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Numerical Methods Root Finding

4. Numerical Methods Root Finding. Fixed-Point Iteration---- Successive Approximation. Many problems also take on the specialized form: g( x )= x , where we seek, x, that satisfies this equation. In the limit, f(x k ) =0, hence x k+1 =x k. f(x)=x. g(x). Simple Fixed-Point Iteration.

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Numerical Methods Root Finding

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  1. 4 Numerical Methods Root Finding

  2. Fixed-Point Iteration---- Successive Approximation • Many problems also take on the specialized form: g(x)=x, where we seek, x, that satisfies this equation. • In the limit, f(xk)=0, hence xk+1=xk f(x)=x g(x)

  3. Simple Fixed-Point Iteration • Rearrange the function f(x)=0 so that x is on the left-hand side of the equation: x=g(x) • Use the new function g to predict a new value of x - that is, xi+1=g(xi) • The approximate error is given by:

  4. Example:

  5. Iterative Solution • Find the root of f(x) = e-x – x • Start with a guess say x1=1, • Generate • x2=e-x1= e-1= 0.368 • x3=e-x2= e-0.368 = 0.692 • x4=e-x3= e-0.692=0.500 In general: After a few more iteration we will get

  6. Problem • Find a root near x=1.0 and x=2.0 • Solution: • Starting at x=1, x=0.292893 at 15th iteration • Starting at x=2, it will not converge • Why? Relate to g'(x)=x. for convergence g'(x) < 1 • Starting at x=1, x=1.707 at iteration 19 • Starting at x=2, x=1.707 at iteration 12 • Why? Relate to

  7. The False-Position Method (Regula-Falsi) • We can approximate the solution by doing a linear interpolation between f(xu) and f(xl) • Find xr such that l(xr)=0, where l(x) is the linear approximation of f(x) between xl and xu • Derive xr using similar triangles

  8. Birge – Vieta Method: • Used for finding roots of polynomial functions. • Uses “synthetic division” of polynomial to extract factor of the given polynomial in the form of (x – p). Problem: Find roots of f (x) = 2x³ – 5x + 1 using Birge – Vieta Method. Solution: Assume that x = 1 is root of the equation. Hence initial approximation of the solution is p0 = 1. Synthetic Division will be performed as below: Let f (x) = a0x3 + a1x2 + a2x + a3 p0 a0 a1 a2 a3 p0b0 p1b1 p2b2 p0 b0 b1=a1+p0b0 b1 b2 b3 p1 = p0 – b3/c2 s i m i l a r l y Repeat synthetic division using p1 c0 c1 c2 c3

  9. Birge-Vieta Method • NR method with f(x) and f'(x) evaluated using Horner’s method • Once a root is found, reduce order of polynomial

  10. Iteration No. 1: 1 2 0 -5 1 2 2 -3 1 2 2 -3 -2 2 4 1 2 4 1 -1 p1 = p0 – b3/c2 = 1 – (-2)/1 = 3 Iteration No. 2: 3 2 0 -5 1 6 18 39 Not required 3 2 6 13 40 6 36 147 2 12 49 187 p2 = p1 – b3/c2 = 3 – 40/49 = 2.1837

  11. the equation x3+x2-3x-3 using Birge-Vieta Method where x0= 2. Using the synthetic division, 2|1 1 -3 -3 | 2 6 6 |1 3 3 3¬f(x0) | 2 10 |1 5 13¬f ’(x0) Now, x1= 2 – 3/13 = 1.7692

  12. Summary

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