puff! the magic dragon, live by the sea… 第 6 章 線性代數方程式 linear algebraic...
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Puff! The magic dragon, live by the sea…
第 6章 線性代數方程式Linear Algebraic Equations
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linear
5x-2y=13
7x+3y=24
• Linear ~ dependent variable 不可以有一次以外的次方
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6.1 Elementary Solution Methods
• Successive elimination of variables
• Gauss elimination
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Gauss elimination
-x+y+2z=2 (1)
3x-y+z=6 (2)
-x+3y+4z=4 (3)
-x+y+2z=2 (1)
2y+7z=12 (4)
2y+2z=2 (5)
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Gauss elimination
-x+y+2z=2 (1)
5z=10 (6)
y=-1 (7)
x=1
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Self test p.334
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Singular & ill-conditioned problems
1. 3x-4y=5 6x-10y=2 Sol. (7,4)
2. 3x-4y=5 6x-8y=10 Infinite solutions
3. 3x-4y=5 6x-8y=3 empty set
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Homogeneous equations
6x+ay=0
2x+4y=0
(a-12)y=0
If a~=12, x=0
If a=12, infinite solutions
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Ill-conditioned equations
3x-4y=5 (1)
6x-8.002y=3 (2)
(11 y=(3x-5)/4
(11 y=(3x-1.5)/4.001
(x,y)=(4668, 3500)
Very sensitive !
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Matrix methods for linear equations
7
5
43
92
7
5
43
92
743
592
2
1
2
1
21
21
bxA
bAx
x
x
x
x
xx
xx
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mn
mnmm
n
n
mn
mnmm
n
n
mnmnmm
nn
nn
b
b
b
x
x
x
aaa
aaa
aaa
b
b
b
x
x
x
aaa
aaa
aaa
bxaxaxa
bxaxaxa
bxaxaxa
2
12
1
21
22221
11211
2
12
1
21
22221
11211
2211
22222121
11212111
bxA
bAx
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Determinants 行列式
379
2106
143
A
A=[3,-4,1;6,10,2;9,-7,3];det(A)ans = 0
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Determinants and singular problems
• If determinant of A=0, then a singular problem.
• If determinant of A ~= 0
• If determinant of A = 0
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Left-division methodA=[3,2,-9;-9,-5,2;6,7,3];det(A)b=[-65;16;5];A\b
ans = 288ans = 2.0000 -4.0000 7.0000
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Right-division method
]193[53
26][ 21
xx
d/Cx
dxC
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Ex.6.2-2 p.342
• Kirchhoff’s law
• Current conservation at a node
532
421
iii
iii
0
0
0
23355
552244
44111
viRiR
iRiRiR
iRiRv
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Puff! The magic dragon, live by the sea… 0
0
0
23355
552244
44111
viRiR
iRiRiR
iRiRv
532
421
iii
iii
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% resist.m% solve the current i_1,i_2,i_3R=[5,100,200,150,250]*1000;v1=100; v2=50;A1=[R(1)+R(4),-R(4),0];A2=[-R(4),R(2)+R(4)+R(5),-
R(5)];A3=[0,R(5),-(R(3)+R(5))];A=[A1;A2;A3];b=[v1;0;v2];current=A\b;disp('The currents are:')disp(current)The currents are: 1.0e-003 *
0.9544 0.3195 0.0664
235325
35254214
124141
)(
0)(
)(
viRRiR
iRiRRRiR
viRiRR
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Exercise p.343-345
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Matrix inverse
bAxxI
bAAxA
IAAAA
bAx
1
11
11
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An example of Statics
• Derivation of equilibrium equations.
• Solve the linear equations by matlab.
• P.346-348
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Ex. 6.2-5
• Check by matlab as well as by hand for an inversion of 2 by 2 matrix.
2x+9y=53x-4y=7
A=[2,9;3,-4];b=[5;7];x=inv(A)*bx = 2.3714 0.0286
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Cramer’s method
• Self test …
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6.4 Underdetermined systems
A=[1,3];
b=6;
x=A\b
x =
0
2
solutionsinfinite
36
631
63
yx
y
x
yx
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Underdetermined systems
0A
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Matrix rank 矩陣的秩
2rank
1579
43
4437
210
0
379
2106
143
A=[3,-4,1;6,10,2;9,-7,3];rank(A)
ans =
2
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Existence and uniqueness of solutions
• Augmented matrix
10862
7935][
10
7,
862
935
][
bA
bA
bA
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Existence and uniqueness of solutions
• Augmented matrix
][rank][rankifonlyandif
unknowns) eqns., (11
bAA
bxA
nmmnnm
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Singular matrix
0A
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Ex 6.4-1
3x-2y+8z=48-6x+5y+z=-129x+4y+2z=24
A=[3,-2,8;-6,5,1;9,4,2];b=[48;-12;24];rank(A)rank([A b])x=A\b
ans =3ans = 3x = 2.0000 -1.0000 5.0000
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Ex 6.4-2
2x-4y+5z=-4-4x-2y+3z=42x+6y-8z=0
• A=[2,-4,5;-4,-2,3;2,6,-8];• b=[-4;4;0];• rank(A)• rank([A b])• x=A\b
• ans =• 2• ans =• 2• Warning: Matrix is singu
lar to working precision.• x =• Inf• Inf• Inf
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pinv command & Euclidean norm
• x=pinv(A)*b
x =
-1.2148
0.2074
-0.1481
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Ex 6.4-3
• Statically indeterminate problem
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1600
400
0510
111][
1600
400,
0510
111
1600
400
0510
111
0510)4(400,
0400,
3
2
1
3
2
1
21
321
Ab
x
bA
0M
0F
T
T
T
T
T
T
TT
TTT
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32
31
12
2480
80
2320
TT
TT
TT
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A=[1,1,1;10,5,0];b=[400;1600];rank(A)rank([A b])A\bans = 2ans = 2ans = 160.0000 0 240.0000
pinv(A)*b
ans =
93.3333
133.3333
173.3333
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Ex 6.4-4, pp.362
2x-4y+5z=-4-4x-2y+3z=42x+6y-8z=0
• x+0y-0.1z=-1.2• 0x+y-1.3z=0.4• 0x+0y+0z=0
A=[2,-4,5;-4,-2,3;2,6,-8];
b=[-4;4;0];
x=pinv(A)*b
rref([A b])
x =
-1.2148
0.2074
-0.1481
ans =
1.0000 0 -0.1000 -1.2000
0 1.0000 -1.3000 0.4000
0 0 0 0
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Underdetermined systems
• P.363-364 ex.6.4-55x+3y+3z=403x+3y+4z=30
• x-0.5z=5• y+1.8333z=5
z
y
x
xBA30
40
433
335
A=[5,3,3;3,3,4];
b=[40;30];
rref([A b])
ans =
1.0000 0 -0.5000 5.0000
0 1.0000 1.8333 5.0000
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Ex 6.4-6 p.364
100+200=f1+f4
f1+f2=300+200
600+400=f2+f3
f3+f4=300+500
4
3
2
1
800
1000
500
300
1100
0110
0011
1001
f
f
f
f
xbA
bAx
800
200
300
0
800
200
300
0000
1100
1-010
1001
b])rref([A
43
42
41
ff
ff
ff
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Puff! The magic dragon, live by the sea…
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Puff! The magic dragon, live by the sea…