1. MATH 175: Numerical Analysis II
Lecturer: Jomar F. Rabajante
IMSP, UPLB
2nd Sem AY

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. r
delta r
delta^2 r
rk-2
rk-1
rk-1-rk-2 rk
rk-rk-1
(rk-rk-1)-(rk-1-rk-2)

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