ECIV 301 Programming & Graphics Numerical Methods for Engineers REVIEW II.
ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 15 Solution of Systems of...
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ECIV 301
Programming & Graphics
Numerical Methods for Engineers
Lecture 15
Solution of Systems of Equations
Objectives
• Introduction to Matrix Algebra
• Express System of Equations in Matrix Form
• Introduce Methods for Solving Systems of Equations
• Advantages and Disadvantages of each Method
Last Time Linear Equations in Matrix Form
10z8y3x5
6z3yx12
24z23y6x10
23610
3112
835
z
y
x
24
6
10 10
z
y
x
835
6
z
y
x
3112
24
z
y
x
23610
# Equations = # Unknowns = n
Square Matrix n x n
Last Time Solution of Linear Equations
10
7
9
500
310
835
z
y
x
Express In Matrix Form
Upper Triangular
What is the characteristic?
Solution by Back Substitution
Last Time Solution of Linear Equations
Objective
Can we express any system of equations in a form
nnnn
n
n
n
b
b
b
b
x
x
x
x
a
aa
aaa
aaaa
3
2
1
3
2
1
333
22322
1131211
000
00
0
0
Last Time Background
Consider
1035 yx(Eq 1)
5810 yx(Eq 2)
Solution
5.7
5.6
y
x
20610 yx2*(Eq 1)
5810 yx(Eq 2)
Solution
5.7
5.6
y
x!!!!!!
Scaling Does Not Change the SolutionScaling Does Not Change the Solution
Last Time Background
Consider
20610 yx(Eq 1)
152 y(Eq 2)-(Eq 1)
Solution
5.7
5.6
y
x!!!!!!
20610 yx(Eq 1)
5810 yx(Eq 2)
Solution
5.7
5.6
y
x
Operations Do Not Change the SolutionOperations Do Not Change the Solution
Last Time Gauss Elimination
10835 zyx
2423610 zyx
6312 zyx
Example
Forward Elimination
Last Time-Correction Gauss Elimination
10835 zyx
24z23y6x10
zyx 835
5
1210
5
12
6312 zyx
-
305
81
5
310 zyx 302.162.60 zyx
Last Time-Correction Gauss Elimination
10835 zyx
24z23y6x10
6312 zyx 302.162.60 zyx
Substitute 2nd eq with new
Last Time-Correction Gauss Elimination
10835 zyx
24z23y6x10
302.162.60 zyx
zyx 835
5
1010
5
10-
439120 zyx
Last Time-Correction Gauss Elimination
10835 zyx
24z23y6x10
302.162.60 zyx
Substitute 3rd eq with new
439120 zyx
Last Time-Correction Gauss Elimination
10835 zyx
302.162.60 zyx
439120 zyx
zy 2.162.6
2.6
12 30
2.6
12-
064.62645.700 zyx
Last Time-Correction Gauss Elimination
10835 zyx
30970 zyx
Substitute 3rd eq with new
439120 zyx 064.62645.700 zyx
Last Time-Correction Gauss Elimination
064.62
30
10
645.700
2.162.60
835
z
y
x
Last Time Gauss Elimination
Forward Elimination
064.62
30
10
645.700
2.162.60
835
z
y
x
Last Time-Correction Gauss Elimination
Back Substitution
118.8645.7/064.62 z
0502.26
2.6
118.82.1630
y
6413.0
5
118.880502.26310
x
064.62
30
10
645.700
2.162.60
835
z
y
x
Gauss Elimination – Potential Problem
10830 zyx
2423610 zyx
6312 zyx
Forward Elimination
Gauss Elimination – Potential Problem
10830 zyx
2423610 zyx
6312 zyx
0
12 Division By Zero!!Operation Failed
Gauss Elimination – Potential Problem
10830 zyx
2423610 zyx
6312 zyx
12
0OK!!
Gauss Elimination – Potential Problem
10830 zyx
2423610 zyx
6312 zyx
Pivoting
6312 zyx
10830 zyx
Partial Pivoting
nn
nnnnn
lll
n
n
n
b
b
b
b
x
x
x
x
aaaa
aaaa
aaaaaaaa
aaaa
3
2
1
3
2
1
321
ln321
3333231
2232221
1131211
a32>a22
al2>a22
NO
YES
Partial Pivoting
nn
nnnnn
n
n
lll
n
b
b
b
b
x
x
x
x
aaaa
aaaa
aaaaaaaa
aaaa
3
2
1
3
2
1
321
2232221
3333231
ln321
1131211
Full Pivoting
• In addition to row swaping
• Search columns for max elements
• Swap Columns
• Change the order of xi
• Most cases not necessary
EXAMPLE
We will work directly on the coefficient matrix