ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 14 Elimination Methods

ECIV 301

Programming & Graphics

Numerical Methods for Engineers

Lecture 14

Elimination Methods

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 Matrix Algebra

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

Rectangular Array of Elements Represented by a single symbol [A]


nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

Row 1

Row 3

Column 2 Column m

n x m Matrix

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A


32a

3rd Row

2nd Column


m321 bbbbB

1 Row, m Columns

Row Vector

B


n

3

2

1

c

c

c

c

C

n Rows, 1 Column

Column Vector

C


5554535251

4544434241

3534333231

2524232221

1514131211

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

A

If n = m Square Matrix

e.g. n=m=5e.g. n=m=5Main Diagonal


9264

2732

6381

4215

A

Special Types of Square Matrices

Symmetric: aSymmetric: aijij = a = ajiji


9000

0700

0080

0005

A

Diagonal: aDiagonal: aijij = 0, i = 0, ijj



1000

0100

0010

0001

I

Identity: aIdentity: aiiii=1.0 a=1.0 aijij = 0, i = 0, ijj


nm

m333

m22322

m1131211

a000

aa00

aaa0

aaaa

A


Upper TriangularUpper Triangular


nm3n2n1n

333231

2221

11

aaaa

0aaa

00aa

000a

A


Lower TriangularLower Triangular


nm

3332

232221

1211

a000

0aa0

0aaa

00aa

A


BandedBanded


Last Time Matrix Operating Rules - Equality

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

[A]mxn=[B]pxq

n=p m=q aij=bij

Last Time Matrix Operating Rules - Addition

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

[C]mxn= [A]mxn+[B]pxq

n=p

m=qcij = aij+bij

Last Time Multiplication by Scalar

nm3n2n1n

m3333231

m2232221

m1131211

gagagaga

gagagaga

gagagaga

gagagaga

AgD

Last Time Matrix Multiplication

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

[A] n x m . [B] p x q = [C] n x q

m=p

n

1kkjikij bac


nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

1nn13113

2112111111

baba

babac

11c

C

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A


pq3p2p1p

q3333231

q2232221

q1131211

bbbb

bbab

bbbb

bbbb

B

3nn23323

2322132123

baba

babac

23c

C

nmm3m2m1

3n332313

2n322212

1n312111

T

aaaa

aaaa

aaaa

aaaa

A

nm3n2n1n

m3333231

m2232221

m1131211

aaaa

aaaa

aaaa

aaaa

A

Last Time Operations - Transpose

Last Time Operations - Inverse

[A] [A]-1

[A] [A]-1=[I]

If [A]-1 does not exist[A] is singular

Last Time Operations - Trace

5554535251

4544434241

3534333231

2524232221

1514131211

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

A

Square Matrix

tr[A] = tr[A] = aaiiii

1035

y

x 1035

y

x

Equations in Matrix Form

1035 yxConsider

Linear Equations in Matrix Form

10z8y3x5

6z3yx12

24z23y6x10


10z8y3x5

10

z

y

x

835


10z8y3x5

6z3yx12

24z23y6x10


6

z

y

x

3112

6z3yx12


10z8y3x5

6z3yx12

24z23y6x10


24

z

y

x

23610

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

Solution of Linear Equations

9835 zyx

7310 zyx

10500 zyx

Consider the system


9835 zyx

730 zyx

10500 zyx

25

10 z


9835 zyx

730 zyx

2z

7230 yx

167 y

Solution of Linear Equations9835 zyx

2z

1y

928135 x

25

1639

x 2x


10

7

9

500

310

835

z

y

x

Express In Matrix Form

Upper Triangular

What is the characteristic?

Solution by Back Substitution

Solution of Linear EquationsObjective

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

BackgroundConsider

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

BackgroundConsider

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

Gauss Elimination

10835 zyx

2423610 zyx

6312 zyx

Example

Forward Elimination

Gauss Elimination

10835 zyx

24z23y6x10

zyx 835

5

1210

5

12

6312 zyx

-

305

81

5

310 zyx 302.162.60 zyx

Gauss Elimination

10835 zyx

24z23y6x10

6312 zyx 302.162.60 zyx

Substitute 2nd eq with new

Gauss Elimination

10835 zyx

24z23y6x10

302.162.60 zyx

zyx 835

5

1010

5

10-

439120 zyx

Gauss Elimination

10835 zyx

24z23y6x10

302.162.60 zyx

Substitute 3rd eq with new

439120 zyx

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

Gauss Elimination

10835 zyx

30970 zyx

Substitute 3rd eq with new

439120 zyx 064.62645.700 zyx

Gauss Elimination

064.62

30

10

645.700

2.162.60

835

z

y

x

Gauss Elimination

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

Documents

ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 14 Elimination Methods