BAP: Lecture 4 (Feb 21, 2002)
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In this lecture, we saw the need for partial or complete pivoting
in Gaussian elimination.
We then examined cases in which pivoting is not necessary.
They included:
- Diagonal dominant matrices (proof from Golub and van Loan section 3.4.1).
- Positive Definite matrices (proof from Wilkinson,
"Error Analysis of Direct Methods of Matrix Inversion", J ACM 8,
pp. 281-330. (available at JSTOR).
We then began an examination of the smoothed analysis of the growth factors
of Gaussian elimination without pivoting.
For this class's matlab example, you need the program noPivot.m,
which produces the LU factorization without pivoting.
Today's matlab example follows. Try it out yourself.
Get the code you need at the BAP matlab code directory.
A = [2^(-50), 1; 1, 1];
[l,u] = noPivot(A);
l*u
Now, if we go machine epsilon, which is
eps
2^(-52)
we get
A = [2^(-55), 1; 1, 1];
[l,u] = noPivot(A);
l*u