Lecture 13
Vector and Matrix norms, and the
consequences of doing Gaussian Elimination in the presence of floating point
error.
This morning I started by re-introducing
the small two by two system of linear equations that we initially saw last week.
Geometrically this can be represented by two lines crossing, or not crossing
when there is no solution. I than use the geometric analogy to describe the case
in the presence of floating point inaccuracies. I drew a picture of a situation
where the presence of small errors could lead to ambiguities on whether there is
no, one, or infinitely many solutions to the system. I described this type of
input as being "almost singular". In this way, using broad strokes, I gave you
an intuitive but imprecise
overview.
In order to put all
of this into a more concrete form we need to be able to apply a scalar magnitude
to the value of a vector, and matrix. To this end we saw the p-norms, and a set
of properties that define any norm in
general.
The two ways of
measuring the discrepancy between an exact solution to an approximation is to
measure the error and the residual. I gave formulae for these two
quantities using vector
norms.
Using matrix and
vector norms I derived an expression that relates the residual to the error.
Although I have not defined condition number yet, we will see (on Friday) that
the error is bounded from above by the residual times the condition number of
the input matrix.
On Thursday
I will go over solutions to the last two homeworks. To get the most benefit from
this please come prepared with your own solutions.
Posted: Tue - October 11, 2005 at 11:38 AM