Bracketing Methods

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Solve nonlinear equations using bracketing methods: Bisection and False Position

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Bracketing Methods

  1. 1. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Roots of Nonlinear Equations
  2. 2. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Understand the need for numerical solutions of nonlinear equations • Be able to use the bisection algorithm to find a root of an equation • Be able to use the false position method to find a root of an equations • Write down an algorithm to outline the method being used • Realize the need for termination criteria
  3. 3. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Root of Nonlinear Equations • Solve   0xf
  4. 4. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Bracketing Methods
  5. 5. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Intermediate Value Theorem • For our specific interest If f(x) is continuous in the interval [a,b], and f(a).f(b)<0, then there exists ‘c’ such that a<c<b and f(c)=0.
  6. 6. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • For the parachutist problem    mct e c mg tv / 1   • Find ‘c’ such that   smv /4010  • Where, kgmsmg 1.68,/8.9 2 
  7. 7. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example (cont’d) • You get  1.68/10 1 8.9*1.68 40 c e c   • OR: • Giving,     401 38.667 147.0   c e c cf     269.216&067.612  ff
  8. 8. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example (cont’d) • Graphically
  9. 9. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Bisection Method
  10. 10. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Algorithm 1. Search for a & b such that f(a).f(b)<0 2. Calculate ‘c’ where c=0.5(a+b) 3. If f(c)=0; end 4. If f(a).f(c)>0 then let a=c; goto step 2 5. If f(b).f(c)>0 then let b=c; goto step 2
  11. 11. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Algorithm (cont’d) • That algorithm will go on forever! • We need to define a termination criterion • Examples of termination criteria: 1. |f(c)|<es 2. |b-a|<es 3. ea=|(cnew -cold)/cnew|<es 4. Number of iterations > N
  12. 12. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Algorithm: Modified • So, let’s modify the algorithm 1. Search for a & b such that f(a).f(b)<0 2. Calculate ‘c’ where c=0.5(a+b) 3. If |f(c)|<es; end 4. If f(a).f(c)>0 then let a=c; goto step 2 5. If f(b).f(c)>0 then let b=c; goto step 2
  13. 13. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com False-Position Method
  14. 14. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The False-Position Method
  15. 15. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Evaluating ‘c’ • The slope of the line joining the two point maybe written as: bc yy mor ac yy m bcac       bc yy ac yy bcac           bcac yyacyybc 
  16. 16. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Evaluating ‘c’      ba yacybc  00 aybycycy baab   ab ab yy byay c           afbf bafabf c   
  17. 17. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com False Position Algorithm 1. Search for a & b such that f(a).f(b)<0 2. Calculate ‘c’ where c=(af(b)-bf(a))/(f(b)-f(a)) 3. If |f(c)|<es; end 4. If f(a).f(c)>0 then let a=c; goto step 2 5. If f(b).f(c)>0 then let b=c; goto step 2
  18. 18. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Conclusion • The need for numerical solution of nonlinear equations led to the invention of approximate techniques! • The bracketing techniques ensure that you will find a solution for a continuous function if the solution exists • A termination criterion should be embedded into the numerical algorithm to ensure its termination!
  19. 19. Numerical Analysis: Bracketing Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Homework #1 • Chapter 5, page 139, numbers: 5.3,5.6,5.7,5.8,5.12 • You are not required to get the solution graphically! • Homework due Next week!

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