Silvana Ilie - MTH510 Lecture Notes
1
Matrix Norms and Condition
A vector X = [x1 , x2 , x3 , . . . , xn ] has the Euclidian norm
n
X
2
x2 + x2 + . . . + x2
n
1
2
x2 =
i
=
i=1
For an n nmatrix A
n
A
n
a2
ij
=
f
i=1 j =1
is called the Frobenius norm.
1.1
A
Silvana Ilie - MTH510 Lecture Notes
Binary Floats
Machine numbers in oating point representation are of the form:
x = (1 + f )2e
where
f is mantissa, f = (0.d1 d2 . . . dt1 dt )2 (in base 2)
t = precision (number of digits, t > 0)
emin = lower bound on
Silvana Ilie - MTH510 Lecture Notes
1
LU Factorization
PROBLEM: Find the solution of the following system of linear equations:
Ax = b
where A is an n n matrix, x and b are n 1 (column vectors).
NOTE: Gaussian elimination used in Chapter 9 to nd solutions