CS205  Class 3
1.
When solving systems of equations, we want
Ax=b
.
We define the
residual
as
r=bAx
, and
note that the residual gives us some measure of the error.
Of course the goal is to attain
r=
0.
2.
Both Gaussian Elimination and the y factorization method are examples of direct methods.
The other kind of method is called an iterative method where one starts with an initial guess
1
x
and iterates through a sequence
1
x
,
2
x
,
3
x
… ending up with a final guess of
m
x
.
a.
Issues here include both finding an initial guess
1
x
and deciding on a stopping criterion
that says that
m
x
, for some m, is good enough.
b.
The best preconditioner, if we could guess it would be
A
1
.
Then we would get
Ix=A
1
b
.
On a sparse matrix we might include incomplete factorizations such as an incomplete
Cholesky preconditioner where we don’t allow fillin of nonzero entries of the
L
matrix
not in the orginal
A
matrix.
3.
So far we have discussed solving Ax=b for square
n
n
×
matrices
A
. For more general
n
m
×
matrices, there are a variety of scenarios.
a.
When
m < n
, the problem is
underdetermined
since there is not enough information to
determine a unique solution for all the variables. Usually m<n implies that there are
infinite solutions
. However, in some cases, there may be contradictory equations
leading to the absence of any solutions.
b.
When
m > n
, the problem is
overdetermined
although this in itself doesn’t tell us
everything about the nature of the solution. For example, if enough equations are linear
combinations of each other, there can still be a unique solution or infinite solutions.
i.
We can use the rank of the matrix to enumerate the possibilities. Recall that the
rank
of a matrix is the number of linearly independent columns that it has. Thus
a
n
m
×
matrix has at most a rank of n.
ii.
If the rank <
n
some columns are linear combinations of others and we say that
the matrix is
rankdeficient
and there may be an infinite number of solutions.
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 Fall '07
 Fedkiw
 Linear Algebra, Linear least squares

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