MATH 175: Numerical Analysis II

1. MATH 175: Numerical Analysis II Lecturer: Jomar F. Rabajante IMSP, UPLB 2ndSem AY 2012-2013

2. ACCELERATING CONVERGENCE:Aitken’s ∆2 process • Used to accelerate linearly convergent sequences, regardless of the method used. • Actually, this acceleration method is not only applicable to root-finding algorithms.

3. ACCELERATING CONVERGENCE:Aitken’s ∆2 process F O R M U L A (update the value of rk)

4. ACCELERATING CONVERGENCE:Aitken’s ∆2 process Actually, Aitken’s process is an extrapolation.

5. ACCELERATING CONVERGENCE:Aitken’s ∆2 process Steffensen’s Method: a modified Aitken’s delta-squared process applied to fixed point iteration For sample computations, see the MS Excel file.

6. Other Method: Muller’s Method • Extension of secant method – instead of using linear interpolation, it uses quadratic interpolation (parabola) • May generate complex zeros (use software that can understand complex arithmetic) • Less sensitive to starting values compared to Newton’s Method • Order of convergence: p≈1.84

7. Other Method: Muller’s Method Initial points: f parabola

8. Other Method: Muller’s Method Formula z1,z2 & z3 came from Newton’s Divided Difference, and xk came from quadratic formula