CDA6530: Performance Models of Computers and Networks Chapter 4: Using Matlab for Performance Analysis and Simulation - TexPoint fonts used in EMF.
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Objective - Learn a useful tool for mathematical analysis and simulation
- Interpreted language, easy to learn
- Use it to facilitate our simulation projects
- A good tool to plot simulation/experiment results figures for academic papers
- More powerful than excel
- Could directly create .eps for Latex
Introduction - MatLab : Matrix Laboratory
- Numerical Computations with matrices
- Why Matlab?
- User Friendly (GUI)
- Easy to work with
- Powerful tools for complex mathematics
- Matlab has extensive demo and tutorials to learn by yourself
Matlab Software Access - all UCF in-campus computers have student-version Matlab installed
- If you have no access to Matlab, you can use Octave, an open-source free software
- http://www.gnu.org/software/octave/
- The programming should be almost identical
Matrices in Matlab - To enter a matrix
- >> A = [2 5 3; 6 4 1]
- >> B = [1:1.5:6; 2 3 4 5]
- >> for i=1:4
- for j=1:3
- C(i,j)=i*j;
- end
- end
- >> D =[]; D=[D;5]; D=[D;6;7]
- >> E = zeros(4, 5)
Basic Mathematical Operations - Remember that every variable can be a matrix!
- Addition:
- >> C = A + B
- Subtraction:
- >> D = A – B
- Multiplication:
- >> E = A * B (Matrix multiplication)
- >> E = A .* B (Element wise multiplication, A and B same size)
- Division:
- Left Division and Right Division
- >> F = A . / B (Element wise division)
- >> F = A / B = A*inv(B) (A * inverse of B)
- >> F = A . \ B (Element wise division)
- >> F = A \ B=inv(A)*B (inverse of A * B)
Generating basic matrices - Matrix with ZEROS:
- >> A = zeros(m, n)
- Matrix with ONES:
- >> B = ones(m, n)
- IDENTITY Matrix:
- >> I = eye(m, n)
- m Rows
- n Columns
- zeros, ones, eye Matlab functions
Obtain Information - Size(A): return [m n]
- Length(A): length of a vector
- B = A(2:4,3:5)
- B is the subset of A from row 2 to row 4, column 3 to column 5
- A(:, 2)=[]
- Inv(A): inverse of A
- Rank(A): rank of matrix A
- A’: transpose of A
- Det(A): determinant
- V= eig(A): eigenvalue vector of A
- [V,D] = eig(A) produces matrices of eigenvalues (D) and eigenvectors (V) of matrix A, so that A*V = V*D
Random Number Generators - Rand(m,n): matrix with each entry ~ U(0,1)
- You can use this for the programming project 1
- Randn(m,n): standard normal distribution
- You cannot use this in programming project 1
- You must use the polar method I introduced!
Basic 2-D Figure Plot - Plot(X, Y):
- Hold: next plot action on the same figure
- Title(‘title text here’)
- Xlabel(‘…’), ylabel(‘…’)
- Axis([XMIN XMAX YMIN YMAX])
- Legend(‘…’)
- Grid
- Example demo
Elementary Math Function - Abs(), sign()
- Sin(), cos(), asin(), acos()
- Exp(), log(), log10()
- Ceil(), floor()
- Sqrt()
- Real(), imag()
Elementary Math Function - Vector operation:
- Max(), min(): max/min element of a vector
- Mean(), median()
- Std(), var(): standard deviation and variance
- Sum(), prod(): sum/product of elements
- Sort(): sort in ascending order
Save/Load Data - Save fname
- Save all workspace data into fname.mat
- Save fname x y z
- Save(fname): when fname is a variable
- Load fname
- No error in data
- You can run simulation intermittently
- Save/load data between runs
Input/Output for Text Files - Input data file for further analysis in Matlab
- Run simulation using C
- matlab is slow in doing many loops
- Use Matlab for post-data processing
- Matrix calculation, utilize Matlab math functions
- Simply use Matlab for figure ploting
- Excel has constraint on data vector length (<300?)
- Functions:
- [A,B…]= Textread(fname, format)
- Use fprintf(), fscanf() similar to C
- Note that variables here can be vectors/matrices
- Show examples here of writing data to text file
Advanced Graph - Subplot(m, n, p)
- breaks the Figure window into an m-by-n matrix of small axes, selects the p-th axes for the current plot, and returns the axis handle.
- Semilogx(), semilogy(), loglog()
3-D plot - x=[0:10]; y=[0:10]; z=x’*y;
- mesh(x,y,z); figure; surf(x,y,z);
M-file - Script or function
- Scripts are m-files containing MATLAB statements
- Functions are like any other m-file, but they accept arguments
- It is always recommended to name function file the same as the function name
- function A = changeSign(B)
- % change sign for each element
- [m,n] = size(B); A = zeros(m,n);
- for i=1:m
- for j=1:n
- A(i,j)= -B(i,j);
- end
- end
- return
Online Tutorials - Matlab itself contains many tutorials
- Other online tutorials:
- http://www.math.siu.edu/matlab/tutorials.html
- http://www.cs.cmu.edu/~ggordon/780/lectures/matlab_tutorial.pdf
- Google search “matlab tutorial ppt” to find a lot more
- Example on Using Matlab for Markov Chain Steady State Calculation
- Discrete-time Markov Chain transition matrix:
- ¼ P = ¼ , ¼ [1 1 1… 1]T = 1
- ¼ (P – I) = 0, But we cannot use it directly
- Replace first column in (P-I) with [1 1..1]T to be A, then we can solve the linear equation set by ¼ = [1 0 0 … 0] A-1
- Another way: P*P*P*P……
- Tutorial on Matlab Simulink
- Graphical programming language
- Powerful modeling tool
- Differential Equations
- Physiological systems
- Control systems
- Transfer functions
- M-file can call a simulink model
- “sim fname”
- Use current workspace variables
- Simulation results can be saved to workspace variables
- Thus can be process after simulink
Example: Internet Worm Propagation Example 2: RC Circuit Save result to workspace variables - the save format is "structure with time".
- Suppose the workspace variable is X_t.
- Then:
- X_t.time saves the simulation step times (vector)
- X_t.signals.values saves the simulation results (vector).
- plot(X_t.time, X_t.signals.values);
- Variable step simulation or fixed step simulation:
- "to workspace" use "-1" for sample time (inherited)
- Then X_t.time has variable size
- "to workspace" use "1" for sample time
- Then each time tick has one result value
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