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ECIV 301
Programming & Graphics
Numerical Methods for Engineers
Lecture 18
LU Decomposition and Matrix Inversion
EXAMPLE
4.71
3.19
85.7
102.03.0
3.071.0
2.01.03
3
2
1
x
x
x
Eliminate Column 1
3
1.0
PIVOTS
4.71
3.19
85.7
102.03.0
3.071.0
2.01.03
3
3.0
1,11
11 ia
apivot i
i
njapivotaa jiijij ,,2,1,11
Eliminate Column 1
6150.70
5617.19
85.7
0200.1019000.00
29333.000333.70
2.01.03
Eliminate Column 2
00333.7
19000.0
PIVOTS
6150.70
5617.19
85.7
0200.1019000.00
29333.000333.70
2.01.03
2,22
22 ia
apivot i
i
njapivotaa jiijij ,,2,1,22
Eliminate Column 2
0843.70
5617.19
85.7
01200.1000
29333.000333.70
2.01.03
UpperTriangular
Matrix[ U ]
ModifiedRHS
{ b }
LU DecompositionPIVOTS
Column 1PIVOTS
Column 2
03333.0
1.0 02713.0
LU Decomposition
As many as, and in the location of, zeros
UpperTriangular
MatrixU
01200.1000
29333.000333.70
2.01.03
LU DecompositionPIVOTS
Column 1
PIVOTSColumn 2
LowerTriangular
Matrix
1
1
1
0
0
0
L
03333.0
1.0 02713.0
LU Decomposition
102713.01.0
0103333.0
001
=
This is the original matrix!!!!!!!!!!
01200.1000
29333.000333.70
2.01.03
102.03.0
3.071.0
2.01.03
LU Decomposition
4.71
3.19
85.7
102713.01.0
0103333.0
001
3
2
1
y
y
y
4.71
3.19
85.7
102.03.0
3.071.0
2.01.03
3
2
1
x
x
x
[ L ] { y } { b }
[ A ] { x } { b }
LU Decomposition
4.71
3.19
85.7
102713.01.0
0103333.0
001
3
2
1
y
y
y
L y b
85.71 y
5617.190333.03.19 12 yy
0843.70)02713.0(1.04.71 213 yyy
LU Decomposition85.71 y
5617.190333.03.19 12 yy
0843.70)02713.0(1.04.71 213 yyy
0843.70
5617.19
85.7
01200.1000
29333.000333.70
2.01.03
ModifiedRHS
{ b }
LU Decomposition
• Ax=b
• A=LU - LU Decomposition
• Ly=b- Solve for y
• Ux=y - Solve for x
Matrix Inversion
4.71
3.19
85.7
102.03.0
3.071.0
2.01.03
3
2
1
x
x
x
bxA
Matrix Inversion
[A] [A]-1
[A] [A]-1=[I]
If [A]-1 does not exist[A] is singular
Matrix Inversion
b xA bxA 1A 1A
I
Matrix Inversion
bAx 1
Solution
Matrix Inversion
[A] [A]-1=[I]
100
010
001
aaa
aaa
aaa
aaa
aaa
aaa
nnn2n1
2n2221
1n1211
nnn2n1
2n2221
1n1211
Matrix Inversion
100
010
001
aaa
aaa
aaa
aaa
aaa
aaa
nnn2n1
2n2221
1n1211
nnn2n1
2n2221
1n1211
Matrix Inversion
100
010
001
aaa
aaa
aaa
aaa
aaa
aaa
nnn2n1
2n2221
1n1211
nnn2n1
2n2221
1n1211
Matrix Inversion
100
010
001
aaa
aaa
aaa
aaa
aaa
aaa
nnn2n1
2n2221
1n1211
nnn2n1
2n2221
1n1211
Matrix Inversion
• To calculate the invert of a nxn matrix solve n times :
nj
2j
1j
nj
2j
1j
nnn2n1
2n2221
1n1211
a
a
a
aaa
aaa
aaa
nj ,,2,1
otherwise
ji if
0
1ij
Matrix Inversion
• For example in order to calculate the inverse of:
102.03.0
3.071.0
2.01.03
Matrix Inversion
• First Column of Inverse is solution of
0
0
1
a
a
a
102.03.0
3.071.0
2.01.03
31
21
11
Matrix Inversion
0
1
0
a
a
a
102.03.0
3.071.0
2.01.03
32
22
12
• Second Column of Inverse is solution of
Matrix Inversion
• Third Column of Inverse is solution of:
1
0
0
a
a
a
102.03.0
3.071.0
2.01.03
33
23
13
Use LU Decomposition
102713.01.0
0103333.0
001
01200.1000
29333.000333.70
2.01.03
102.03.0
3.071.0
2.01.03
A
Use LU Decomposition – 1st column
• Forward SUBSTITUTION
0
0
1
y
y
y
102713.01.0
0103333.0
001
31
21
11
111 y
03333.00333.00 1121 yy
1009.002713.01.00 211131 yyy
Use LU Decomposition – 1st column
• Back SUBSTITUTION
1009.0
0333.0
1
a
a
a
01200.1000
29333.000333.70
2.01.03
31
21
11
010078.0012.10/1009.0a31
00518.000333.7/a2933.00333.0a 3121
332489.03/a2.0a1.01a 312111
Use LU Decomposition – 2nd Column
• Forward SUBSTITUTION
0
1
0
y
y
y
102713.01.0
0103333.0
001
32
22
12
012 y
122 y
02713.002713.01.00 221232 yyy
Use LU Decomposition – 2nd Column
• Back SUBSTITUTION
02713.0
1
0
a
a
a
01200.1000
29333.000333.70
2.01.03
32
22
12
002709.0012.10/02713.0a32
1429.000333.7/a2933.01a 3222
004944.03/a2.0a1.00a 322212
Use LU Decomposition – 3rd Column
• Forward SUBSTITUTION
1
0
0
y
y
y
102713.01.0
0103333.0
001
33
23
13
013 y
023 y
102713.01.01 231333 yyy
Use LU Decomposition – 3rd Column
• Back SUBSTITUTION
1
0
0
a
a
a
01200.1000
29333.000333.70
2.01.03
33
23
13
09988.0012.10/1a33
004183.000333.7/a2933.00a 3323
006798.03/a2.0a1.00a 332313
Result
102.03.0
3.071.0
2.01.03
A
09988.000271.001008.0
004183.0142903.000518.0
006798.0004944.0332489.0
A 1
Test It
09988.000271.001008.0
004183.0142903.000518.0
006798.0004944.0332489.0
102.03.0
3.071.0
2.01.03
11046.30
1047.31106736.8
0108.11
18
1818
18
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