Numerical methods ppt

Numerical methods ppt

• 1.
  NUMERICAL METHODS
• 3.
  DEFINITION  The bisectionmethod is a root finding method that applies to any continuous functions for which one knows two values with opposite signs.  The method consist of repeatedly bisecting the interval defined by these values and then selecting the subinterval in the function changes sign and therefore must contain the root.
• 4.
  POLYNOMIALS For polynomials moreelaborated method exists for testing the existence of a root in an interval.  Descarte’s Rule of Signs  Sturm’s theorem  Budan’s theorem They allow existing bisection method into efficient algorithms for finding all real roots of a polynomial.
• 5.
  Example Suppose that thebisection method is used to find a root of the polynomial 𝑓 𝑥 = 𝑥3 − 𝑥 − 2 Put x=1,2,… 𝑓 1 = 13 − 1 − 2 = −2 𝑓 2 = 23 − 2 − 2 = 4 Both had –ve points So the midpoint is 𝑐 = 2=1 2 = 1.5 𝑓 1 = 1.53 − 1.5 − 2 = −0.125
• 6.
  ITERATION 𝒂 𝒏𝒃 𝒏 𝒄 𝒏 𝒇(𝒄 𝒏) 1 1 2 1.5 -0.125 2 1.5 2 1.75 1.609 3 1.5 1.75 1.625 1.666 4 1.5 1.625 1.5625 0.252 5 1.5 1.5625 1.5312500 0.059 6 1.5 1.5312500 1.5156250 -0.034 7 1.5156250 1.5312500 1.5234375 0.0122 8 1.5156250 1.5234375 1.5195313 -0.0109 9 1.5195313 1.5214844 1.5205078 0.0006 10 1.5195313 1.5214844 1.5209961 -0.0051 11 1.5205078 1.5214844 1.5212402 -0.0022 12 1.5209961 1.5214844 1.5212402 -0.0008 13 1.5212402 1.5214844 1.5213623 -0.0001 14 1.5213623 1.5214844 1.5214233 0.0002 15 1.5213623 1.5214233 1.5213928 0.00078
• 7.
  RESULT 1.521 is aReal root of the equation 𝑓 𝑥 = 𝑥3 − 𝑥 − 2
• 8.
  THANK YOU