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Numerical Methods Root Finding

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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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slide1

4

Numerical Methods

Root Finding

fixed point iteration successive approximation
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)

simple fixed point iteration
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:
iterative solution
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

problem
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
the false position method regula falsi
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
slide8

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

birge vieta method
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
slide10

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

slide11

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