Initial Value Problems for Ordinary
Dierential Equations
A dierential equations if an equation that involves one or
more derivatives
DEs are used to model a large number of systems in physics
and engineering (among others).
The solution to a DE is a eq
Numerical Dierentiation
Recall
f (x0 + h) f (x0 )
h0
h
Provided the limit exists.
f (x0 ) = lim
We can use this as an approximation to the derivative at x0 .
Namely,
f (x0 + h) f (x0 )
f (x0 )
h
Note that h = x1 x0
This formula gives us no error term
Chapter 3, Interpolation and Polynomial
Approximation
Interpolation - determine a function that represents a collection
of data points.
Polynomials are a natural choice for interpolation
(well-behaved functions)
Weierstrass Approximation Theorem.
Let
Solutions of Equations of one Variable
Let f be a real-valued function dened for some subset of the
real numbers.
We say f has a root at x = p provided f (p) = 0.
Note that the equation f (x) = 0 has an x-intercept at the point
(p, 0).
1
>
f := x -> x^
Roundo Error and Computer
Arithmetic
Computer arithmetic is done using a nite set of numbers.
Consequences
There is a largest possible number
There is a smallest possible number
There is a smallest possible positive number
The computer hardware dict