EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen
SECTION 4: INTRODUCTION TO BASIC NUMERICAL LINEAR ALGEBRA IN R
Section 4 culminates with two factorization methods for solving various types of systems
of linear equations.
The first of these methods is LU factorization, and the second is
LU factorization is important for solving large systems of linear
equations, especially those for which there are many “right-hand-side” vectors.
is quite large.
For instance, such problems arise in large linear programming
Choleski factorization is important for a special class of
“positive definite” matrices.
We begin by laying the linear algebraic foundation for these
Recall an earlier example:
(1 10 )
The exact answer is
However, the vector
results in residuals of 0
for the first and second equations, respectively.
Therefore, if one didn't know
the exact solution it might appear that the vector
is not a bad solution for the system of
However, as we've already seen, the relative error is
a percentage error of 100%.
Consider the system of equations
The exact solution of this system of equations is
However, let's use
arithmetic to solve this system by Gauss elimination.
That is we "pivot"
Upon doing so, in (10,4) arithmetic, the new