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第 6 章 線性代數方程式 Linear Algebraic Equations. linear. 5x-2y=13 7x+3y=24 Linear ~ dependent variable 不可以有一次以外的次方. 6.1 Elementary Solution Methods. Successive elimination of variables Gauss elimination. Gauss elimination. -x+y+2z=2(1) 3x-y+z=6(2) -x+3y+4z=4(3) -x+y+2z=2(1) - PowerPoint PPT Presentation
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第 6章 線性代數方程式Linear Algebraic Equations
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linear
5x-2y=137x+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 solutions3. 3x-4y=5 6x-8y=3 empty set
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Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Homogeneous equations
6x+ay=02x+4y=0(a-12)y=0If a~=12, x=0If 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
75
4392
75
4392
743592
2
1
2
1
21
21
bxA
bAx
xx
xx
xxxx
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mn
mnmm
n
n
mn
mnmm
n
n
mnmnmm
nn
nn
b
bb
x
xx
aaa
aaaaaa
b
bb
x
xx
aaa
aaaaaa
bxaxaxa
bxaxaxabxaxaxa
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
3792106143
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[5326
][ 21
xx
d/CxdxC
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Ex.6.2-2 p.342
• Kirchhoff’s law
• Current conservation at a node
532
421
iiiiii
00
0
23355
552244
44111
viRiRiRiRiR
iRiRv
Puff! The magic dragon, live by the sea… 0
00
23355
552244
44111
viRiRiRiRiR
iRiRv
532
421
iiiiii
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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)(
)(
viRRiRiRiRRRiR
viRiRR
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Exercise p.343-345
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Matrix inverse
bAxxIbAAxAIAAAA
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
solutionsinfinite36
631
63
yxyxyx
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Underdetermined systems
0A
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Matrix rank 矩陣的秩
2rank
157943
4437210
03792106143
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
108627935
][
107
,862935
][
bA
bA
bA
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Existence and uniqueness of solutions
• Augmented matrix
][rank][rankifonlyandifunknowns) eqns., (11
bAAbxA
nmmnnm
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Singular matrix0A
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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 6.4-22x-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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1600400
0510111
][
1600400
,0510111
1600400
0510111
0510)4(400,0400,
3
2
1
3
2
1
21
321
Ab
x
bA
0M0F
TTT
TTT
TTTTT
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32
31
12
248080
2320
TTTT
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)*bans = 93.3333 133.3333 173.3333
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Ex 6.4-4, pp.3622x-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)*brref([A b])x = -1.2148 0.2074 -0.1481ans = 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
zyx
xBA3040
433335
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+f4f1+f2=300+200600+400=f2+f3f3+f4=300+500
4
3
2
1
8001000500300
1100011000111001
ffff
xbA
bAx
800200300
0800200300
000011001-0101001
b])rref([A
43
42
41
ffffff
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Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…
Puff! The magic dragon, live by the sea…