Matlab Basics

programming_language

Fundamental

1. Plotting

   • 2-D
     Assume that Y is the function of X.
     plot(X, Y, 'o')
     To add text to the graph, specify "title", "legend", "xlabel", "ylabel".
     "hold on" and "hold off" to control the same figure.

   • 3-D
     Assume that Z is the function of X and Y.
     surf(X, Y, Z)
     Dimensionality reduction to 2-D.

     figure
     [c,h] = contour(X,Y,Z);
     clabel(c,h);

   • subgraph
     subplot(n, m, i)
     Piece the whole figure into n rows & m columns. i refers to the i-th sub-figure.

2. Loop

   for k = 1:m
       sth...
   end

3. Condiction

   if condiction_1
       do_sth
   elseif condiction_2
       do_sth
   else
       do_sth
   end

4. Get help
   doc + command shows the document of the command.
   help + command shows the instruction of the command.

5. Matrix Manipulation

   • create
     a = [1 2 3; 4,5,6]
     A comma or space between numbers specifies columns.
     The semicolon specifies rows.
     "ones", "zeros" and "rand" can generate matrix with a pair of parameters (row, col).
     "magic(x)" generates a x*x matrix.
   • operations
     a' transpose
     inv(a) invertion
     Arithmatic operations with . before mean "element-wise".
   • concatenate
     [a, a] extends horizontally.
     [a; a] extends vertically.

   • indexing
     Matlab index from ONE, not zero.
     To access an element in a matrix, a tuple (row, col) is needed.
     The usage of the colon　　start:end or start:step:end
     A single colon means "ALL".

   • workspace
     whos examines the info of all variables in the workspace.
     save + file_name.mat saves the workspace.
     load + file_name.mat loads it.

Optimization

1. Minimizing Functions of One Variable
   fminbnd(@fun_name, lower, upper, opt)
   A chosen opt means setting like optimset('Display','iter').
   Or options = optimset('PlotFcns',@optimplotfval) to plot the iteration.
   Standard return: [optimal_x, optimal_f, exit_flag, output_massage]

2. Minimizing Functions of Several Variables
   fminsearch(@fun_name, init_values)
   It it better to define the input variable of a function as a compressed one and decompose it inside the function for convenience of initialization.

3. Finding the zero point
   fzero(@fun_name, [low high], opt)

4. Unconstraint minimization
   fminunc(@f, init_value)

5. Min-Max Optimization
   fminimax(@f, init_value, A, b, Aeq, beq, lb, ub, nonlcon, opt)
   

   $$Ax\leq b \\ Aeq*x=beq \\ lb\leq x\leq ub$$
   f is a vector-value function. Min-Max optimization returns the maximum value among the function values with the minimum x.
   return [x, fval, maxfval, exitflag, output]

6. Constraint Optimization
   fmincon(@f, init, A, b, Aeq, beq, lb, ub, nonlcon, opt)

7. Nonlinear Least Square Optimization
   lsqnonlin(@f, init, lb, ub, opt)
   return [x, f^2, f, flag, output]

8. Linear Programming
   linprog(@f, A, b, Aeq, beq, lb, ub, init, opt)
   

   $$\min f^Tx$$

9. Integer Programming
   intlinprog(@f, intcon, A, b, Aeq, beq, lb, ub, opt)
intcon is the indices of integers in matrix A.

10. Quadratic Programming
    quadprog(H, f, A, b, Aeq, beq, lb, ubm init, opt)
    

    $$\min \frac{1}{2} x^T H x + f^Tx$$

Interplotation

One other interesting characteristic is that the interpolating function passes through the data points. This is an important distinction between interpolation and curve/surface fitting. In fitting, the function does not necessarily pass through the sample data points.

1. Cubic Spline Interplotation.
   Xq = (start:step:end)
Vq = interp1(X, Y, Xq, 'spline')
   So the Vq=F(Xq) specifies a interplotation function.

2. 2-D Cubic Spline Interplotation.
   set up a 2-D mesh grid
   [Xq, Yq] = meshgrid(start: step: end);
Vq = interp2(X, Y, Z, Xq, Yq, 'spline')

3. Interpolating Scattered Data
   Zq = griddata(X, Y, Z, Xq, Yq);

Symbolic computation

Basic

1. Define symbolic variables
   syms x y z
   Symbolic variables can be used to define expressions, like x+y and 4*z^2-1.
   Define a function of x using sym f(x).
   f(x) = x^2 + x + 1
   Matrices, of course.

2. Solve an equation.
   solve(x^4 == 1)
   Solve with assumptions. assume(x, x>0)
   The second assumption. assumeAlso(x, 'real').
   Clear all assumptions on x. assume(x, 'clear')

3. Substitution
   subs(x^2, y) substitute all x with y.
   subs(x+y, [x y], [1 2])

4. Simplify an expression
   simplify(x^2+x*2+1)
   Similarily, combine(2*cos(x)*sin(x), 'sincos')

5. Factorization
   factor(x^3-y^3)

6. Expand an expression.
   expand((x - y)^3)

7. Functional composition
   f(x) = and g(x)=.
   compose(f, g, x)

Calculus

1. differentiation
   diff(f(x), x)

2. integration
   int(f(x), x)
int(f(x), x, 0, 1) x from 0 to 1

3. Taylor Expansion
   taylor(f(x))

4. Limitation
   limit(f(x), x, c, 'left') the left limitation of f(x) at x=c

Ordinary differential equations

${\partial y\over \partial x} = -ay$
dsolve(diff(y, x) == -a*y)
with initial condition $y(0) = c$
dsolve(diff(y, x) == -a*y, y(0)==c)