Numerical Analysis

Math 370 Fall 1998  MWF 11:30am - 12:25pm
\begin{matrix}c \\ ◯ \end{matrix}1998 Ron Buckmire  Fowler 127

Class 7: Friday September 18

SUMMARY Rates of Convergence of Functions and Sequnences CURRENT READING Burden & Faires (6th edition,) pages 36-37

Rates of Convergence of Functions If we know that lim_{h→0} F (h) = L and and lim_{h→0} G(h) = 0 If a positive constant K exists with

|F (h) − L| ≤ KG(h), for sufficiently small h

then we write F (h) = L + O(G(h))

This can also be computed using the idea that F (h) = L + O(G(h)) if

lim_{h→0} \frac{|F (h) − L|}{|G(h)|} = K

where K is some positive, finite constant. Taylor Expansions Another approach to figuring out rates of convergence of functions is to think about Taylor’s Series Approximations Recall that if you have a function f(x) near a point x = a and f(x) is infinitely-differentiable, you can write down

f(x) =

or you can truncate this series and write down

f(x) ≈

\underline{Exercise} Write down the following Taylor Series Approximations (for small h):

sin(h) ≈

cos(h) ≈

e^{h} ≈

(1 + h)^{p} ≈

\underline{Example}

We can use this new way to re-do our previous exercises and how that cos(h) + \frac{h^{2}}{2} = 1 + O(h^{4})

What is the rate of convergence of sin(h^{3}) as h → 0?