Download 1.Topics

Topics
1.
2.
3.
4.
Computer Algebra Systems
Input-Output Analysis
Accuracy of Computations
Analyzing Networks
1.1. Computer Algebra Systems
Computer systems/packages that can do linear algebra:
1. Library of routines: LINPACK, IMSL.
2. Symbolic systems: Maple, Mathematica, Reduce, MATLAB.
3. Free packages: SciLab, MuPAD, Octave.
2. Input-Output Analysis
3. Accuracy of Computations
FORTRAN code for Gaussian forward elimination:
DO I = 1, N
DO J = I+1, N
F = A[ J, I ] / A[ I, I ]
DO K = I, N
A[ J, K ] = A[ J, K ]  F*A[ I, K ]
ENDDO
ENDDO
ENDDO
Caveat Emptor:
• Partial pivoting prevents errors caused by small pivots.
• Solutions are unreliable for ill-conditioned systems.
4. Analyzing Networks
Electrical network of a car :
Highest load: Both hi-beam headlights & brake lights are on.
Kirchhoff’s laws:
Current Law: For any point in a network, the flow in equals the flow out.
Voltage Law: Around any circuit the total drop equals the total rise.
Example:
Current law:
i 0  i1  i 2  0
Voltage law:
12i1  20
8i 2  20
i 0  i 1  i 2  0
12i1  8i 2  0
 1 1 1 0 
 1 1 1 0 
 1 0 0 25/ 6 
3 /8 
3 1 
 0 12 0 20  
 


0
1
0
5/
3

0
1
0
5/
3
2 1 

 2 /12 


 0 0 8 20 
 0 0 1 5/ 2 
 0 0 1 5/ 2 






Wheatstone bridge:
Current law:
Voltage law:
5i1  10i3  10
Top node:
i 0  i1  i 2
Left leg:
Left node:
i1  i 3  i 5
Right leg:
Right node:
i2  i5  i4
Upper loop:
5i1  50i 5  2i 2
Bottom node:
i3  i4  i0
Lower loop:
50i 5  4i 4  10i 3
Outer loop:
5i1  10i3  2i2  4i 4
 1 1 1 0 0 0 0 
 0 1 0 1 0 1 0 


 0 0 1 0 1 1 0 


0
5
0
10
0
0
10


 0 0 2 0 4 0 10 


 0 5 2 0 0 50 0 
2i 2  4i 4  10