matrix methods systems of linear equations student notes
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MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes. ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier Dr. B.A. DeVantier. Specific Study Objectives. Review basic definitions Review basic matrix operations - PowerPoint PPT PresentationTRANSCRIPT
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MATRIX METHODSSYSTEMS OF LINEAR EQUATIONSStudent NotesENGR 351 Numerical Methods for EngineersSouthern Illinois University CarbondaleCollege of EngineeringDr. L.R. ChevalierDr. B.A. DeVantier
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• Review basic definitions • Review basic matrix operations
• Addition/subtraction• Multiplication• Determinant
• 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
Specific Study Objectives
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• Apply matrix inversion • Understand that 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
• Understand the rationale behind relaxation and how to apply this technique
Specific Study Objectives
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How to represent a system of linear equations as a matrix
[A]{x} = {c}
where {x} and {c} are both column vectors
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44.067.001.0
5.03.01.09.115.0
152.03.0
}{}{
44.05.03.01.067.09.15.0
01.052.03.0
3
2
1
321
321
321
xxx
cxA
xxxxxx
xxx
How to represent a system of linear equations as a matrix
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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
30
90
60
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1000 kg
30
90
60
F1
H2
V2 V3
2
3
1
FREE BODY DIAGRAM F
FH
v
0
0
F2
F3
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Node 1 F1,V
F1,H
F3F1
6030
F F F F
F F F F
F F
F F
H H
V V
0 30 60
0 30 60
30 60 0
30 60 1000
1 3 1
1 3 1
1 3
1 3
cos cos
sin sin
cos cos
sin sin
,
,
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F H F F
F V FH
V
0 30
0 302 2 1
2 1
cos
sin
Node 2
F2
F1
30
H2V2
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F F F
F F VH
V
0 60
0 603 2
3 3
cos
sin
Node 3
F2
F3
60
V3
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060sin
060cos
030sin
030cos
100060sin30sin
060cos30cos
33
23
12
122
31
31
VF
FF
FV
FFH
FF
FF
SIX EQUATIONSSIX UNKNOWNS
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F1 F2 F3 H2 V2 V3
1
2
3
4
5
6
-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
0
-1000
0
0
0
0
Do some book keeping
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This is the basis for your matrices and the equation[A]{x}={c}
0 866 0 0 5 0 0 00 5 0 0 866 0 0 0
0 866 1 0 1 0 00 5 0 0 0 1 00 1 0 5 0 0 00 0 0 866 0 0 1
010000000
1
2
3
2
2
3
. .. .
..
..
FFFHVV
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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
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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
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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 aa a a a
a a a a
n
n
m m m mn
11 12 13 1
21 22 23 2
1 2 3
...
.... . . .
...
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mnmmm
n
n
aaaa
aaaaaaaa
A
.......
...
...
321
2232221
1131211
This matrix has m rows and n column.
It has the dimensions m by n (m x n)
notesubscript
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A
a a a aa a a a
a a a a
n
n
m m m mn
11 12 13 1
21 22 23 2
1 2 3
...
.... . . .
...
row 2
column 3Note the consistentscheme with subscriptsdenoting row,column
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Row vector: m=1
Column vector: n=1 Square matrix: m = n
B b b bn 1 2 .......
C
cc
cm
1
2
.
.
Aa a aa a aa a a
11 12 13
21 22 23
31 32 33
Types of Matrices
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• Symmetric matrix• Diagonal matrix• Identity matrix• Inverse of a matrix• Transpose of a matrix• Upper triangular matrix• Lower triangular matrix• Banded matrix
Definitions
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Symmetric Matrix
aij = aji for all i’s and j’s
A
5 1 21 3 72 7 8
Does a23 = a32 ?
Yes. Check the other elementson your own.
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Diagonal Matrix
A square matrix where all elements off the main diagonal are zero
A
aa
aa
11
22
33
44
0 0 00 0 00 0 00 0 0
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Identity Matrix
A diagonal matrix where all elements on the main diagonal are equal to 1
A
1 0 0 00 1 0 00 0 1 00 0 0 1
The symbol [I] is used to denote the identify matrix.
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Inverse of [A]
IAAAA 11
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Transpose of [A]
A
a a aa a a
a a a
t
m
m
n n mn
11 21 1
12 22 2
1 2
. . .
. . .. . . . . .. . . . . .. . . . . .
. . .
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Upper Triangle Matrix
Elements below the main diagonal are zero
Aa a a
a aa
11 12 13
22 23
33
00 0
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Lower Triangular Matrix
All elements above the main diagonal are zero
A
5 0 01 3 02 7 8
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Banded Matrix
All elements are zero with the exception of a band centered on the main diagonal
A
a aa a a
a a aa a
11 12
21 22 23
32 33 34
43 44
0 00
00 0
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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]
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Matrix Operating Rules• Multiplication of a matrix [A] by a
scalar g is obtained by multiplying every element of [A] by g
B g A
ga ga gaga ga ga
ga ga ga
n
n
m m mn
11 12 1
21 22 2
1 2
. . .
. . .. . . . . .. . . . . .. . . . . .
. . .
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Matrix Operating Rules• The product of two matrices is
represented as [C] = [A][B]• Basic algorithm
c a bij ik kjk
N
1
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[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
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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
N
1
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ExampleDetermine [C] given [A][B] = [C]
203123
142
320241231
B
A
Strategy
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Strategy• A 3x3 times a 3x3 is a 3x3• For c11=(-1)(-2)+(3)(3)+(2)(3)
203123
142
320241231
BA
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Strategy• c12
• c13
203123
142
320241231
BA
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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]
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Determinants
Denoted as det A or lAl
for a 2 x 2 matrix
bcaddcba
bcaddcba
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Determinants
254329132
For a 3 x 3
254329132
254329132
+ - +
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516234971
A
Problem
Determine the determinant of the matrix.
Strategy
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Strategy
516234971
516234971
A
516234971
516234971
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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
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Matrix Methods• Graphical• Cramer’s Rule• Gauss elimination• Matrix inversion • Gauss Seidel/Jacobi
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Graphical Method2 equations, 2 unknowns
a x a x ca x a x c
xaa
xca
xaa
xca
11 1 12 2 1
21 1 22 2 2
211
121
1
12
221
221
2
22
x2
x1
( x1, x2 )
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3 2 182 2
32
9
12
1
1 2
1 2
2 1
2 1
x xx x
x x
x x
x2
x1
( 4 , 3 )
3
2
2
1
9
1
Check: 3(4) + 2(3) = 12 + 6 = 18
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When does this not work?• No solution• Infinite solution• Ill-conditioned
x2
x1
( x1, x2 )
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a) No solution - same slope f(x)
xb) infinite solution f(x)
x
-1/2 x1 + x2 = 1-x1 +2x2 = 2
c) ill conditionedso close that the points ofintersection are difficult todetect visually
f(x)
x
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Ill Conditioned: What do I mean?• The equations are not independent• The equations are not unique• The solution isn’t worth 2¢
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• If the determinant is zero, the slopes are identical
a x a x ca x a x c
11 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
The Determinant as a Tool
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x aa
x ca
x aa
x ca
211
121
1
12
221
221
2
22
If the slopes are nearly equal (ill-conditioned)
aa
aa
a a a aa a a a
11
12
21
22
11 22 21 12
11 22 21 12 0
a aa a
A11 12
21 22
det
Isn’t this the determinant?
The Determinant as a Tool
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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.
The Determinant as a Tool
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ExampleDetermine whether the following matrix is ill-conditioned.
12
225.22.197.42.37
2
1
xx
Strategy Take the determinant or graph the equations
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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.
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Cramer’s Rule• Not efficient for solving large
numbers of linear equations• Useful for explaining some
inherent problems associated with solving linear equations.
bxAbbb
xxx
aaaaaaaaa
3
2
1
3
2
1
333231
232221
131211
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Cramer’s Rule
xA
b a ab a ab a a
1
1 12 13
2 22 23
3 32 33
1
to solve forxi - place {b} inthe ith column
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Cramer’s Rule
to solve forxi - place {b} inthe ith column
33231
22221
11211
3
33331
23221
13111
2
33323
23222
13121
1
1
11
baabaabaa
Ax
abaabaaba
Ax
aabaabaab
Ax
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EXAMPLE
2 3 55
2 31 1
55
1 2
1 2
1
2
x xx x
xx
Strategy
Use Cramer’s rule to solve
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Strategy• Put in matrix form Ax=b• Calculate |A|
33231
22221
11211
3
33331
23221
13111
2
33323
23222
13121
1
1
11
baabaabaa
Ax
abaabaaba
Ax
aabaabaab
Ax
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Elimination of Unknowns( algebraic approach)
2112221111121
1212122111121
112222121
211212111
2222121
1212111
caxaaxaaSUBTRACTcaxaaxaa
acxaxaacxaxa
cxaxacxaxa
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21122211
2121221
11222112
2111212
1122112112222111
2112221111121
1212122111121
aaaacacax
aaaacacax
acacxaaxaa
caxaaxaaSUBTRACTcaxaaxaa
NOTE: same result asCramer’s Rule
Elimination of Unknowns( algebraic approach)
![Page 62: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/62.jpg)
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
![Page 63: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/63.jpg)
Two Phases of Gauss Elimination
a a a ca a a ca a a c
a a a ca a c
a c
11 12 13 1
21 22 23 2
31 32 33 3
11 12 13 1
22 23 2
33 3
00 0
|||
|||
' ' '
' ' ' '
ForwardElimination
Note: the prime indicatesthe number of times the element has changed from the original value.
![Page 64: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/64.jpg)
Two Phases of Gauss Elimination
11
31321211
'22
3123
'2
2
''33
''3
3
''3
''33
'2
'23
'22
1131211
|00|0|
axaxacx
axacx
acx
cacaacaaa
Back substitution
![Page 65: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/65.jpg)
Rules• 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.
![Page 66: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/66.jpg)
Example
2 3 14 4 7 12 5 9 3
1 2 3
1 2 3
1 2 3
x x xx x xx x x
Perform Gauss Elimination of the following matrix.
Work with class
![Page 67: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/67.jpg)
Solution2 3 14 4 7 12 5 9 3
1 2 3
1 2 3
1 2 3
x x xx x xx x x
Multiply the first equation by a21/ a11 = 4/2 = 2
Note: a11 is called the pivot element
2624 321 xxx
![Page 68: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/68.jpg)
2 3 14 4 7 12 5 9 3
1 2 3
1 2 3
1 2 3
x x xx x xx x x
2624 321 xxx
a21 / a11 = 4/2 = 2
![Page 69: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/69.jpg)
2 3 14 4 7 12 5 9 3
1 2 3
1 2 3
1 2 3
x x xx x xx x x
3952
17442624
321
321
321
xxxxxxxxx
a21 / a11 = 4/2 = 2
![Page 70: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/70.jpg)
Subtract the revised first equation from the second equation
2 3 14 4 7 12 5 9 3
1 2 3
1 2 3
1 2 3
x x xx x xx x x
a21 / a11 = 4/2 = 2
395217442624
321
321
321
xxxxxxxxx
![Page 71: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/71.jpg)
4 4 4 2 7 6 1 20 2 1
1 2 3
1 2 3
x x xx x x
2 3 14 4 7 12 5 9 3
1 2 3
1 2 3
1 2 3
x x xx x xx x x
a21 / a11 = 4/2 = 2
Subtract the revised first equation from the second equation
395217442624
321
321
321
xxxxxxxxx
![Page 72: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/72.jpg)
4 4 4 2 7 6 1 20 2 1
1 2 3
1 2 3
x x xx x x
2 3 14 4 7 12 5 9 3
1 2 3
1 2 3
1 2 3
x x xx x xx x x
a21 / a11 = 4/2 = 2
Subtract the revised first equation from the second equation
395217442624
321
321
321
xxxxxxxxx
3952120
132
321
321
321
xxxxxxxxx
NEWMATRIX
![Page 73: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/73.jpg)
4 4 4 2 7 6 1 20 2 1
1 2 3
1 2 3
x x xx x x
2 3 14 4 7 12 5 9 3
1 2 3
1 2 3
1 2 3
x x xx x xx x x
a21 / a11 = 4/2 = 2
Subtract the revised first equation from the second equation
395217442624
321
321
321
xxxxxxxxx
3952120
132
321
321
321
xxxxxxxxx
NOW LET’SGET A ZEROHERE
![Page 74: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/74.jpg)
Multiply equation 1 by a31/a11 = 2/2 = 1and subtract from equation 3
2 2 5 1 9 3 3 10 4 6 2
1 2 3
1 2 3
x x xx x x
2 3 14 4 7 12 5 9 3
1 2 3
1 2 3
1 2 3
x x xx x xx x x
Solution
![Page 75: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/75.jpg)
2 3 14 4 7 12 5 9 3
2 3 12 14 6 2
1 2 3
1 2 3
1 2 3
1 2 3
2 3
2 3
x x xx x xx x x
x x xx xx x
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
![Page 76: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/76.jpg)
2 3 12 14 6 2
2 3 12 1
4 4
1 2 3
2 3
2 3
1 2 3
2 3
3
x x xx xx x
x x xx x
x
THIS DERIVATION OFAN UPPER TRIANGULAR MATRIXIS CALLED THE FORWARDELIMINATION PROCESS
Solution
![Page 77: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/77.jpg)
From the system we immediately calculate:
x344
1
Continue to back substitute
2 3 12 1
4 4
1 2 3
2 3
3
x x xx x
x
x
x
2
1
1 12
1
1 3 12
12
THIS SERIES OFSTEPS IS THEBACK SUBSTITUTION
Solution
![Page 78: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/78.jpg)
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
![Page 79: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/79.jpg)
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
![Page 80: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/80.jpg)
For example
Pivoting is used here to avoid division by zero
2 3 84 6 7 32 6 5
2 3
1 2 3
1 2 3
x xx x xx x x
4 6 7 32 3 8
2 6 5
1 2 3
2 3
1 2 3
x x xx x
x x x
![Page 81: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/81.jpg)
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
![Page 82: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/82.jpg)
Use Gauss Elimination to solve the following setof linear equations. Employ partial pivoting when necessary.3 13 502 6 454 8 4
2 3
1 2 3
1 3
x xx x xx x
Example
Work with class
![Page 83: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/83.jpg)
3 13 502 6 454 8 4
2 3
1 2 3
1 3
x xx x xx x
First write in matrix form, employing short hand presented in class.
0 3 13 502 6 1 454 0 8 4
We will clearly run intoproblems of divisionby zero.
Use partial pivoting
Solution
![Page 84: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/84.jpg)
0 3 13 502 6 1 454 0 8 4
Pivot with equationwith largest an1
![Page 85: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/85.jpg)
501330451624804
480445162501330
![Page 86: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/86.jpg)
501330433604804
501330451624804
480445162501330
Begin developingupper triangular matrix
![Page 87: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/87.jpg)
4 0 8 40 6 3 430 3 13 50
4 0 8 40 6 3 430 0 14 5 28 5
28 514 5
1 96643 3 1 966
68149
4 8 1 9664
2 931
3 8149 13 1 966 50
3 2
1
. .
.
..
..
..
. .
x x
x
CHECKokay
...end ofproblem
![Page 88: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/88.jpg)
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
![Page 89: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/89.jpg)
Graphical Depiction of the Gauss-Jordan Method with Matrix Inversion
A Ia a aa a aa a a
a a aa a aa a a
I A
11 12 13
21 22 23
31 32 33
111
121
131
211
221
231
311
321
331
1
1 0 00 1 00 0 1
1 0 00 1 00 0 1
Note: the superscript“-1” denotes thatthe original valueshave been convertedto the matrix inverse,not 1/aij
![Page 90: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/90.jpg)
• All rows are normalized by dividing them by their pivot elements
• Elimination step results in an identity matrix rather than an UT matrix
Gauss-Jordan
1 0 00 1 00 0 1
1
1
2
3
1
2 2
3 3
|||
ccc
x c
x c
x c
n
n
n
n
n
n
a a a ca a a ca a a c
ccc
n
n
n
11 12 13 1
21 22 23 2
31 32 33 3
2
3
1 0 00 1 00 0 1
1
|||
|||
' ' '
' ' ' '
![Page 91: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/91.jpg)
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}
![Page 92: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/92.jpg)
WHEN IS THE INVERSE MATRIX USEFUL?
CONSIDER STIMULUS-RESPONSE CALCULATIONS THAT ARE SO COMMON IN ENGINEERING.
![Page 93: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/93.jpg)
Stimulus-Response Computations• Conservation Laws
massforceheatmomentum
• We considered the conservation of force in the earlier example of a truss
![Page 94: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/94.jpg)
• [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
![Page 95: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/95.jpg)
ExampleGiven the following, determine {x} for the two different loads {c}
174
321
413362112
1
T
T
c
c
A
cAx
Strategy
![Page 96: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/96.jpg)
Strategy• Ax=c can be solved as A-1c=x• Note that c is given as cT.
[ 1.29 0.14 0.430.81 0.24 0.38
−0.76 − 0.05 −0.48 ]{𝑥1
𝑥2
𝑥3}={123 }
321
413362112
1
Tc
A
[ 2 −1 1−2 6 3−3 1 − 4]{
123 }={𝑥1
𝑥2
𝑥3}
![Page 97: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/97.jpg)
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
Example Spreadsheet
![Page 98: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/98.jpg)
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
![Page 99: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/99.jpg)
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 xa
n n1
1 12 2 13 3 1
11
To assure that you understand this, write the equation for x2
![Page 100: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/100.jpg)
x c a x a x a xa
x c a x a x a xa
x c a x a x a xa
x c a x a x a xa
n n
n n
n n
nn n n nn n
nn
11 12 2 13 3 1
11
22 21 1 23 3 2
22
33 31 1 32 2 3
33
1 1 3 2 1 1
![Page 101: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/101.jpg)
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
![Page 102: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/102.jpg)
x c a x a x ax c a x a x ax c a x a x a
x c a a a ca x
x c a x a a xx 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
111
2 2 21 1 23 22 2
3 3 31 1 32 2 33 3
0 0
0
///
/ '
' / '' ' / '
Gauss-Seidel Method
![Page 103: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/103.jpg)
Example
2 3 1 24 1 2 23 2 1 1
Given the following augmented matrix, complete one iteration of the Gauss Seidel method. Start with an initial estimate of xT={0,0,0}
Strategy
![Page 104: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/104.jpg)
Strategy
33323213133
2222312122
22232312122
111
111131211
111
11131321211
'/''
'/0''/'
'/00
'/
xaxaxacx
xaaxacxxaxaxacx
xacaaacx
xacaxaxacx
![Page 105: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/105.jpg)
Jacobi Method
• 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
![Page 106: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/106.jpg)
FIRST ITERATION
x c a x a x a x c a x a x a
x c a x a x a x c a x a x a
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 a x a
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
/ /
/ /
/ /
/ /
/
a x a
x c a x a x a x c a x a x a
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
![Page 107: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/107.jpg)
2 3 1 24 1 2 23 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. Start with an initial estimate of xT={0,0,0}
Example
Strategy
![Page 108: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/108.jpg)
Strategy
333323133
33323213133
222232122
22222312122
111
111131211
111
11131321211
'/00'/
'/00'/
'/00
'/
xaaacxxaxaxacx
xaaacxxaxaxacx
xacaaacx
xacaxaxacx
![Page 109: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/109.jpg)
Gauss-Seidel Methodconvergence criterion
a iij
ij
ij s
x xx,
1
100
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
![Page 110: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/110.jpg)
Convergence criteria
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.
![Page 111: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/111.jpg)
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,
439
xxx
9.05.01.04.08.02.04.02.01
3
2
1
![Page 112: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/112.jpg)
x1
Note: we are convergingon the solution
v x xu x x::
11 9 9911 13 286
1 2
1 2
CONVERGENCE
x2
![Page 113: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/113.jpg)
x1
Change the order ofthe equations: i.e. changedirection of initial estimates
u x xv x x::
11 13 28611 9 99
1 2
1 2
DIVERGENCE
x2
![Page 114: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/114.jpg)
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
inew
iold 1
![Page 115: MATRIX METHODS SYSTEMS OF LINEAR EQUATIONS Student Notes](https://reader035.vdocuments.mx/reader035/viewer/2022062323/568166e6550346895ddb24bc/html5/thumbnails/115.jpg)
Improvement of Convergence Using Relaxation
• if = 1 unmodified• 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
inew
iold 1
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• Review basic definitions • Review basic matrix operations
• Addition/subtraction• Multiplication• Determinant
• 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
Specific Study Objectives
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• Apply matrix inversion • Understand that 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
• Understand the rationale behind relaxation and how to apply this technique
Specific Study Objectives