matrix

MATRIX METHODSSYSTEMS OF LINEAR EQUATIONS

ENGR 351 Numerical Methods for EngineersSouthern Illinois University CarbondaleCollege of EngineeringDr. L.R. ChevalierDr. B.A. DeVantier

Permission is granted to students at Southern Illinois University at Carbondaleto make one copy of this material for use in the class ENGR 351, NumericalMethods for Engineers. No other permission is granted.

All other rights are reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,electronic, mechanical, photocopying, recording, or otherwise, withoutthe prior written permission of the copyright owner.

Systems of Linear Algebraic EquationsSpecific Study Objectives

• Understand the graphic interpretation of ill-conditioned systems and how it relates to the determinant

• Be familiar with terminology: forward elimination, back substitution, pivot equations and pivot coefficient

• Apply matrix inversion to evaluate stimulus-response computations in engineering

• Understand why the Gauss-Seidel method is particularly well-suited for large sparse systems of equations

• Know how to assess diagonal dominance of a system of equations and how it relates to whether the system can be solved with the Gauss-Seidel method

Specific Study Objectives

• Understand the rationale behind relaxation and how to apply this technique

Specific Study Objectives

How to represent a system of linear equations as a matrix

[A]{x} = {c}

where {x} and {c} are both column vectors

5.03.01.0

9.115.0

152.03.0

44.05.03.01.0

67.09.15.0

01.052.03.0

How to represent a system of linear equations as a matrix

Practical application

• Consider a problem in structural engineering

• Find the forces and reactions associated with a statically determinant truss

hinge: transmits bothvertical and horizontalforces at the surface

roller: transmitsvertical forces

1000 kg

FREE BODY DIAGRAM F

Node 1 F1,V

F F F F

0 30 60

30 60 0

30 60 1000

cos cos

sin sin

cos cos

sin sin

F H F F

Node 2

Node 3

060sin

060cos

030sin

030cos

100060sin30sin

060cos30cos

SIX EQUATIONSSIX UNKNOWNS

F1 F2 F3 H2 V2 V3

-cos30 0 cos60 0 0 0

-sin30 0 -sin60 0 0 0 cos30 1 0 1 0 0

sin30 0 0 0 1 0

0 -1 -cos60 0 0 0

0 0 sin60 0 0 1

Do some book keeping

This is the basis for your matrices and the equation[A]{x}={c}

0 866 0 0 5 0 0 0

0 5 0 0 866 0 0 0

0 866 1 0 1 0 0

0 5 0 0 0 1 0

0 1 0 5 0 0 0

0 0 0 866 0 0 1

System of Linear Equations

• We have focused our last lectures on finding a value of x that satisfied a single equation• f(x) = 0

• Now we will deal with the case of determining the values of x1, x2, .....xn, that simultaneously satisfy a set of equations

System of Linear Equations• Simultaneous equations

• f1(x1, x2, .....xn) = 0

• f2(x1, x2, .....xn) = 0 • .............• fn(x1, x2, .....xn) = 0

• Methods will be for linear equations• a11x1 + a12x2 +...... a1nxn =c1

• a21x1 + a22x2 +...... a2nxn =c2

• ..........

• an1x1 + an2x2 +...... annxn =cn

Mathematical BackgroundMatrix Notation

• a horizontal set of elements is called a row• a vertical set is called a column• first subscript refers to the row number• second subscript refers to column number

a a a a

m m m mn

11 12 13 1

21 22 23 2

. . . .

2232221

1131211

This matrix has m rows an n column.

It has the dimensions m by n (m x n)

2232221

1131211

This matrix has m rows and n column.

It has the dimensions m by n (m x n)

notesubscript

a a a a

m m m mn

11 12 13 1

21 22 23 2

. . . .

column 3Note the consistentscheme with subscriptsdenoting row,column

Row vector: m=1

Column vector: n=1 Square matrix: m = n

B b b bn 1 2 .......

11 12 13

21 22 23

31 32 33

Types of Matrices

• Symmetric matrix• Diagonal matrix• Identity matrix• Inverse of a matrix• Transpose of a matrix• Upper triangular matrix• Lower triangular matrix• Banded matrix

Definitions

Symmetric Matrix

aij = aji for all i’s and j’s

2 7 8Does a23 = a32 ?

Yes. Check the other elementson your own.

Diagonal Matrix

A square matrix where all elements off the main diagonal are zero

Identity Matrix

A diagonal matrix where all elements on the main diagonal are equal to 1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

The symbol [I] is used to denote the identify matrix.

Inverse of [A]

IAAAA 11

Transpose of [A]

n n mn

11 21 1

12 22 2

. . . . . .

Upper Triangle Matrix

Elements below the main diagonal are zero

11 12 13

Lower Triangular Matrix

All elements above the main diagonal are zero

Banded Matrix

All elements are zero with the exception of a band centered on the main diagonal

21 22 23

32 33 34

Matrix Operating Rules

• Addition/subtraction• add/subtract corresponding terms• aij + bij = cij

• Addition/subtraction are commutative• [A] + [B] = [B] + [A]

• Addition/subtraction are associative• [A] + ([B]+[C]) = ([A] +[B]) + [C]

Matrix Operating Rules• Multiplication of a matrix [A] by a

scalar g is obtained by multiplying every element of [A] by g

ga ga ga

m m mn

11 12 1

21 22 2

. . . . . .

Matrix Operating Rules

• The product of two matrices is represented as [C] = [A][B]

• n = column dimensions of [A]• n = row dimensions of [B]

c a bij ik kjk

[A] m x n [B] n x k = [C] m x k

interior dimensionsmust be equal

exterior dimensions conform to dimension of resulting matrix

Simple way to check whether matrix multiplication is possible

Recall the equation presented for matrix multiplication• The product of two matrices is

represented as [C] = [A][B]

• n = column dimensions of [A]• n = row dimensions of [B]

c a bij ik kjk

ExampleDetermine [C] given [A][B] = [C]

Matrix multiplication• If the dimensions are suitable,

matrix multiplication is associative• ([A][B])[C] = [A]([B][C])

• If the dimensions are suitable, matrix multiplication is distributive• ([A] + [B])[C] = [A][C] + [B][C]

• Multiplication is generally not commutative• [A][B] is not equal to [B][A]

Determinants

Denoted as det A or lAl

for a 2 x 2 matrix

bcaddc

Determinants

For a 3 x 3

Problem

Determine the determinant of the matrix.

Properties of Determinants

• det A = det AT

• If all entries of any row or column is zero, then det A = 0

• If two rows or two columns are identical, then det A = 0

• Note: determinants can be calculated using mdeterm function in Excel

Excel Demonstration

• Excel treats matrices as arrays• To obtain the results of

multiplication, addition, and inverse operations, you hit control-shift-enter as opposed to enter.

• The resulting matrix cannot be altered…let’s see an example using Excel in class

matrix.xls

Matrix Methods

• Cramer’s Rule• Gauss elimination• Matrix inversion • Gauss Seidel/Jacobi

11 12 13

21 22 23

31 32 33

Graphical Method2 equations, 2 unknowns

a x a x c

11 1 12 2 1

21 1 22 2 2

( x1, x2 )

3 2 18

( 4 , 3 )

Check: 3(4) + 2(3) = 12 + 6 = 18

Special Cases

• No solution• Infinite solution• Ill-conditioned

( x1, x2 )

a) No solution - same slope f(x)

xb) infinite solution f(x)

-1/2 x1 + x2 = 1-x1 +2x2 = 2

c) ill conditionedso close that the points ofintersection are difficult todetect visually

• If the determinant is zero, the slopes are identical

a x a x c

a x a x c11 1 12 2 1

21 1 22 2 2

Rearrange these equations so that we have an alternative version in the form of a straight line:

i.e. x2 = (slope) x1 + intercept

Ill Conditioned Systems

If the slopes are nearly equal (ill-conditioned)

a a a a

11 22 21 12

11 22 21 12 0

a aA11 12

Isn’t this the determinant?

If the determinant is zero the slopes are equal.This can mean:

- no solution- infinite number of solutions

If the determinant is close to zero, the system is illconditioned.

So it seems that we should use check the determinant of a system before any further calculations are done.

Let’s try an example.

Example

Determine whether the following matrix is ill-conditioned.

5.22.19

7.42.37

37 2 4 7

19 2 2 537 2 2 5 4 7 19 2

. .. . . .

What does this tell us? Is this close to zero? Hard to say.

If we scale the matrix first, i.e. divide by the largesta value in each row, we can get a better sense of things.

Solution

0 5 10 15

This is further justifiedwhen we consider a graphof the two functions.

Clearly the slopes arenearly equal

1 0126

1 01300 004

Solution

Another Check

• Scale the matrix of coefficients, [A], so that the largest element in each row is 1. If there are elements of [A]-1 that are several orders of magnitude greater than one, it is likely that the system is ill-conditioned.

• Multiply the inverse by the original coefficient matrix. If the results are not close to the identity matrix, the system is ill-conditioned.

• Invert the inverted matrix. If it is not close to the original coefficient matrix, the system is ill-conditioned.

We will consider how to obtain an inverted matrix later.

Cramer’s Rule

• Not efficient for solving large numbers of linear equations

• Useful for explaining some inherent problems associated with solving linear equations.

333231

232221

131211

Cramer’s Rule

b a a1

1 12 13

2 22 23

3 32 33

to solve forxi - place {b} inthe ith column

Cramer’s Rule

to solve forxi - place {b} inthe ith column

EXAMPLE Use of Cramer’s Rule

2 1 3 1 2 3 5

55 1 3 5

52 5 5 1

Solution

Elimination of Unknowns( algebraic approach)

2112221111121

1212122111121

112222121

211212111

2222121

1212111

caxaaxaa

SUBTRACTcaxaaxaa

acxaxa

21122211

2121221

11222112

2111212

1122112112222111

2112221111121

1212122111121

acacxaaxaa

caxaaxaa

SUBTRACTcaxaaxaa

NOTE: same result asCramer’s Rule

Elimination of Unknowns( algebraic approach)

Gauss Elimination

• One of the earliest methods developed for solving simultaneous equations

• Important algorithm in use today• Involves combining equations in

order to eliminate unknowns and create an upper triangular matrix

• Progressively back substitute to find each unknown

Two Phases of Gauss Elimination

a a a c

11 12 13 1

21 22 23 2

31 32 33 3

11 12 13 1

22 23 2

' ' ' '

ForwardElimination

Note: the prime indicatesthe number of times the element has changed from the original value.

Two Phases of Gauss Elimination

31321211

1131211

xaxacx

Back substitution

• Any equation can be multiplied (or divided) by a nonzero scalar

• Any equation can be added to (or subtracted from) another equation

• The positions of any two equations in the set can be interchanged.

EXAMPLE

4 4 7 1

2 5 9 3

Perform Gauss Elimination of the following matrix.

Solution

4 4 7 1

2 5 9 3

Multiply the first equation by a21/ a11 = 4/2 = 2

Note: a11 is called the pivot element

2624 321 xxx

4 4 7 1

2 5 9 3

2624 321 xxx

a21 / a11 = 4/2 = 2

4 4 7 1

2 5 9 3

a21 / a11 = 4/2 = 2

Subtract the revised first equation from the second equation

4 4 7 1

2 5 9 3

a21 / a11 = 4/2 = 2

4 4 4 2 7 6 1 2

0 2 11 2 3

4 4 7 1

2 5 9 3

a21 / a11 = 4/2 = 2

4 4 4 2 7 6 1 2

0 2 11 2 3

4 4 7 1

2 5 9 3

a21 / a11 = 4/2 = 2

NEWMATRIX

4 4 4 2 7 6 1 2

0 2 11 2 3

4 4 7 1

2 5 9 3

a21 / a11 = 4/2 = 2

NOW LET’SGET A ZEROHERE

Multiply equation 1 by a31/a11 = 2/2 = 1and subtract from equation 3

2 2 5 1 9 3 3 1

0 4 6 21 2 3

4 4 7 1

2 5 9 3

Solution

4 4 7 1

2 5 9 3

Following the same rationale, subtract the 3rd equation from the first equation

Continue thecomputation by multiplying the second equationby a32’/a22’ = 4/2 =2

Subtract the third equation of the newmatrix

Solution

THIS DERIVATION OFAN UPPER TRIANGULAR MATRIXIS CALLED THE FORWARDELIMINATION PROCESS

Solution

From the system we immediately calculate:

Continue to back substitute

THIS SERIES OFSTEPS IS THEBACK SUBSTITUTION

Solution

Pitfalls of the Elimination Method

• Division by zero• Round off errors

• magnitude of the pivot element is small compared to other elements

• Ill conditioned systems

Pivoting

• Partial pivoting• rows are switched so that the pivot element is

not zero• rows are switched so that the largest element

is the pivot element

• Complete pivoting• columns as well as rows are searched for the

largest element and switched• rarely used because switching columns

changes the order of the x’s adding unjustified complexity to the computer program

For example

Pivoting is used here to avoid division by zero

4 6 7 3

Another Improvement: Scaling

• Minimizes round-off errors for cases where some of the equations in a system have much larger coefficients than others

• In engineering practice, this is often due to the widely different units used in the development of the simultaneous equations

• As long as each equation is consistent, the system will be technically correct and solvable

Use Gauss Elimination to solve the following setof linear equations. Employ partial pivoting when necessary.

3 13 50

2 6 45

Example (solution in notes)

3 13 50

2 6 45

First write in matrix form, employing short hand presented in class.

0 3 13 50

2 6 1 45

4 0 8 4

We will clearly run intoproblems of divisionby zero.

Use partial pivoting

Solution

0 3 13 50

2 6 1 45

4 0 8 4

Pivot with equationwith largest an1

501330

Begin developingupper triangular matrix

4 0 8 4

0 6 3 43

0 3 13 50

4 0 8 4

0 6 3 43

0 0 14 5 28 5

14 51 966

43 3 1 966

4 8 1 966

42 931

3 8149 13 1 966 50

okay...end ofproblem

Gauss-Jordan

• Variation of Gauss elimination

• Primary motive for introducing this method is that it provides a simple and convenient method for computing the matrix inverse.

• When an unknown is eliminated, it is eliminated from all other equations, rather than just the subsequent one

• All rows are normalized by dividing them by their pivot elements

• Elimination step results in an identity matrix rather than an UT matrix

11 12 13

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Gauss-Jordan

Graphical depiction of Gauss-Jordan

a a a c

11 12 13 1

21 22 23 2

31 32 33 3

' ' ' '

a a a c

11 12 13 1

21 22 23 2

31 32 33 3

' ' ' '

Graphical depiction of Gauss-Jordan

Matrix Inversion• [A] [A] -1 = [A]-1 [A] = I• One application of the inverse is to

solve several systems differing only by {c}• [A]{x} = {c}• [A]-1[A] {x} = [A]-1{c}• [I]{x}={x}= [A]-1{c}

• One quick method to compute the inverse is to augment [A] with [I] instead of {c}

Graphical Depiction of the Gauss-Jordan Method with Matrix Inversion

11 12 13

21 22 23

31 32 33

Note: the superscript“-1” denotes thatthe original valueshave been convertedto the matrix inverse,not 1/aij

WHEN IS THE INVERSE MATRIX USEFUL?

CONSIDER STIMULUS-RESPONSE CALCULATIONS THAT ARE SO COMMON IN ENGINEERING.

Stimulus-Response Computations• Conservation Laws

massforceheatmomentum

• We considered the conservation of force in the earlier example of a truss

• [A]{x}={c}• [interactions]{response}={stimuli}• Superposition

• if a system subject to several different stimuli, the response can be computed individually and the results summed to obtain a total response

• Proportionality• multiplying the stimuli by a quantity results

in the response to those stimuli being multiplied by the same quantity

• These concepts are inherent in the scaling of terms during the inversion of the matrix

Stimulus-Response Computations

Example

Given the following, determine {x} for the two different loads {c}

Solution

{c}T = {1 2 3}x1 = (2)(1) + (-1)(2) + (1)(3) = 3x2 = (-2)(1) + (6)(2) + (3)(3) = 19x3 = (-3)(1) + (1)(2) + (-4)(3) = -13

{c} T= {4 -7 1)x1 = (2)(4) + (-1)(-7) + (1)(1)=16x2 = (-2)(4) + (6)(-7) + (3)(1) = -47x3 = (-3)(4) + (1)(-7) + (-4)(1) = -23

Gauss Seidel Method

• An iterative approach• Continue until we converge within some pre-

specified tolerance of error• Round off is no longer an issue, since you control

the level of error that is acceptable• Fundamentally different from Gauss elimination.

This is an approximate, iterative method particularly good for large number of equations

Gauss-Seidel Method

• If the diagonal elements are all nonzero, the first equation can be solved for x1

• Solve the second equation for x2, etc.x

c a x a x a x

11 12 2 13 3 1

To assure that you understand this, write the equation for x2

xc a x a x a x

nn n n nn n

11 12 2 13 3 1

22 21 1 23 3 2

33 31 1 32 2 3

1 1 3 2 1 1

Gauss-Seidel Method

• Start the solution process by guessing values of x

• A simple way to obtain initial guesses is to assume that they are all zero

• Calculate new values of xi starting with• x1 = c1/a11

• Progressively substitute through the equations

• Repeat until tolerance is reached

x c a x a x a

x c a a a ca x

x c a x a a x

x c a x a x a x

1 1 12 2 13 3 11

2 2 21 1 23 3 22

3 3 31 1 32 2 33

1 1 12 13 111

2 2 21 1 23 22 2

3 3 31 1 32 2 33 3

' ' / '

Gauss-Seidel Method

Example

2 3 1 2

4 1 2 2

3 2 1 1

Given the following augmented matrix, complete one iteration of the Gauss Seidel method.

2 3 1 2

4 1 2 2

3 2 1 1

x c a a a ca x

x c a x a a x

x c a x a x a x

1 1 12 13 111

2 2 21 1 23 22 2

3 3 31 1 32 2 33 3

2 3 0 1 0

2 4 1 2 0

1 3 1 2 6

1 3 12

' ' / '

GAUSS SEIDEL

Jacobi Iteration

• Iterative like Gauss Seidel• Gauss-Seidel immediately uses the

value of xi in the next equation to predict x i+1

• Jacobi calculates all new values of xi’s to calculate a set of new xi values

FIRST ITERATION

x c a x a x a x c a x a x a

SECOND ITERATION

x c a x a x a x c a x

1 1 12 2 13 3 11 1 1 12 2 13 3 11

2 2 21 1 23 3 22 2 2 21 1 23 3 22

3 3 31 1 32 2 33 3 3 31 1 32 2 33

1 1 12 2 13 3 11 1 1 12 2 13 3 11

2 2 21 1 23 3 22 2 2 21 1

23 3 22

3 3 31 1 32 2 33 3 3 31 1 32 2 33

Graphical depiction of difference between Gauss-Seidel and Jacobi

2 3 1 2

4 1 2 2

3 2 1 1

Note: We worked the Gauss Seidel method earlier

Given the following augmented matrix, complete one iteration of the Gauss Seidel method and the Jacobi method.

Example

Gauss-Seidel Methodconvergence criterion

as in previous iterative procedures in finding the roots,we consider the present and previous estimates.

As with the open methods we studied previously with onepoint iterations

1. The method can diverge2. May converge very slowly

Class question:where do theseformulas come from?

Convergence criteria for two linear equations

u x xc

v x xc

consider the partial derivatives of u and v

Convergence criteria for two linear equations cont.

Criteria for convergencewhere presented earlierin class materialfor nonlinear equations.

Noting that x = x1 andy = x2

Substituting the previous equation:

This is stating that the absolute values of the slopes mustbe less than unity to ensure convergence.

Extended to n equations:

a a where j n excluding j iii ij 1,

This condition is sufficient but not necessary; for convergence.

When met, the matrix is said to be diagonally dominant.

Diagonal Dominance

9.05.01.0

4.08.02.0

4.02.01

To determine whether a matrix is diagonally dominant you need to evaluate the values on the diagonal.

Diagonal Dominance

Now, check to see if these numbers satisfy the following rule for each row (note: each row represents a unique equation).a a where j n excluding j iii ij 1,

9.05.01.0

4.08.02.0

4.02.01

Review the conceptsof divergence andconvergence by graphicallyillustrating Gauss-Seidelfor two linear equations

11 13 286

11 9 991 2

Note: we are convergingon the solution

11 9 99

11 13 2861 2

CONVERGENCE

Change the order ofthe equations: i.e. changedirection of initial estimates

11 13 286

11 9 991 2

DIVERGENCE

Improvement of Convergence Using Relaxation

This is a modification that will enhance slow convergence.

After each new value of x is computed, calculate a new valuebased on a weighted average of the present and previousiteration.

x x xinew

iold 1

Improvement of Convergence Using Relaxation

• if = 1unmodified• if 0 < < 1 underrelaxation

• nonconvergent systems may converge• hasten convergence by dampening out

oscillations

• if 1<< 2 overrelaxation• extra weight is placed on the present value• assumption that new value is moving to the

correct solution by too slowly

x x xinew

iold 1

matrix

f f f f f v h v

f h f f f v f h v

f f f h v v

n column

n n mn

square matrix

matrix transpose

matrix inversion

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