eciv 301 programming & graphics numerical methods for engineers lecture 20 solution of linear...
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ECIV 301
Programming & Graphics
Numerical Methods for Engineers
Lecture 20
Solution of Linear System of Equations - Iterative Methods
Iterative Methods
Recall Techniques for Root finding of Single Equations
Initial Guess
New Estimate
Error Calculation
Repeat until Convergence
Gauss Seidel
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
11
31321211 a
xaxabx
22
32312122 a
xaxabx
33
23213133 a
xaxabx
Gauss Seidel
11
1
11
1312111
00
a
b
a
aabx
22
23112121
2
0
a
axabx
33
1232
113131
3 a
xaxabx
First Iteration: 0,0,0 321 xxx
Better Estimate
Better Estimate
Better Estimate
Gauss Seidel
11
1313
121212
1 a
xaxabx
22
1323
212122
2 a
xaxabx
33
2232
213132
3 a
xaxabx
Second Iteration: 13
12
11 ,, xxx
Better Estimate
Better Estimate
Better Estimate
Gauss SeidelIteration Error:
%1001
, ji
ji
ji
ia x
xx
s
Convergence Criterion:
n
jij
ijii aa1
nnn2n1
2n2221
1n1211
aaa
aaa
aaa
Jacobi Iteration
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
11
31321211 a
xaxabx
22
32312122 a
xaxabx
33
23213133 a
xaxabx
Jacobi Iteration
11
1
11
1312111
00
a
b
a
aabx
22
2321212
00
a
aabx
33
3231313
00
a
aabx
First Iteration: 0,0,0 321 xxx
Better Estimate
Better Estimate
Better Estimate
Jacobi Iteration
11
1313
121212
1 a
xaxabx
22
1323
112122
2 a
xaxabx
33
1232
113132
3 a
xaxabx
Second Iteration: 13
12
11 ,, xxx
Better Estimate
Better Estimate
Better Estimate
Jacobi Iteration
Iteration Error:
%1001
, ji
ji
ji
ia x
xx
s
Example
4.71
3.19
85.7
102.03.0
3.071.0
2.01.03
3
2
1
x
x
x
3
2.01.085.7 321
xxx
7
3.01.03.19 312
xxx
10
2.03.04.71 213
xxx
Determinants
nnn2n1
2n2221
1n1211
aaa
aaa
aaa
A
nnn2n1
2n2221
1n1211
aaa
aaa
aaa
det
AA
Are composed of same elements
Completely Different Mathematical Concept
Determinants
2221
1211
aa
aaA
Defined in a recursive form
2x2 matrix
122122112221
1211det aaaaaa
aaA
DeterminantsDefined in a recursive form
3x3 matrix
3231
222113
3331
232112
3332
232211
det
aa
aaa
aa
aaa
aa
aaa
A
333231
232221
131211
aaa
aaa
aaa
333231
232221
131211
aaa
aaa
aaa
Determinants
3332
232211 aa
aaa
3231
222113
3331
232112 aa
aaa
aa
aaa
3332
2322
aa
aaMinor a11
333231
232221
131211
aaa
aaa
aaa
Determinants
3331
2321
aa
aaMinor a12
3332
232211 aa
aaa
3331
232112 aa
aaa
3231
222113 aa
aaa
333231
232221
131211
aaa
aaa
aaa
Determinants
3231
2221
aa
aaMinor a13
3332
232211 aa
aaa
3331
232112 aa
aaa
3231
222113 aa
aaa
Solution of Small Systems of Equations – Cramer’s Rule
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
333231
232221
131211
det
aaa
aaa
aaa
D A
1. Compute
Solution of Small Systems of Equations – Cramer’s Rule
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
33323
23222
131211det1
aab
aab
aab
D A
2. Compute
Solution of Small Systems of Equations – Cramer’s Rule
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
33331
23221
131112det2
aba
aba
aba
D A
3. Compute
Solution of Small Systems of Equations – Cramer’s Rule
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
33231
22221
112113det3
baa
baa
baa
D A
4. Compute
Solution of Small Systems of Equations – Cramer’s Rule
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
D
Dx
11
D
Dx
22
D
Dx
33
If D=0 solution does NOT exist
Singular Matrices
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
If D=0 solution does NOT exist
Regardless of Method
Singular Matrices
0det if AD
3
2
1
3
2
1
333231
232221
131211
b
b
b
x
x
x
aaa
aaa
aaa
bxA
bAx 1
For Example
{x} does not exist
[A]-1 does not exist
Determinants and LU Decomposition
064.62
30
10
645.700
2.162.60
835
z
y
x
24
6
10
23610
3112
835
z
y
x
{x} is not affected
Determinants and LU Decomposition
{x} is not affected
)operations pivoting no (if detdet UA D
Determinants and LU Decomposition
nnaaaaD 332211det U
33
2322
131211
00
0
a
aa
aaa
Example
23610
3112
835
610
1128
2310
3123
236
315
237)1072(8)30276(3)1823(5
Example
645.700
2.162.60
835
After Elimination [A] becomes
995.236 )645.7)(2.6)(5(
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