Computer Arithmetic Continued and Root Finding Algorithms | MATH 441, Study notes of Mathematical Methods for Numerical Analysis and Optimization

Download Computer Arithmetic Continued and Root Finding Algorithms | MATH 441 and more Study notes Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity! MATH 441/541 - Numerical Analysis Sixth Meeting: Computer Arithmetic Continued & Root Finding Algorithms Thursday, September 27th Müller’s Method 1. Given the continuous function f(x) and three reasonable guesses to the root, x0, x1, and x2. 2. A Visual Explanation: 3. The Formula for the New Guess for the Root: Given three guesses, x0, x1, and x2 generate the new approximation, x3 as: x3 = x2 − 2c b+ sgn(b) √ b2 − 4ac where c = f(x2) b = (x0 − x2)2(f(x1)− f(x2))− (x1 − x2)2(f(x0)− f(x2)) (x0 − x2)(x1 − x2)(x0 − x1) a = (x1 − x2)2(f(x0)− f(x1))− (x0 − x2)2(f(x1)− f(x2)) (x0 − x2)(x1 − x2)(x0 − x1) 4. Strengths and Weaknesses: 5. Rates of Convergence for Methods1: Bisection Secant Newton False Position Fixed Point Müller’s α = 2 α ≈ 1 α = 1 + √ 5 2 ≈ 1.62 (for simple roots) α = 1 α ≈ 1.839 α = 1 (for multiple roots) 1For Müller’s Method rate in particular see Michael T. Heath. Scientific Computing: An Introductory Survey, 2nd Ed.. McGraw Hill, 2002. Page 234 1