Chapter 9 Slide

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Part 3 -

Chapter 9


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Part 3Linear Algebraic Equations

An equation of the form ax+by+c=0 or equivalently ax+by=-

c is called a linear equation inx andy variables.

ax+by+cz=dis a linear equation in three variables,x, y, and

z.

Thus, a linear equation in n variables is

a1x1+a2x2+ +anxn = b

A solution of such an equation consists of real numbers c1, c2,

c3, , cn. If you need to work more than one linearequations, a system of linear equations must be solved

simultaneously.


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Noncomputer Methods for Solving

Systems of Equations

For small number of equations (n 3) linear

equations can be solved readily by simple

techniques such as method of elimination.

Linear algebra provides the tools to solve such

systems of linear equations.

Nowadays, easy access to computers makes

the solution of large sets of linear algebraic

equations possible and practical.


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Fig. pt3.5


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Gauss EliminationChapter 9

Solving Small Numbers of Equations

There are many ways to solve a system of

linear equations:

Graphical method

Cramers rule

Method of elimination

Computer methods

For n 3


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Graphical Method

For two equations:

Solve both equations for x2:

2222121

1212111

bxaxa

bxaxa

22

21

22

212

1212

1

112

11

2 intercept(slope)

a

bx

a

ax

xxa

b

xa

a

x


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Plot x2 vs. x1

on rectilinear

paper, theintersection of

the lines

present the

solution.

Fig. 9.1


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Figure 9.2



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Determinants and Cramers Rule

Determinant can be illustrated for a set of three

equations:

Where [A] is the coefficient matrix:

BxA

333231

232221

131211

aaa

aaaaaa

A



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Assuming all matrices are square matrices,

there is a number associated with each squarematrix [A] called the determinant, D, of [A].

If [A] is order 1, then [A] has one element:

[A]=[a11]D=a11

For a square matrix of order 3, the minorof

an element aij is the determinant of the matrixof order 2 by deleting row i and columnj of

[A].



11

22313221

3231

2221

13

23313321

3331

2321

12

23323322

3332

2322

11

333231

232221

131211

aaaaaa

aaD

aaaaaa

aaD

aaaaaa

aa

D

aaa

aaa

aaa

D



12

3231

2221

13

3331

2321

12

3332

2322

11aa

aaa

aa

aaa

aa

aaaD

Cramers rule expresses the solution of a

systems of linear equations in terms of ratios

of determinants of the array of coefficients ofthe equations. For example, x1 would be

computed as:

D

aab

aab

aab

x33323

23222

13121

1



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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 becomesextremely tedious to solve by hand.



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Naive Gauss Elimination

Extension ofmethod of elimination to largesets of equations by developing a systematicscheme or algorithm to eliminate unknowns

and to back substitute. As in the case of the solution of two equations,

the technique forn equations consists of twophases:

Forward elimination of unknowns

Back substitution



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Fig. 9.3


16/20Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.Operation Counting

1 1

1 1 1

1

2

1

32 2 2 2 2 2

1

( ) ( )

( ) ( ) ( ) ( )

1 1 1 1 1 ... 1

1 1

( 1)1 2 3 ... ( )

2 2

( 1)(2 1)1 2 3 ... ( )

6 3

( )

m m

i i

m m m

i i i

m

i

m

i k

m

i

m

i

n

cf i c f i

f i f i f i g i

m

m k

m m mi m O m

m m m mi m O m

O m

means terms of order mnand lower.



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Pitfalls of Elimination Methods

Division by zero. It is possible that during bothelimination and back-substitution phases a division

by zero can occur, hence called naive.

Round-off errors.

Ill-conditioned systems. Systems where small changesin coefficients result in large changes in the solution.Alternatively, it happens when two or more equationsare nearly identical, resulting a wide ranges of

answers to approximately satisfy the equations. Sinceround off errors can induce small changes in thecoefficients, these changes can lead to large solutionerrors.



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Singular systems. When two equations are

identical, we would loose one degree offreedom and be dealing with the impossiblecase ofn-1 equations forn unknowns. Forlarge sets of equations, it may not be obvious

however. The fact that the determinant of asingular system is zero can be used and testedby computer algorithm after the eliminationstage. If a zero diagonal element is created,calculation is terminated.



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Techniques for Improving Solutions

Use of more significant figures.

Pivoting. If a pivot element is zero,normalization step leads to division by zero.

The same problem may arise, when the pivotelement is close to zero. Problem can beavoided:

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

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


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Gauss-Jordan

It is a variation of Gauss elimination. Themajor differences are:

When an unknown is eliminated, it is eliminated

from all other equations rather than just thesubsequent ones.

All rows are normalized by dividing them by theirpivot elements.

Elimination step results in an identity matrix.Consequently, it is not necessary to employ back

substitution to obtain solution.

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