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Pivoted LU Factorization (Crout version)
Ref:
(S&H) Applied Numerical Methods for Engineers
,R
.
Schilling and S. Harris, Brooks/Cole, 2000.
The purpose of these notes is to show how to compute the
factors of a square matrix,
A
, of order n so that:
(1)
PA
=
LU
(Crout version)
where
P
is a (general) permutation matrix of order n;
L
is
lower triangular (i.e.
L
=[
l
ij
] and l
ij
= 0 for j>i) and
U
is
unit upper triangular; i.e.
U
u
ij
] with u
ii
=1.0 and u
ij
=0.0
for i>j.
It is a useful property of
P
that it is nonsingular
and
(2)
P
1
=
P
T
See separate notes for more details on permutation matrices.
The Crout version of LU decomposition assigns 1.0 to the
values on the diagonal of
U
.
In the Doolittle version, the L
matrix has 1.0 values on the diagonals:
(3)
PA
=
L
U
(Doolittle version)
Equations (1) or (3) are useful for solving sets of systems
of equations:
(4)
Ax
k
=
b
k
for k=1,2,.
..K
I.e. solving
Ax
=
b
when solutions for a set of various
b
vectors is sought.
In LU decomposition we do not repeat the
steps of Gaussian elimination for a set of augmented matrices
with last column (
b
vector) being replaced for each new
problem, but rather we reuse the LU factorization.
Here are the steps in the Crout method for solving one set of
the equations in (4).
1.
Obtain the factors
L
,
U
, and
P
so that
PA
=
LU
.
Note that
Ax
=
b
can then be rewritten as
PAx
=
LUx
=
Pb
≡
c
.
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 Winter '10
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 Matrices, Triangular matrix, AC3, AC3 LC1 LC2

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