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The University of Jordan Mechatronics Engineering Department Dr. Osama M. Al-Habahbeh 2012

The University of Jordan Mechatronics Engineering Department Dr. Osama M. Al-Habahbeh 2012. Chapter 5. Bracketing Methods Two initial guesses for the root are required. • Graphical Techniques

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The University of Jordan Mechatronics Engineering Department Dr. Osama M. Al-Habahbeh 2012

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  1. The University of JordanMechatronics Engineering DepartmentDr. Osama M. Al-Habahbeh2012

  2. Chapter 5 Bracketing Methods Two initial guesses for the root are required. • Graphical Techniques They are not precise. They provide rough estimates of roots starting guesses for numerical methods.

  3. 1 root Figure (a) XL : Lower bound. Xu : Upper bound.

  4. 3 roots Figure (b) If f(XL) and f(XU) have opposite signs odd number of roots.

  5. Figure (c)

  6. Figure (d) If f(XL) and f(Xu) have same sign No roots or even number of roots .

  7. Exceptions: tangential and discontinuous functions: Figure(e): tangential to the X-axis

  8. Figure(f): discontinuous function

  9. Bisection Method • If f(x) is real and continuous on [XL, Xu] and f(XL)f(Xu) < 0 opposite signs then there is at least one real root between XL and Xu . • Search internal ,then subinterval (half of previous interval) . Function value at midpoint is evaluated sign change subinterval repeat evaluation.

  10. Algorithm 1. Choose XL and Xu such that f(XL)f(Xu) < 0 2. Estimate root by 3. If f(XL)f(Xr) < 0 root is at lower subinterval set Xu = Xr and go to step 2 . If f(XL)f(Xr) > 0 root is at upper subinterval set XL = Xr ,and go to step 2 . If f(XL)f(Xr) = 0 root = Xr end of computation . Xr = XL + Xu 2

  11. Termination criteria and error estimates : a = Xrnew - Xrold × 100% Xrnew Xrnew : Root of present iteration. Xrold : Root from previous iteration. a : Relative Error . • When a < s Terminate the computation . s: Stopping criterion.

  12. Ean =  X 2n • Each iteration halves the error Ean :Error at iteration (n) .  X : Xu- XL  iteration(0) Zero iteration. n: Number of iterations. • If Ea,d is the desired error Required number of iterations (n) is : n = log ( X / Ea,d) = log2 ( X / Ea,d) log 2

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