The Islamic University of Gaza Faculty of Engineering Civil Engineering Department

The Islamic University of GazaFaculty of Engineering

Civil Engineering Department

Numerical Analysis ECIV 3306

Chapter 9

Gauss Elimination

1

Introduction

0)( xfRoots of a single equation:A general set of equations: - n equations, - n unknowns.

0),,(

0),,(0),,(

21

212

211

nn

n

n

xxxf

xxxfxxxf

2

Linear Algebraic Equations

nnnnnn

nnx

nn

bxaxaxa

bxxaeaxa

bxaxxaxa

2211

23

22223

121

15

12112111

/)()(

)(2

nnnnnn

nn

nn

bxaxaxa

bxaxaxabxaxaxa

2211

22222121

11212111

Nonlinear Equations

3

Review of Matrices

m n nm2n1n

m22221

m11211

aaa

aaaaaa

]A[

2nd row

mth column

Elements are indicated by a i j

row column

Row vector: Column vector:

n 1 n21 rrr]R[

1 m

m

2

1

c

cc

]C[

Square matrix: - [A]nxm is a square matrix if n=m. - A system of n equations with n unknonws has a square coefficient matrix. 4

Review of Matrices

• Main (principle) diagonal: [A]nxn consists of elements aii ; i=1,...,n

• Symmetric matrix: If aij = aji [A]nxn is a symmetric matrix

• Diagonal matrix: [A]nxn is diagonal if aij = 0 for all i=1,...,n ; j=1,...,n and ij

• Identity matrix: [A]nxn is an identity matrix if it is diagonal with aii=1 i=1,...,n . Shown as [I]

5

Review of Matrices

• Upper triangular matrix:

[A]nxn is upper triangular if aij=0 i=1,...,n ; j=1,...,n and i>j

• Lower triangular matrix:

[A]nxn is lower triangular if aij=0 i=1,...,n ; j=1,...,n and i<j

• Inverse of a matrix:

[A]-1 is the inverse of [A]nxn if [A]-1[A] = [I]

• Transpose of a matrix:

[B] is the transpose of [A]nxn if bij=aji Shown as [A] or [A]T

6

Special Types of Square Matrices

88 6 39 166 9 7 2

39 7 3 116 2 1 5

][A

nn a aa

D

22

11

][

1 1

1 1

][I

Symmetric Diagonal Identity

nn

n

n

a

aa aaa

A

][ 222

11211

nnn a a

aaa

A

1

2221

11

][

Upper Triangular Lower Triangular

7

Review of Matrices

• Matrix multiplication:

r

1kkjikij b a cNote: [A][B] [B][A]

8

Review of Matrices

• Augmented matrix: is a special way of showing two matrices together.

For example augmented with the

column vector is

• Determinant of a matrix: A single number. Determinant of [A] is shown as |A|.

2221

1211

aaaa

A

2

1

bb

B

22221

11211

baabaa

9

Part 3- Objectives

10

Solving Small Numbers of Equations

There are many ways to solve a system of linear equations:

• Graphical method• Cramer’s rule• Method of elimination• Numerical methods for solving larger number of linear equations: - Gauss elimination (Chp.9) - LU decompositions and matrix inversion(Chp.10)

For n ≤ 3

11

1. Graphical method• For two equations (n = 2):

• Solve both equations for x2: the intersection of the lines presents the solution.

• For n = 3, each equation will be a plane on a 3D coordinate system. Solution is the point where these planes intersect.

• For n > 3, graphical solution is not practical.

2222121

1212111

bxaxabxaxa

22

21

22

212

1212

11

12

112 intercept(slope)

abx

aax

xxabx

aax

12

Graphical Method -Example

• Solve:

• Plot x2 vs. x1, the intersection of the lines presents the solution.

221823

21

21

xxxx

13

Graphical Method

No solution Infinite solution ill condition (sensitive to round-off errors)

14

2.Determinants and Cramer’s Rule

Determinant can be illustrated for a set of three equations:

Where [A] is the coefficient matrix:

BxA .

333231

232221

131211

aaaaaaaaa

A

15

223132213231

222113

233133213331

232112

233233223332

232211

333231

232221

131211

aaaaaaaa

D

aaaaaaaa

D

aaaaaaaa

D

aaaaaaaaa

D

Graphical MethodMathematically• Coefficient matrices of (a) & (b) are singular. There

is no unique solution for these systems. Determinants of the coefficient matrices are zero and these matrices can not be inverted.

• Coefficient matrix of (c) is almost singular. Its inverse is difficult to take. This system has a unique solution, which is not easy to determine numerically because of its extreme sensitivity to round-off errors.

17

Cramer’s Rule

3231

222113

3331

232112

3332

232211 aa

aaa

aaaa

aaaaa

aD

Daabaabaab

x 33323

23222

13121

1 D

abaabaaba

x 33331

23221

13111

2 D

baabaabaa

x 33231

22221

11211

3

18

3

2

1

3

2

1

333231

232221

131211

bbb

xxx

aaaaaaaaa

BAX

Cramer’s Rule

• For a singular system D = 0 Solution can not be obtained.

• For large systems Cramer’s rule is not practical because calculating determinants is costly.

21

3. Method of Elimination

• The basic strategy is to successively solve one of the equations of the set for one of the unknowns and to eliminate that variable from the remaining equations by substitution.

• The elimination of unknowns can be extended to systems with more than two or three equations. However, the method becomes extremely tedious to solve by hand.

22

Elimination of Unknowns MethodGiven a 2x2 set of equations:

2.5x1 + 6.2x2 = 3.0

4.8x1 - 8.6x2 = 5.5

• Multiply the 1st eqn by 8.6 and the 2nd eqn by 6.2

21.50x1 + 53.32x2=25.8

29.76x1 – 53.32x2=34.1

• Add these equations 51.26 x1 + 0 x2 = 59.9

• Solve for x1 : x1 = 59.9/51.26 = 1.168552478

• Using the 1st eqn solve for x2 : x2 =(3.0–2.5*1.168552478)/6.2 = 0.01268045242

• Check if these satisfy the 2nd eqn: 4.8*1.168552478–8.6*0.01268045242 = 5.500000004 (Difference is due to the round-off errors). 23

Naive Gauss Elimination Method

• It is a formalized way of the previous elimination technique to large sets of equations by developing a systematic scheme or algorithm to eliminate unknowns and to back substitute.

• As in the case of the solution of two equations, the technique for n equations consists of two phases: 1. Forward elimination of unknowns. 2. Back substitution.

24

Naive Gauss Elimination Method• Consider the following system of n equations.

a11x1 + a12x2 + ... + a1nxn = b1 (1)a21x1 + a22x2 + ... + a2nxn = b2 (2)

...an1x1 + an2x2 + ... + annxn = bn (n)

Form the augmented matrix of [A|B].

Step 1 : Forward Elimination: Reduce the system to an upper triangular system.

1.1- First eliminate x1 from 2nd to nth equations. - Multiply the 1st eqn. by a21/a11 & subtract it from the 2nd equation. This is the new 2nd eqn. - Multiply the 1st eqn. by a31/a11 & subtract it from the 3rd equation. This is the new 3rd eqn. ... - Multiply the 1st eqn. by an1/a11 & subtract it from the nth equation. This is the new nth eqn. 25

Note: - In these steps the 1st eqn is the pivot equation and a11 is the pivot element. - Note that a division by zero may occur if the pivot element is zero. Naive-Gauss Elimination does not check for this.

The modified system is

indicates that the system is modified once.

n

3

2

1

n

3

2

1

nn3n2n

n33332

n22322

n1131211

b

bbb

x

xxx

aaa0

aaa0aaa0aaaa

Naive Gauss Elimination Method (cont’d)

26


1.2- Now eliminate x2 from 3rd to nth equations.

The modified system is

n

3

2

1

n

3

2

1

nn3n

n333

n22322

n1131211

b

bbb

x

xxx

aa00

aa00aaa0aaaa

Repeat steps (1.1) and (1.2) upto (1.n-1).11 12 13 1 1 1

22 23 2 2 2

33 3 3 3

( 1) ( 1)

0 0 0

0 0 0 0

n

n

n

n nnn n n

a a a a x ba a a x b

a a x b

a x b

we will get this upper triangular system

27


Step 2 : Back substitution Find the unknowns starting from the last equation.1. Last equation involves only xn. Solve for it.

2. Use this xn in the (n-1)th equation and solve for xn-1.

...3. Use all previously calculated x values in the 1st eqn

and solve for x1.

)1(

)1(

nnn

nn

n abx

121for )1(1

1)1(

, ..., , n-n- ia

xabx i

ii

n

ijj

iij

ii

i

28

Summary of Naive Gauss Elimination Method

29

Pseudo-code of Naive Gauss Elimination Method

(a) Forward Elimination (b) Back substitution

k,j

30

Naive Gauss Elimination Method Example 1

Solve the following system using Naive Gauss Elimination.

6x1 – 2x2 + 2x3 + 4x4 = 1612x1 – 8x2 + 6x3 + 10x4 = 26 3x1 – 13x2 + 9x3 + 3x4 = -19-6x1 + 4x2 + x3 - 18x4 = -34

Step 0: Form the augmented matrix

6 –2 2 4 | 1612 –8 6 10 | 26 R2-2R1 3 –13 9 3 | -19 R3-0.5R1-6 4 1 -18 | -34 R4-(-R1)

31


Step 2: Back substitution

Find x4 x4 =(-3)/(-3) = 1

Find x3 x3 =(-9+5*1)/2 = -2

Find x2 x2 =(-6-2*(-2)-2*1)/(-4) = 1

Find x1 x1 =(16+2*1-2*(-2)-4*1)/6= 3

33

Naive Gauss Elimination Method Example 2(Using 6 Significant Figures)

3.0 x1 - 0.1 x2 - 0.2 x3 = 7.850.1 x1 + 7.0 x2 - 0.3 x3 = -19.3 R2-(0.1/3)R10.3 x1 - 0.2 x2 + 10.0 x3 = 71.4 R3-(0.3/3)R1

Step 1: Forward elimination

3.00000 x1- 0.100000 x2 - 0.200000 x3 = 7.85000

7.00333 x2 - 0.293333 x3 = -19.5617 - 0.190000 x2 + 10.0200 x3 = 70.6150

3.00000 x1- 0.100000 x2 - 0.20000 x3 = 7.85000

7.00333 x2 - 0.293333 x3 = -19.5617 10.0120 x3 = 70.0843

34


Step 2: Back substitution

x3 = 7.00003 x2 = -2.50000 x1 = 3.00000

Exact solution: x3 = 7.0 x2 = -2.5 x1 = 3.0

35

Pitfalls of Gauss Elimination Methods

1. Division by zero 2 x2 + 3 x3 = 84 x1 + 6 x2 + 7 x3 = -3 2 x1 + x2 + 6 x3 = 5

It is possible that during both elimination and back-substitution phases a division by zero can occur.

2. Round-off errors In the previous example where up to 6 digits were kept

during the calculations and still we end up with close to the real solution.

x3 = 7.00003, instead of x3 = 7.0

a11 = 0 (the pivot element)

36

Pitfalls of Gauss Elimination (cont’d)

3. Ill-conditioned systems x1 + 2x2 = 10

1.1x1 + 2x2 = 10.4

x1 + 2x2 = 10

1.05x1 + 2x2 = 10.4

Ill conditioned systems are those where small changes in coefficients result in large change in solution. Alternatively, it happens when two or more equations are nearly identical, resulting a wide ranges of answers to approximately satisfy the equations. Since round off errors can induce small changes in the coefficients, these changes can lead to large solution errors.

x1 = 4.0 & x2 = 3.0

x1 = 8.0 & x2 = 1.0

37

Pitfalls of Gauss Elimination (cont’d)4. Singular systems. • When two equations are identical, we would be

dealing with case of n-1 equations for n unknowns.

To check for singularity:• After getting the forward elimination process and

getting the triangle system, then the determinant for such a system is the product of all the diagonal elements. If a zero diagonal element is created, the determinant is Zero then we have a singular system.

• The determinant of a singular system is zero.38

Techniques for Improving Solutions1. Use of more significant figures to solve for

the round-off error.2. Pivoting. If a pivot element is zero, elimination

step leads to division by zero. The same problem may arise, when the pivot element is close to zero. This Problem can be avoided by:

Partial pivoting. Switching the rows so that the largest element is the pivot element.

Complete pivoting. Searching for the largest element in all rows and columns then switching.

3. Scaling Solve problem of ill-conditioned system. Minimize round-off error

39

Partial Pivoting

Before each row is normalized, find the largest available coefficient in the column below the pivot element. The rows can then be switched so that the largest element is the pivot element so that the largest coefficient is used as a pivot.

40

Use of more significant figures to solve for the round-off error :Example.

Use Gauss Elimination to solve these 2 equations: (keeping only 4 sig. figures)

0.0003 x1 + 3.0000 x2 = 2.00011.0000 x1 + 1.0000 x2 = 1.000

0.0003 x1 + 3.0000 x2 = 2.0001 - 9999.0 x2 = -6666.0

Solve: x2 = 0.6667 & x1 = 0.0

The exact solution is x2 = 2/3 & x1 = 1/3 41

Use of more significant figures to solve for the round-off error :Example (cont’d).

32

2 x0003.0

)3/2(30001.21

x

Significant Figures x2 x1

3 0.667 -3

4 0.6667 0.000

5 0.66667 0.3000

6 0.666667 0.33000

7 0.6666667 0.333000

42

Pivoting: ExampleNow, solving the pervious example using the partial pivoting technique:

1.0000 x1+ 1.0000 x2 = 1.0000.0003 x1+ 3.0000 x2 = 2.0001The pivot is 1.01.0000 x1+ 1.0000 x2 = 1.000 2.9997 x2 = 1.9998

x2 = 0.6667 & x1=0.3333Checking the effect of the # of significant digits: # of dig x2 x1 Ea% in x1

4 0.6667 0.3333 0.01 5 0.66667 0.33333 0.001

43

Scaling: Example• Solve the following equations using naïve gauss elimination: (keeping only 3 sig. figures)

2 x1+ 100,000 x2 = 100,000

x1 + x2 = 2.0

• Forward elimination:

2 x1+ 10.0×104 x2 = 10.0×104

- 5.00 ×104 x2 = - 5.00 ×104

Solve x2 = 1.00 & x1 = 0.00

• The exact solution is x1 = 1.00002 & x2 = 0.99998

44

Scaling: Example (cont’d)B) Using the scaling algorithm to solve:

2 x1+ 100,000 x2 = 100,000 x1 + x2 = 2.0

Scaling the first equation by dividing by 100,000:0.00002 x1+ x2 = 1.0

x1+ x2 = 2.0Rows are pivoted:

x1 + x2 = 2.0 0.00002 x1+ x2 = 1.0

Forward elimination yield: x1 + x2 = 2.0 x2 = 1.00 Solve: x2 = 1.00 & x1 = 1.00

The exact solution is x1 = 1.00002 & x2 = 0.99998 45

Scaling: Example (cont’d)C) The scaled coefficient indicate that pivoting is necessary.We therefore pivot but retain the original coefficient to give:

x1 + x2 = 2.02 x1+ 100,000 x2 = 100,000

Forward elimination yields: x1 + x2 = 2.0 100,000 x2 = 100,000 Solve: x2 = 1.00 & x1 = 1.00

Thus, scaling was useful in determining whether pivoting was necessary, but the equation themselves did not require scaling to arrive at a correct result.

46

Determinate Evaluation

The determinate can be simply evaluated at the end of the forward elimination step, when the program employs partial pivoting:

47

' '' ( 1)11 22 33

where: represents the number of times that ro

1

ws are pivoted

pnnnD a a a a

p

Example: Gauss Elimination

2 4

1 2 3 4

1 2 4

1 2 3 4

2 02 2 3 2 24 3 76 6 5 6

x xx x x xx x xx x x x

a) Forward Elimination

0 2 0 1 0 6 1 6 5 62 2 3 2 2 2 2 3 2 2

1 44 3 0 1 7 4 3 0 1 76 1 6 5 6 0 2 0 1 0

R R

48

Example: Gauss Elimination (cont’d)6 1 6 5 62 2 3 2 2 2 0.33333 14 3 0 1 7 3 0.66667 10 2 0 1 0

6 1 6 5 60 1.6667 5 3.6667 4

2 30 3.6667 4 4.3333 110 2 0 1 0

6 1 6 5 60 3.6667 4 4.3333 110 1.6667 5 3.6667 40 2 0 1 0

R RR R

R R

Example: Gauss Elimination (cont’d)6 1 6 5 60 3.6667 4 4.3333 110 1.6667 5 3.6667 4 3 0.45455 20 2 0 1 0 4 0.54545 2

6 1 6 5 60 3.6667 4 4.3333 110 0 6.8182 5.6364 9.00010 0 2.1818 3.3636 5.9999 4 0.32000 3

6 1 60 3.6667 40 0 6.80 0

R RR R

R R

5 64.3333 11

182 5.6364 9.00010 1.5600 3.1199

50

Example: Gauss Elimination (cont’d)

b) Back Substitution

6 1 6 5 60 3.6667 4 4.3333 110 0 6.8182 5.6364 9.00010 0 0 1.5600 3.1199

4

3

2

1

3.1199 1.99991.56009.0001 5.6364 1.9999

0.333256.8182

11 4.3333 1.9999 4 0.333251.0000

3.66676 5 1.9999 6 0.33325 1 1.0000

0.500006

x

x

x

x

51

Gauss-Jordan Elimination

• It is a variation of Gauss elimination. The major differences are:– When an unknown is eliminated, it is eliminated

from all other equations rather than just the subsequent ones.

– All rows are normalized by dividing them by their pivot elements.

– Elimination step results in an identity matrix.– It is not necessary to employ back substitution to

obtain solution.

52

Gauss-Jordan Elimination- Example

0 2 0 1 0 1 0.16667 1 0.83335 12 2 3 2 2 2 2 3 2 21 44 3 0 1 7 4 3 0 1 74 / 6.06 1 6 5 6 0 2 0 1 0

R R

R

1 0.16667 1 0.83335 12 2 3 2 2 2 2 14 3 0 1 7 3 4 10 2 0 1 0

R RR R

53

1 0.16667 1 0.83335 10 1.6667 5 3.6667 20 3.6667 4 4.3334 70 2 0 1 0

Dividing the 2nd row by 1.6667 and reducing the second column. (operating above the diagonal as well as below) gives:

1 0 1.5 1.2000 1.40000 1 2.9999 2.2000 2.40000 0 15.000 12.400 19.8000 0 5.9998 3.4000 4.8000

Divide the 3rd row by 15.000 and make the elements in the 3rd Column zero.

54

1 0 0 0.04000 0.580000 1 0 0.27993 1.55990 0 1 0.82667 1.32000 0 0 1.5599 3.1197

Divide the 4th row by 1.5599 and create zero above the diagonal in the fourth column.

1 0 0 0 0.499990 1 0 0 1.00010 0 1 0 0.333260 0 0 1 1.9999

Note: Gauss-Jordan method requires almost 50 % more operations than Gauss elimination; therefore it is not recommended to use it.

55

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department

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