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THE UNIVERSITY OF TEXAS AT AUSTIN

DEPARTMENT OF AEROSPACE ENGINEERING AND ENGINEERING

MECHANICS

ASE 311 ENGINEERING COMPUTATIONFALL 2013

Instructor:

Danial FaghihiThe Institute for Computational Engineering and Science

ACES 4.122, danial@ices.utexas.edu

September 13, 2013

Lecture 6:

Linear Algebraic Equations and Matrices

Chapter 8

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HintforProblems4and5,hw1:

Example4.10thebook:usingtheMaclaurinseriesexpansionofex,howmanyterms

requiredfores9ma9ngthevaluee0.5?

Solu5on:

exactvalue: errorcriterionforthreesignificantfigures

Zeroorder(n=0):Percenttotalerror:

Firstorder(n=1):

Percenttotalerror:

PercentrelaEveerror:

Secondorder(n=2):..

x=0.5

Thestoppingcriteria

isnotsaEsfied

ex

= 1 + x +x2

2+

x3

3!+ . . . +

xn

n!e0.5

= 1.648721

"s = (0.5 1023)% = 0.05%

e

x

' 1

"t = |1.648721 1

1.648721| = 39.3

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HintforProblems4and5,hw1:

Solu5oncon5nue:

UsetheZeroorderM-fileofFigure4.2ofthebook,IterMeth.mtoconfirmtheresults.

>> [approxval, ea, iter] = IterMeth(.5,0.05,100)!approxval =!

1.6487!ea =!

0.0158!iter =!

6!!>> trueval=exp(.5)!trueval =!

1.6487!>> et=abs((trueval- approxval)/trueval)*100!et =!

0.0014 !

ex

= 1 + x +x2

2+

x3

3!+ . . . +

xn

n!

e0.5

=? and "s = 0.05

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Chapter8:

LinearAlgebraicquaEons

andMatrices

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MATRIXALGEBRAOVERVIEW

KnowledgeofmatricesisessenEalforunderstandingthesoluEonof

linearalgebraicequaEons.

rowvectors columnvectors

aij =

m by n dimension

matrix

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MATRIXALGEBRAOVERVIEW

Defini4ons: square matrices: m=n

Principle elementsor main diagonal

Symmetric matrix: aij=aji

diagonal matrix: a square matrix

where all elements off the maindiagonal are equal to zero

identity matrix: a diagonal matrix where

all elements on the main diagonal are

equal to 1

Property of [ I]

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MATRIXALGEBRAOVERVIEW

MatrixOpera4on

AddiEonandSubtracEon:

MulEplicaEonofmatrix[A]byascalarg

cij = aij bij

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MATRIXALGEBRAOVERVIEW

Matrices Product

matrix multiplication is

associative distributive commutative

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MATRIXALGEBRAOVERVIEW

Inverseofamatrix[A]

-1

Ifamatrix[A]issquareandnonsingular,thereisanothermatrix[A]1,

calledtheinverseof[A]:

Transposeofamatrix[A]T

Transposeofmatrix[A]involvestransformingitsrowsintocolumns

anditscolumnsintorows

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MATRIXALGEBRAOVERVIEW

augmenta5on. AmatrixisaugmentedbytheaddiEonofacolumn(orcolumns)to

theoriginalmatrix.

Forexample,supposewehavea3*3matrixofcoefficients.Wemightwishtoaugmentthismatrix[A]witha3*3idenEtymatrixtoyielda

3*6dimensionalmatrix:

PerformingasetofidenEcaloperaEonsontherowsoftwomatrices.

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MATLABMATRIXManipula4on

create[A]

transposeof[A]:

A =

264

1 5 6

7 4 2

3 6 7

375

create{x},{y},and{z}vectors:

combine{x},{y},and{z}toform[B]

x =

h8 6 9

i; y =

h5 8 1

i

z =

h4 8 2

i

B =

2

64

8 6 9

5 8 1

4 8 2

3

75

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MATLABMATRIXManipula4on

[C]=[A]+[B]=?

A =264

1 5 6

7 4 2

3 6 7

375 B = 2648 6 9

5 8 1

4 8 2

375

>> C = A + B!

C =!9 11 15!2 12 3!1 14 9!

>> A = C - B!A =!

1 5 6!7 4 2!-3 6 7!

[A]=[C]-[B]=?

[C1]=[A][B]=?

>> C1 = A*B!

C1 =!7 94 26!44 90 71!

-26 86 -7!

[C1]T

=?>> C1!ans =!

7 44 -26!94 90 86!26 71 -7!

[C1]-1=?

>> inv(C1)!

ans =!0.2073 -0.0890 -0.1334!0.0473 -0.0193 -0.0199!-0.1884 0.0937 0.1079!

[C1]-T

=?>> inv(C1')!>> inv(C1)!ans =!

0.2073 0.0473 -0.1884!-0.0890 -0.0193 0.0937!-0.1334 -0.0199 0.1079!

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MATLABMATRIXManipula4on

[A][D]=?

A =264

1 5 6

7 4 2

3 6 7

375

>> A*D!

Error using * !Inner matrix dimensions

must agree.!

>> D*A!ans =!

20 39 35!58 63 53!

[D][A]=?

[AI]=[A]-1=?

>> AI = inv(A)!

AI =!0.2462 0.0154 -0.2154

-0.8462 0.3846 0.6154!0.8308 -0.3231 -0.4769!

[A][AI]=?>> A*AI!ans =!1.0000 -0.0000 -0.0000!0.0000 1.0000 -0.0000!0.0000 -0.0000 1.0000!

[AI]-1=?

>> inv(AI)!

ans =!1.0000 5.0000 6.0000!7.0000 4.0000 2.0000!-3.0000 6.0000 7.0000!

([AI][B])-T

=?>> inv(AI*B)!ans =!-1.2520 -0.0660 1.2680!0.2320 0.6560 -0.0880!0.5880 0.6540 -0.2920!

="1 4 3

5 8 1#

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Represen4ngLinearAlgebraicEqua4onsinMatrixForm

Systemsoflinearequa4ons:

[A]{x} = {b}

264a11 a12 a13

a21 a22 a23

a31 a32 a33

375264x1

x2

x3

375 =

264b1

b2

b3

375

can be expressed as

matrix of coefficients vector of constantsvector of unknowns

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Represen4ngLinearAlgebraicEqua4onsinMatrixForm

Systemsoflinearequa4ons:

[A]{x} = {b}Solving the system of equation: determining (solving) the vector {x} for given[A] and {b} in the following relation

multiply each side of the equation by [A]-1

[A]1[A]{x} = [A]1{b}

[A]1[A] = [I]

[I]{x} = {x}

{x} = [A]1{b}

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Represen4ngLinearAlgebraicEqua4onsinMatrixForm

Systemsoflinearequa4ons:

Solving the system of equation: determining (solving) the vector {x} for given[A] and {b} in the following relation

MATLAB provides two direct ways to solve systems of linear algebraic

equations.

matrix inversion:>> x = inv(A)*b!

The most efficient way is to employ the backslash>> x = A\b!

{x} = [A]1{b}

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Example8.2ofthebook

Solving the system of equation using MATLAB:

enter the coefficient matrix and the right-hand-side vector

8>>:

(k1 + k2)x1 k2x2 = m1g

k2x1 + (k2 + k3)x2 k3x3 = m2g

k3x2 + k3x3 = m3g

m1 = 60 (kg) k1 = 50 (N/m)

m2 = 70 (kg) k2 = 100 (N/m)

m3 = 80 (kg) k3 = 50 (N/m)