ELEC2102
ELEC2102 -
Lec
Lec 7
Matrices/ numerical methods
introduction
Text 3.2, 6.1-6.3
LU Matrix
LU Matrix
decomposition
decomposition
•
L
and
U
matrices
are easy to invert
•
P
swaps rows
•
P
.
M
=
L
.
U

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LU decomposition 2
LU decomposition 2
•
P rearranges rows to minimise numerical
error
•
For a set of equations
M
.
x
=
y
L
.
U
.
x
=
P
.
y
or
L
.
z
=
P
.
y
where
z
=
U
.
x
solve
z
=
L
\
P
.
y
then
x
=
U
\
z
•
Matlab uses this method with
x
=
M
\
y
but also some other methods depending on
M
and
y
QR Matrix decomposition
QR Matrix decomposition
•
M
=
Q
.
R
•
R
is upper triangular and
Q
is a square
orthonormal matrix:
•
Different columns have an inner product of
zero and the inner product of a column with
itself is one
col(n)
T
x col(m)=0
(m not equal to n)
col(n)
T
x col(n)=1
Q
T
x
Q
=
I
•
Note: col() is not a Matlab command
use “:”
to do this
•
Use
QR
for data fitting – see later