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Easy systems
Gaussian elimination
LU factorisation
LU Factorisation
Dhavide Aruliah
UOIT
MATH 2070U
c
±
D. Aruliah (UOIT)
LU Factorisation
MATH 2070U
1 / 27
Easy systems
Gaussian elimination
LU factorisation
LU Factorisation
1
Easytosolve systems
2
Gaussian elimination
3
LU factorisation
c
±
D. Aruliah (UOIT)
LU Factorisation
MATH 2070U
2 / 27
Easy systems
Gaussian elimination
LU factorisation
Linear systems of equations
Rule for matrix multiplication permits representation of linear
systems of equations using matrices and vectors
e.g., linear system of equations
2
x
1
+
x
2
+
x
3
=
4
4
x
1
+
3
x
2
+
3
x
3
+
x
4
=
11
8
x
1
+
7
x
2
+
9
x
3
+
5
x
4
=
29
6
x
1
+
7
x
2
+
9
x
3
+
8
x
4
=
30
can be written as
A
x
=
b
with
2
1
1
0
4
3
3
1
8
7
9
5
6
7
9
8

{z
}
A
x
1
x
2
x
3
x
4

{z
}
x
4
11
29
30

{z
}
b
c
±
D. Aruliah (UOIT)
LU Factorisation
MATH 2070U
4 / 27
Easy systems
Gaussian elimination
LU factorisation
Solving linear systems of equations in M
ATLAB
Want to solve
n
×
n
system of linear equations
A
x
=
b
Backslash operator does this in M
ATLAB
:
>> A = [ 1 3 ; 4 7 ]
>> b = [ 4; 11 ]
>> x = A \ b
Read
x = A \ b
as
x
=
A

1
b
c
±
D. Aruliah (UOIT)
LU Factorisation
MATH 2070U
5 / 27
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View Full Document Easy systems
Gaussian elimination
LU factorisation
Solving linear systems in M
ATLAB
Present goal: to understand what
\
does
I
Gaussian elimination
I
LU
factorisation
I
Pivoting
See
doc mldivide
←
left matrix division
Solution of
A
x
=
b
Never
solve linear systems by computing
A

1
and
x
=
A

1
b
!
Use backslash (
mldivide
) or some solver enclosing sensible algorithms
c
±
D. Aruliah (UOIT)
LU Factorisation
MATH 2070U
6 / 27
Easy systems
Gaussian elimination
LU factorisation
Diagonal systems
Given vector
b
= (
b
1
, . . . ,
b
n
)
T
∈
R
n
and diagonal matrix
D
=
diag
(
d
)
with
d
= (
d
1
, . . . ,
d
n
)
T
∈
R
n
, wish to solve linear
system of equations
D
x
=
b
, i.e.,
d
1
d
2
.
.
.
d
n
x
1
x
2
.
.
.
x
n
=
b
1
b
2
.
.
.
b
n
Solution of
D
x
=
b
directly computable:
x
k
=
b
k
d
k
(
d
k
6
=
0,
k
=
1:
n
)
c
±
D. Aruliah (UOIT)
LU Factorisation
MATH 2070U
7 / 27
Easy systems
Gaussian elimination
LU factorisation
Diagonal system example
Solve the linear system of equations
2
3

4
x
1
x
2
x
3
=
5
9
1
2
x
1
=
5
⇒
x
1
=
5
2
3
x
2
=
9
⇒
x
2
=
9
3
=
3

4
x
3
=
1
⇒
x
3
=

1
4
Equations are completely decoupled
c
±
D. Aruliah (UOIT)
LU Factorisation
MATH 2070U
8 / 27
Easy systems
Gaussian elimination
LU factorisation
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This note was uploaded on 02/23/2010 for the course MATH 2070 taught by Professor Aruliahdhavidhe during the Spring '10 term at UOIT.
 Spring '10
 aruliahdhavidhe
 Systems Of Equations, Gaussian Elimination, Linear Systems

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