engr/math/physics 25

[email protected] • ENGR-25_Linear_Equations-1.ppt1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Engr/Math/Physics 25

Chp8 LinearAlgebraic

Eqns-1Bruce Mayer, PERegistered Electrical & Mechanical Engineer

[email protected]



Learning Goals Define Linear Algebraic Equations Solve Systems of Linear Equations

by Hand using• Gaussian Elimination (Elem. Row Ops)• Cramer’s Method

Distinguish between Equation System Conditions: Exactly Determined, OverDetermined, UnderDetermined

Use MATLAB to Solve Systems of Eqns



Linear Equations Example In Many Engineering Analyses

(e.g. ENGR36 & ENGR43) The Engineer Must Solve Several Equations in Several Unknowns; e.g.:

3146252267512141436

zyxzyxzyx

Contains 3 Unknowns (x,y,z) in the 3 Equations



Linear Systems - Characteristics Examine the

System of Equations

14625267512

41436

zyxzyx

zyx

We notice These Characteristics that DEFINE Linear Systems

ALL the Variables are Raised EXACTLY to the Power of ONE (1)

COEFFICIENTS of the Variables are all REAL Numbers

The Eqns Contain No Transcendental Functions (e.g. ln, cos, ew)



Gaussian Elimination – ERO’s A “Well Conditioned” System of Eqns

can be Solved by Elementary Row Operations (ERO):• Interchanges: The vertical position of

two rows can be changed• Scaling: Multiplying a row by a

nonzero constant• Replacement: The row can be replaced by

the sum of that row and a nonzero multiple of any other row



ERO Example - 1 Let’s Solve The

System of Eqns 146253

2675122414361

zyxzyx

zyx

INTERCHANGE, or Swap, positions of Eqns (1) & (2)

Next SCALE by using Eqn (1) as the PIVOT To Multiply• (2) by 12/6• (3) by 12/[−5]

146253

4143622675121

zyxzyxzyx



ERO Example - 2 The Scaling

Operation

146255

123

414366

122

2675121

zyx

zyx

zyx

6.334.148.4123

82861222675121

zyxzyxzyx

Note that the 1st Coeffiecient in the Pivot Eqn is Called the Pivot Value• The Pivot is used to

SCALE the Eqns Below it

Next Apply REPLACEMENT by Subtracting Eqs• (2) – (1)• (3) – (1)



ERO Example - 3 The Replacement

Operation Yields 6.74.78.903

1081511022675121

zyxzyx

zyx

Or

Note that the x-variable has been ELIMINATED below the Pivot Row• Next Eliminate in

the “y” Column We can use for the

y-Pivot either of −11 or −9.8• For the best numerical

accuracy choose theLARGEST pivot

6.74.78.93

108151122675121

zyzy

zyx



ERO Example - 4 Our Reduced Sys

6.74.78.93

108151122675121

zyzy

zyx

Since |−11| > |−9.8| we do NOT need to interchange (2)↔(3)

Scale by Pivot against Eqn-(3)

6.74.78.98.9

113

108151122675121

zy

zyzyx

Or

531.8306.8113

108151122675121

zyzy

zyx



ERO Example - 5 Perform

Replacement by Subtracting (3) – (2)

531.116306.233

108151122675121

zzy

zyx

Now Easily Find the Value of z from Eqn (3)

5306.23531.116 z

The Hard Part is DONE

Find y & x by BACK SUBSTITUTION

From Eqn (2)

31133

1175108

1115108

y

zy



ERO Example - 6 BackSub into (1)

21224

12261535

122657267512

x

yzx

zyx

Thus the Solution Set for Our Linear System

146253

2675122414361

zyxzyx

zyx

x = 2 y = −3 z = 5



Importance of Pivoting Computers use finite-precision arithmetic A small error is introduced in each arithmetic

operation, AND… error propagates When the pivot element is very small, then the

multipliers will be even smaller Adding numbers of widely differing magnitude

can lead to a loss of significance. To reduce error, row interchanges are

made to maximize the magnitude of the pivot element



Gaussian Elimination Summary INTERCHANGE Eqns Such that

the PIVOT Value has the Greatest Magnitude

SCALE the Eqns below the Pivot Eqn using the Pivot Value ratio’ed against the Corresponding Value below

REPLACE Eqns Below the Pivot by Subtraction to leave ZERO Coefficients Below the Pivot Value



Poorly Conditioned Systems For Certain Systems Guassian

Elimination Can Fail by• NO Solution → Singular System• Numerically Inaccurate Results →

ILL-Conditioned System In a SINGULAR SYSTEM Two or More

Eqns are Scalar Multiples of each other In ILL-Conditioned Systems 2+ Eqns are

NEARLY Scalar Multiples of each other



A Singular (Inconsistent) Sys Consider 2-Eqns

in 2-Unknowns 5422

421

yx

yx

Perform Elimination by• Swapping Eqns• Mult (2) by 2/1• Subtract (2) – (1)

422

5421

yxyx

8422

5421

yxyx

3002

5421

yxyx

????302



Singular System - Geometry Plot This System

on the XY Plane 5422

421

yx

yx

The Lines do NOT CROSS to Define a A Solution Point

Singular Systems Have at least Two “PARALLEL” Eqns

y



ILL-Conditioned Systems A small deviation in one or more of the

CoEfficients causes a LARGE DEVİATİON in the SOLUTİON.

47.199.048.0321

yxyx

11

yx

47.199.049.0321

yxyx

03

yx



ILL-Conditioned Systems - 2 Systems in Which a

Small Change in a CoEfficient Produces Large Changes in the Solution are said to be STIFF• Essentially the Lines

Have very nearly Equal SLOPES

• “Tilting” The Equations just a bit Dramatically Shifts the Solution (Crossing Point)

Tilt Region



Matrix Methods for LinSys - 1 Consider the

Electrical Ckt Shown at Right

The Operation of this Ckt May be Described in Terms of the • Mesh Currents, I1-I4

• Sources: 4 mA, 12 V• Resistors: 1 & 2 kΩ

Notice Mesh Currents I1 & I2 are Defined by SOURCES



Matrix Methods for LinSys - 3 Using Techniques

from ENGR43 find

Recall Matrix Multiplication to Write the Equation system in Matrix Form

mAIIImAIII

IIImAI

12208230

004000

432

432

321

1

12804

21102310

01110001

4

3

2

1

IIII

A x b



Matrix Methods for LinSys - 3 Thus The (linear)

Ckt Can be Described by

bAx Where

• A Coefficient Matrix– m-Rows x n-Colunms

• b Constraint Vector

• x Solution Vector

This Can Be Written in Std Math Notation

mnnim

iniii

ni

aaa

aaa

aaa

1

1

1111

A

m

i

x

x

x

1

x

m

i

b

b

b

1

b



Determinants - 1 If we Solve a

LinSys by Elimination we may do a Lot of work Before Discovering that the system is Singular or Very-Stiff

Determinants Can Alert us ahead of time to these Difficulties

Determinants are Defined only for SQUARE Arrays

The 2x2 Definition

122122112221

12112 aaaa

aaaa

D

D2 is Sometimes called the “Basic Minor”



Determinants - 2 Calculating Larger-Dimension DETs

becomes very-Tedious very-Quickly• Consider a 3x3 Det

313212111

333231

232221

131211

3 detMinordetMinordetMinor aaaaaaaaaaaa

D

2231322113233133211223323322113 aaaaaaaaaaaaaaaD

• Example

51132137

694

3

exD

87696261348

39776635922654



Determinants - 3 A Determinant, no matter what its size,

Returns a SINGLE Value Matrix vs. Determinant

• For Square Matrix A the Notation

AAA det MATLAB vs det

• The det Calc is quite Painful, but MATLAB’s “det” Fcn Makes it Easy

For the D3ex

>> A = [-4,9,6; 7,13,-2; -3,11,5];>> D3ex = det(A)D3ex = 87



Determinant Indicator - 1 The LARGER the Magnitude of the

Determinant relative to the Coefficients, The LESS-Stiff the System

If det=0, then the System is SINGULAR 5422

421

yx

yx SINGULAR0det

47.199.048.0321

yxyx

STIFF03.0det

47.199.049.0321

yxyx

STIFF01.0det



Determinant Indicator - 2 Consider this

System 23271325

yxyx

Check the “Stiffness” 242725

D

Thus The system appears NON-Stiff Find Solution by Elimination as

13 yx



MATLAB Left Division MATLAB has a

very nice Utility for Solving Well-Conditioned Linear Systems of the Form bAx

Well Conditioned →• Square System →

No. of Eqns & Unknwns are Equal

• det 0

The Syntax is Quite Simple• the hassle is

entering the Matrix-A and Vector-b

x = A\b



Left-Div Example - 1 Consider a 750 kg

Crate suspended by 3 Ropes or Cables

03506.04706.036.081.9*7507792.08824.08.0

05195.0048.0

ADACAB

ADACAB

ADACAB

TTTTTT

TTT

Using Force Mechanics from ENGR36 Find 3 Eqns in 3 Unknowns

wT A

form MATRIXIn



Left-Div Example - 2 The MATLAB

Command Window Session

Or• TAB = 2.625 kN• TAC = 3.816 kN• TAD = 2.426 kN

>> A = [-0.48, 0, 0.5195;...0.8, 0.8824, 0.7792;...-0.36, 0.4706, -.3506];>> w = [0; 9.81*750; 0]

>> T = A\wT =

1.0e+003 * 2.6254 3.8157 2.4258



Matrix Inverse - 1 Recall The Matrix

Formulation for n-Eqns in n-UnknownsbAx

In Matrix Land

Abx To Isolate x, employ

the Matrix Inverse A-1 as Defined by

IAA 1

xIx Use A-1 in Matrix Eqn

bAxbAIxbAΑxA

1

1

11

or

or

Note that the IDENTITY Matrix , I, Has Property



Matrix Inverse - 2 Thus the Matrix

Shorthand for the Solution

bAxbAx

1

Determining the Inverse is NOT Trivial (Ask your MTH6 Instructor)

bAx 1

In addition A-1 is, in general, Less Numerically Accurate Than Pivoted Elimination

However

is Symbolically Elegant and Will be Useful in that regard



Compare MatInv & LeftDiv% Bruce Mayer, PE% ENGR25 * 21Oct09% file = Compare_MatInv_LeftDiv_0910%A = [3 -7 8; 7 6 -5; -9 0 2]b = [13; -29; 37]Ainv = inv(A)xinv = Ainv*bxleft = A\b%% CHECK Both by b = A*xCHKinv = A*xinvCHKleft = A*xleft



All Done for Today

MatrixInversionby Adjoint

||)(1

AAA Adj

The “Adjoint” of a matrix is the transpose of the matrix made up of the “CoFactors” of the original matrix.

Given A, Find A-1



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Appendix 6972 23 xxxxf

engr/math/physics 25

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