xlcontinuous Cells(9, 4).Interior.Color = RGB(255, 190, 218) 'J=1 J = 1 K = 1 Cells(4, 12) = J Cells(4, 14) = Cells(3, 3) - Cells(4, 4) / 2 ^ J Cells(4, 15) = Cells(3, 3) Cells(4, 16) = Cells(3, 3) + Cells(4, 4) / 2 ^ J Cells(4, 3).Select Selection.Copy Cells(5, 14).Select Cells(5, 15).Select Cells(5, 16).Select 'Row J=1 Cells(10, 2) = J Cells(10, 2).Borders.LineStyle = xlcontinuous Cells(10, 2).Interior.Color = RGB(255, 255, 0) Cells(10, 3) = Cells(4, 4) / 2 ^ J Cells(10, 3).Borders.LineStyle = xlcontinuous Cells(10, 3).Interior.Color = RGB(255, 100, 160) Cells(10, 4) = Round((Cells(5, 16) - Cells(5, 14)) / (2 * (Cells(4, 4) / 2 ^ J)), Cells(2, 19)) Cells(10, 4).Borders.LineStyle = xlcontinuous Cells(10, 4).Interior.Color = RGB(255, 190, 218) 191

8 Cells(10, 5) = Round((4 ^ K * Cells(10, 4) - Cells(9, 4)) / (4 ^ K - 1), Cells(2, 19)) Cells(10, 5).Borders.LineStyle = xlcontinuous Cells(10, 5).Interior.Color = RGB(255, 190, 218) Do Until Abs(Cells(9 + J, 4 + K) - Cells(9 + J, 4 + K - 1)) < Cells(2, 20) Or Abs(Cells(9 + J, 4 + K) - Cells(9 + J - 1, 4 + K - 1)) < Cells(2, 20) J = J + 1 K = K + 1 'K value Cells(8, 4 + K) = K Cells(8, 4 + K).Borders.LineStyle = xlcontinuous Cells(8, 4 + K).Interior.Color = RGB(255, 255, 0) Cells(4, 12) = J Cells(4, 14) = Cells(3, 3) - Cells(4, 4) / 2 ^ J Cells(4, 15) = Cells(3, 3) Cells(4, 16) = Cells(3, 3) + Cells(4, 4) / 2 ^ J Cells(4, 3).Select Selection.Copy Cells(5, 14).Select Cells(5, 15).Select Cells(5, 16).Select 'J value Cells(9 + J, 2) = J Cells(9 + J, 2).Borders.LineStyle = xlcontinuous Cells(9 + J, 2).Interior.Color = RGB(255, 255, 0) Cells(9 + J, 3) = Cells(4, 4) / 2 ^ J Cells(9 + J, 3).Borders.LineStyle = xlcontinuous Cells(9 + J, 3).Interior.Color = RGB(255, 100, 160) Cells(9 + J, 4) = Round((Cells(5, 16) - Cells(5, 14)) / (2 * (Cells(4, 4) / 2 ^ J)), Cells(2, 19)) Cells(9 + J, 4).Borders.LineStyle = xlcontinuous Cells(9 + J, 4).Interior.Color = RGB(255, 190, 218) For C = 1 To K Cells(9 + J, 4 + C) = Round((4 ^ C * Cells(9 + J, 4 + C - 1) - Cells(9 + J - 1, 4 + C - 1)) / (4 ^ C - 1), Cells(2, 19)) Cells(9 + J, 4 + C).Borders.LineStyle = xlcontinuous Cells(9 + J, 4 + C).Interior.Color = RGB(255, 190, 218) Next C Loop Cells(4, 5) = Round(Cells(9 + J, 4 + K), Cells(2, 19)) Cells(4, 5).Select End Sub 192

9 Function Eval(Ref As String) Application.Volatile Eval = Evaluate(Ref) End Function 193