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Numerical Methods Part: Simpson Rule For Integration. http://numericalmethods.eng.usf.edu

Numerical Methods Part: Simpson Rule For Integration. http://numericalmethods.eng.usf.edu. For more details on this topic Go to http://numericalmethods.eng.usf.edu Click on Keyword. You are free. to Share – to copy, distribute, display and perform the work

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Numerical Methods Part: Simpson Rule For Integration. http://numericalmethods.eng.usf.edu

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  1. Numerical MethodsPart: Simpson Rule For Integration.http://numericalmethods.eng.usf.edu

  2. For more details on this topic • Go to http://numericalmethods.eng.usf.edu • Click on Keyword

  3. You are free • to Share – to copy, distribute, display and perform the work • to Remix – to make derivative works

  4. Under the following conditions • Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). • Noncommercial — You may not use this work for commercial purposes. • Share Alike — If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.

  5. Chapter 07.08: Simpson Rule For Integration. Lecture # 1 Major: All Engineering Majors Authors: Duc Nguyen http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates 4/1/2014 http://numericalmethods.eng.usf.edu 5

  6. Introduction The main objective in this chapter is to develop appropriated formulas for obtaining the integral expressed in the following form: (1) where is a given function. Most (if not all) of the developed formulas for integration is based on a simple concept of replacing a given (oftently complicated) function by a simpler function (usually a polynomial function) where represents the order of the polynomial function.

  7. In the previous chapter, it has been explained and illustrated that Simpsons 1/3 rule for integration can be derived by replacing the given function with the 2nd –order (or quadratic) polynomial function , defined as: (2) http://numericalmethods.eng.usf.edu

  8. In a similar fashion, Simpson rule for integration can be derived by replacing the given function with the 3rd-order (or cubic) polynomial (passing through 4 known data points) function defined as (3) which can also be symbolically represented in Figure 1.

  9. Method 1 The unknown coefficients (in Eq. (3)) can be obtained by substituting 4 known coordinate data points into Eq. (3), as following (4) http://numericalmethods.eng.usf.edu

  10. Eq. (4) can be expressed in matrix notation as (5) The above Eq. (5) can be symbolically represented as (6) http://numericalmethods.eng.usf.edu

  11. Thus, (7) Substituting Eq. (7) into Eq. (3), one gets (8) http://numericalmethods.eng.usf.edu

  12. Remarks As indicated in Figure 1, one has (9) With the help from MATLAB [2], the unknown vector (shown in Eq. 7) can be solved. http://numericalmethods.eng.usf.edu

  13. Method 2 Using Lagrange interpolation, the cubic polynomial function that passes through 4 data points (see Figure 1) can be explicitly given as (10) http://numericalmethods.eng.usf.edu

  14. Simpsons Rule For Integration Thus, Eq. (1) can be calculated as (See Eqs. 8, 10 for Method 1 and Method 2, respectively): Integrating the right-hand-side of the above equations, one obtains (11) http://numericalmethods.eng.usf.edu

  15. Since hence , and the above equation becomes: (12) The error introduced by the Simpson 3/8 rule can be derived as [Ref. 1]: (13) , where http://numericalmethods.eng.usf.edu

  16. Example 1 (Single Simpson rule) Compute by using a single segment Simpson rule Solution In this example: http://numericalmethods.eng.usf.edu

  17. http://numericalmethods.eng.usf.edu

  18. http://numericalmethods.eng.usf.edu

  19. Applying Eq. (12), one has: The “exact” answer can be computed as http://numericalmethods.eng.usf.edu

  20. 3. Multiple Segments for Simpson Rule Using = number of equal (small) segments, the width can be defined as (14) Notes: = multiple of 3 = number of small segments http://numericalmethods.eng.usf.edu

  21. The integral, shown in Eq. (1), can be expressed as (15) http://numericalmethods.eng.usf.edu

  22. Substituting Simpson rule (See Eq. 12) into Eq. (15), one gets (16) (17) http://numericalmethods.eng.usf.edu

  23. Example 2 (Multiple segments Simpson rule) Compute using Simple multiple segments rule, with number (of ) segments = = 6 (which corresponds to 2 “big” segments). http://numericalmethods.eng.usf.edu

  24. Solution In this example, one has (see Eq. 14): http://numericalmethods.eng.usf.edu

  25. http://numericalmethods.eng.usf.edu

  26. Applying Eq. (17), one obtains: http://numericalmethods.eng.usf.edu

  27. Example 3 (Mixed, multiple segments Simpson and rules) Compute using Simpson 1/3 rule (with 4 small segments), and Simpson 3/8 rule (with 3 small segments). Solution: In this example, one has: http://numericalmethods.eng.usf.edu

  28. http://numericalmethods.eng.usf.edu

  29. Similarly: http://numericalmethods.eng.usf.edu

  30. For multiple segments using Simpson rule, one obtains (See Eq. 19): http://numericalmethods.eng.usf.edu

  31. For multiple segments using Simpson 3/8 rule, one obtains (See Eq. 17): The mixed (combined) Simpson 1/3 and 3/8 rules give: http://numericalmethods.eng.usf.edu

  32. Remarks: (a) Comparing the truncated error of Simpson 1/3 rule (18) With Simple 3/8 rule (See Eq. 13), the latter seems to offer slightly more accurate answer than the former. However, the cost associated with Simpson 3/8 rule (using 3rd order polynomial function) is significant higher than the one associated with Simpson 1/3 rule (using 2nd order polynomial function). http://numericalmethods.eng.usf.edu

  33. (b) The number of multiple segments that can be used in the conjunction with Simpson 1/3 rule is 2,4,6,8,.. (any even numbers). (19) http://numericalmethods.eng.usf.edu

  34. However, Simpson 3/8 rule can be used with the number of segments equal to 3,6,9,12,.. (can be either certain odd or even numbers). (c) If the user wishes to use, say 7 segments, then the mixed Simpson 1/3 rule (for the first 4 segments), and Simpson 3/8 rule (for the last 3 segments). http://numericalmethods.eng.usf.edu

  35. 4. Computer Algorithm For Mixed Simpson 1/3 and 3/8 rule For Integration Based on the earlier discussions on (Single and Multiple segments) Simpson 1/3 and 3/8 rules, the following “pseudo” step-by-step mixed Simpson rules can be given as Step 1 User’s input information, such as Given function integral limits = number of small, “h” segments, in conjunction with Simpson 1/3 rule. http://numericalmethods.eng.usf.edu

  36. = number of small, “h” segments, in conjunction with Simpson 3/8 rule. Notes: = a multiple of 2 (any even numbers) = a multiple of 3 (can be certain odd, or even numbers) http://numericalmethods.eng.usf.edu

  37. Step 2 Compute http://numericalmethods.eng.usf.edu

  38. Step 3 Compute “multiple segments” Simpson 1/3 rule (See Eq. 19) (19, repeated) http://numericalmethods.eng.usf.edu

  39. Step 4 Compute “multiple segments” Simpson 3/8 rule (See Eq. 17) (17, repeated) Step 5 (20) and print out the final approximated answer for I. http://numericalmethods.eng.usf.edu

  40. The End http://numericalmethods.eng.usf.edu

  41. Acknowledgement This instructional power point brought to you by Numerical Methods for STEM undergraduate http://numericalmethods.eng.usf.edu Committed to bringing numerical methods to the undergraduate

  42. For instructional videos on other topics, go to http://numericalmethods.eng.usf.edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

  43. The End - Really

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