Puff! The magic dragon, live by the sea… 第 6 章線性代數方程式 Linear Algebraic Equations.

Puff! The magic dragon, live by the sea… 第 6 章線性代數方程式 Linear Algebraic Equations

Puff! The magic dragon, live by the sea… linear 5x-2y=13 7x+3y=24 Linear ~ dependent variable 不可以有一次以外的次方

Puff! The magic dragon, live by the sea… 6.1 Elementary Solution Methods Successive elimination of variables Gauss elimination

Puff! The magic dragon, live by the sea… 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)

Puff! The magic dragon, live by the sea… Gauss elimination -x+y+2z=2(1) 5z=10(6) y=-1(7) x=1

Puff! The magic dragon, live by the sea… Self test p.334

Puff! The magic dragon, live by the sea… 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

Puff! The magic dragon, live by the sea…

Puff! The magic dragon, live by the sea… Homogeneous equations 6x+ay=0 2x+4y=0  (a-12)y=0  If a~=12, x=0 If a=12, infinite solutions

Puff! The magic dragon, live by the sea… Ill-conditioned equations 3x-4y=5(1) 6x-8.002y=3(2) (1)  y=(3x-5)/4 (2)  y=(3x-1.5)/4.001 (x,y)=(4668, 3500) Very sensitive !

Puff! The magic dragon, live by the sea… Matrix methods for linear equations

Puff! The magic dragon, live by the sea… Determinants 行列式 A=[3,-4,1;6,10,2;9,-7,3]; det(A) ans = 0

Puff! The magic dragon, live by the sea… Determinants and singular problems If determinant of A=0, then a singular problem. If determinant of A ~= 0 If determinant of A = 0

Puff! The magic dragon, live by the sea… Left-division method A=[3,2,-9;-9,-5,2;6,7,3]; det(A) b=[-65;16;5]; A\b ans = 288 ans = 2.0000 -4.0000 7.0000

Puff! The magic dragon, live by the sea… Right-division method

Puff! The magic dragon, live by the sea… Ex.6.2-2 p.342 Kirchhoff’s law Current conservation at a node

Puff! The magic dragon, live by the sea… % resist.m % solve the current i_1,i_2,i_3 R=[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

Puff! The magic dragon, live by the sea… Exercise p.343-345

Puff! The magic dragon, live by the sea… Matrix inverse

Puff! The magic dragon, live by the sea… An example of Statics Derivation of equilibrium equations. Solve the linear equations by matlab. P.346-348

Puff! The magic dragon, live by the sea… Ex. 6.2-5 Check by matlab as well as by hand for an inversion of 2 by 2 matrix. 2x+9y=5 3x-4y=7 A=[2,9;3,-4]; b=[5;7]; x=inv(A)*b x = 2.3714 0.0286

Puff! The magic dragon, live by the sea… Cramer’s method Self test …

Puff! The magic dragon, live by the sea… 6.4 Underdetermined systems A=[1,3]; b=6; x=A\b x = 0 2

Puff! The magic dragon, live by the sea… Underdetermined systems

Puff! The magic dragon, live by the sea… Matrix rank 矩陣的秩 A=[3,-4,1;6,10,2;9,-7,3]; rank(A) ans = 2

Puff! The magic dragon, live by the sea… Existence and uniqueness of solutions Augmented matrix

Puff! The magic dragon, live by the sea… Singular matrix

Puff! The magic dragon, live by the sea… Ex 6.4-1 3x-2y+8z=48 -6x+5y+z=-12 9x+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 = 3 ans = 3 x = 2.0000 5.0000

Puff! The magic dragon, live by the sea… Ex 6.4-2 2x-4y+5z=-4 -4x-2y+3z=4 2x+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 singular to working precision. x = Inf

Puff! The magic dragon, live by the sea… pinv command & Euclidean norm x=pinv(A)*b x = -1.2148 0.2074 -0.1481

Puff! The magic dragon, live by the sea… Ex 6.4-3 Statically indeterminate problem

Puff! The magic dragon, live by the sea… A=[1,1,1;10,5,0]; b=[400;1600]; rank(A) rank([A b]) A\b ans = 2 ans = 2 ans = 160.0000 0 240.0000 pinv(A)*b ans = 93.3333 133.3333 173.3333

Puff! The magic dragon, live by the sea… Ex 6.4-4, pp.362 2x-4y+5z=-4 -4x-2y+3z=4 2x+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

Puff! The magic dragon, live by the sea… Underdetermined systems P.363-364 ex.6.4-5 5x+3y+3z=40 3x+3y+4z=30 x-0.5z=5 y+1.8333z=5 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

Puff! The magic dragon, live by the sea… Ex 6.4-6 p.364 100+200=f1+f4 f1+f2=300+200 600+400=f2+f3 f3+f4=300+500

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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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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Puff! The magic dragon, live by the sea… 第 6 章線性代數方程式 Linear Algebraic Equations.

Presentation on theme: "Puff! The magic dragon, live by the sea… 第 6 章線性代數方程式 Linear Algebraic Equations."— Presentation transcript: