8 - 8 Linear Algebraic Equations 8 Gauss Elimination...

8: Linear Algebraic Equations
8: Linear Algebraic Equations
Gauss Elimination
Gauss Elimination

Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 2
Objectives
●
understand concepts of singularity and ill-condition
●
implement forward elimination and back substitution as in Gauss elimination
●
count flops as a measure of algorithmic efficiency
●
implement partial pivoting
●
solve a tridiagonal system efficiently

Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 3
Introduction
●
Consider the following system of linear algebraic equations
[ A ]{ x } = { b }
●
MATLAB provides two direct methods for solving the above system of
equations:
–
left-division:
>> x = A\b
–
matrix-inversion:
>> x = inv(A)*b
●
An understanding of how MATLAB obtains the solution is needed
●
Gauss elimination involves combining equations to “eliminate” unknowns

Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 4
8.1 Solving Small Numbers of Equations
●
Besides Gauss elimination, several methods are appropriate for solving
small ( n ≤ 3 ) sets of equations:
–
graphical method
–
Cramer's rule
–
elimination of unknowns

Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 5
8.1.1: The Graphical Method (1)
●
used for solving two linear equations in two unknowns
●
plot each equation as a straight line
●
the solution is the intersection of the two lines
●
Example:
3x
1
 2x
2
= 18
− x
1
 2x
2
= 2
⇒
x
2
=−
3
2
x
1
 9
x
2
=
1
2
x
1
 1
Figure 8.1

Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 6
The Graphical Method (2)
●
problems that can be encountered:
– (a) singular system – no solution
–
(b) singular system – infinite number of solutions
–
(c) ill-conditioned system – solution exists but sensitive to round-off error
Figure 8.2

Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 7
8.1.2: Determinants and Cramer's Rule (1)
●
consider the following system of linear algebraic equations
[ A ]{ x } = { b }
●
[ A ] is the coefficient matrix
●
the determinant of this system is represented by
[
A
]
=
[
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
]
D =
∣
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
∣
= a
11
∣
a
22
a
23
a
32
a
33
∣
− a
12
∣
a
21
a
23
a
31
a
33
∣
 a
13
∣
a
21
a
22
a
31
a
32
∣
= a
11

a
22
a
23
− a
32
a
23

− a
12

a
21
a
33
− a
31
a
23

 a
13

a
21
a
32
− a
31
a
22


Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 8
Determinants and Cramer's Rule (2)
●
Example 8.1:
– Fig. 8.1: non-singular well-conditioned:
–
Fig. 8.2a: singular:
–
Fig. 8.2b: singular:
– Fig. 8.2c: non-singular ill-conditioned:
D =
∣
3 2
− 1 2
∣
= 3  2 − 2 − 1 = 8
D =
∣
− 1 / 2 1
− 1 / 2 1
∣
=− 1 / 2  1 − 1 − 1 / 2 = 0
D =
∣
− 1 / 2 1
− 1 2
∣
=− 1 / 2  2 − 1 − 1 = 0
D =
∣
− 1 / 2 1
− 2.3 / 5 1
∣
=− 1 / 2  1 − 1 − 2.3 / 5 =− 0.04

Oct 23, 2006 18:16 ECOR2606 -- Hassan & Holtz 9
Determinants and Cramer's Rule (3)
●
Cramer's rule:
●
As the size of the system of equations increases, computing the
determinants becomes very time consuming and Cramer's rule becomes
less practical
●
Example:
x
1
=
∣
b
1
a
12
a
13
b
2
a
22
a
23
b
3
a
32
a
33
∣
D
, x
2
=
∣
a
11
b
1
a
13
a
21
b
2
a
23
a
31
b
3
a
33
∣
D
, x
3
=
∣
a
11
a
12
b
1
a
21
a
22
b
2
a
31
a
32
b
3
∣
D
[
0.3 0.52