Lecture 11
Gaussian Elimination (No pivot) as reported by
Chris McAloney
We began the class by discussing how a linear
system of equations
could be written in the
form of a single matrix equation Ax = b,
where
A is the matrix formed by taking the
coefficients of the variables in
the linear
system, x is a column vector of unknown values, and b is
a
column vector of known values -- the
solutions to the linear
equations. We then
considered two specific types of linear
systems;
namely diagonal systems and
triangular systems and discussed how
they
could be solved -- a direct solution in
the case of diagonal systems,
and forward or
backward substitution in the case of
triangular
systems.
The
remainder of the class was spent outlining the
Gaussian
Elimination algorithm (Algorithm 8.4
in the Recktenwald book) whereby
an augmented
matrix (the matrix formed by adding the column vector
b
to the coefficient matrix A) could be
reduced through a series of row
operations
(specifically, the addition of a multiple of one row
to
another row) into a matrix in upper
triangular form, which can then be
solved for
the vector x through backward substitution. Also, an
error
in the Recktenwald version of the
algorithm was discussed.
Posted: Thu - October 7, 2004 at 10:04 AM