THE SECANT METHOD
Newtons method was based on using the line tangent
to the curve of y = f (x), with the point of tangency
(x0, f (x0). When x0 , the graph of the tangent
line is approximately the same as the graph of y =
f (x) around x = . We then used t
PROPAGATION OF ERROR
Suppose we are evaluating a function f (x) in the machine. Then the result is generally not f (x), but rather
e
an approximate of it which we denote by f (x). Now
suppose that we have a number xA xT . We want
e
to calculate f (xT ), b
SOME DEFINITIONS
Let xT denote the true value of some number, usually
unknown in practice; and let xA denote an approximation of xT .
The error in xA is
error(xA) = xT xA
The relative error in xA is
error(xA)
xT xA
=
rel(xA) =
xT
xT
Example: xT = e, xA =
COMPUTING ANOMALIES
These examples are meant to help motivate the study
of machine arithmetic.
1. Calculator example : Use an HP-15C calculator,
which contains 10 digits in its display. Let
x1 = x2 = x3 = 98765
There are keys on the calculator for the mea
EVALUATING A POLYNOMIAL
Consider having a polynomial
p(x) = a0 + a1x + a2x2 + + anxn
which you need to evaluate for many values of x. How
do you evaluate it? This may seem a strange question,
but the answer is not as obvious as you might think.
The standa
SUMMATION
How should we compute a sum
S = a1 + a2 + + an
with a sequence of machine numbers cfw_a1, ., an. Should
we add from largest to small, should we add from
smallest to largest, or should we just add the numbers
based on their original given order?
ROOTFINDING
We want to nd the numbers x for which
f (x) = 0, with f a given function. Here, we denote
such roots or zeroes by the Greek letter . Rootnding problems occur in many contexts. Sometimes they
are a direct formulation of some physical situtation
INTERPOLATION
Interpolation is a process of nding a formula (often
a polynomial) whose graph will pass through a given
set of points (x, y ).
As an example, consider dening
x1 = ,
x0 = 0,
4
and
yi = cos xi,
x2 =
2
i = 0, 1, 2
This gives us the three point
MULTIPLE ROOTS
We study two classes of functions for which there is
additional diculty in calculating their roots. The rst
of these are functions in which the desired root has a
multiplicity greater than 1. What does this mean?
Let be a root of the functi
FIXED POINT ITERATION
We begin with a computational example. Consider
solving the two equations
E1: x = 1 + .5 sin x
E2: x = 3 + 2 sin x
Graphs of these two equations are shown on accompanying graphs, with the solutions being
E1: = 1.49870113351785
E2: =
ROOTFINDING : A PRINCIPLE
We want to nd the root of a given function f (x).
Thus we want to nd the point x at which the graph of
y = f (x) intersects the x-axis. One of the principles
of numerical analysis is the following.
If you cannot solve the given p
THE TAYLOR POLYNOMIAL ERROR FORMULA
Let f (x) be a given function, and assume it has derivatives around some point x = a (with as many derivatives as we nd necessary). For the error in the Taylor
polynomial pn(x), we have the formulas
1
f (x) pn(x) =
(x a