04.07.1
Chapter 04.07
LU Decomposition
After reading this chapter, you should be able to:
1.
identify when LU decomposition is numerically more efficient than Gaussian
elimination,
2.
decompose a nonsingular matrix into LU, and
3.
show how LU decomposition is used to find the inverse of a matrix.
I hear about LU decomposition used as a method to solve a set of simultaneous linear
equations.
What is it?
We already studied two numerical methods of finding the solution to simultaneous linear
equations – Naïve Gauss elimination and Gaussian elimination with partial pivoting.
Then,
why do we need to learn another method?
To appreciate why LU decomposition could be a
better choice than the Gauss elimination techniques in some cases, let us discuss first what
LU decomposition is about.
For a nonsingular matrix
[ ]
A
on which one can successfully conduct the Naïve Gauss
elimination forward elimination steps, one can always write it as
[ ] [ ][ ]
U
L
A
=
where
[ ]
L
= Lower triangular matrix
[ ]
U
= Upper triangular matrix
Then if one is solving a set of equations
]
[
]
][
[
C
X
A
=
,
then
[ ][ ][ ] [ ]
C
X
U
L
=
as
[ ][ ]
( )
U
L
A
]
[
=
Multiplying both sides by
[ ]
1
−
L
,
[ ] [ ][ ][ ] [ ] [ ]
C
L
X
U
L
L
1
1
−
−
=
[ ][ ][ ]
X
U
I
=
[ ] [ ]
C
L
1
−
as
[ ] [ ]
( )
]
[
1
I
L
L
=
−
[ ][ ] [ ] [ ]
C
L
X
U
1
−
=
as
[ ][ ]
( )
]
[
U
U
I
=
Let
[ ] [ ] [ ]
Z
C
L
=
−
1

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