Lecture 10.
Gaussian Elimination and LU
decomposition.
Quiz 1 was returned today. In general
people have done extremely well on this quiz.
Today we began looking at
solving a system of linear equations. The method that we use is Gaussian
elimination. It appears that everyone in the class has taken a course in linear
algebra and has already seen Gaussian elimination at least once. Nevertheless I
went through several simple 2 by 2 problems to illustrate the possible outcomes
when trying to solve a system of linear equations. Also through the use of an
example we saw how Gaussian elimination followed by back substitution can be
used to solve a system of linear equations. Actually, today I presented a slight
variant of Gaussian elimination where we save the multipliers so that we can
compute a unit lower triangular matrix L and an upper triangular matrix U such
that LU = A. In this way we can use L and U to solve a system Ax = b. We also
observed that having L and U at hand allows us to subsequently solve for Ax=
bi for any other column vectors bi.
We have yet to see how
floating point arithmetic impacts on the problem. However, we will do this
within the next few days.
Posted: Tue - October 4, 2005 at 10:49 AM