Week 2 Flashcards
Rate and order of convergence of a sequence x_k of a numerical approximation of some exact quantity x*
x_k converges with order q to x* if
(For rate of convergence 0<μ<inf)
If q = 1? What does this mean for μ?
q=1 means linearly convergent
This means we must have 0<μ<=1 for convergence otherwise sequence diverges
1 sublinear convergence
1=μ
q=1
1 súper linear convergence
q>=1 and μ=0
Adjust Taylor series for f(x) for rate of convergence of newton method
Include quadratic term
Taylor series of Newton Method with quadratic
Two consecutive iterations
Rewrite Taylor series of newton method in terms of ε
Simplify ε Taylor expansion of newton method
Meaning of quadratic convergence in terms of error
Each successive ε reduces size of error by 2 orders of magnitude
ε Taylor’s expansion of newton method: double zero case
f(x) for Secant method formula
Secant q=?
(1+sqrt(5))/2
Find x_n using secant
Preferable value for q?
Higher
Eg: Newton > Secant
Taylor expansion of f(x) about x_0
Recursion relation for Newton method
Draw diagram of Newton method labelling for 4 points
Draw diagram of secant method with first 5 points
Draw bisection method
