Final Exam Review

MATLAB

Create a function:

function [y1, y2] = name(x1, x2, x3)

Anonymous Functions:

y = @(x,z) 3*x + 6*z

Open files

% opens as write only
 fileID = fopen(filename, 'w');

% write to file
fprintf(fileID, '%.6f %.6f\n', x(i), y(i));
   
% close file
fclose(fileID);

Random

rand(1)

Generates a random decimal number between 0 and 1.

Graphing

2D Plotting If x and y are a vectors of values:

plot(x,y)

Using linspace and an equation:

x = linspace(1,10,100);
y = (2./x).^3;
plot(x,y,'go-') % green, circle marker, solid line

3D Plotting

plot3(x,y,z)

Plotting area

xlim(lower_bound, upper_bound);

Labling Use hold on to add stuff to the same figure.

% add a title
title('title');

% label axis
xlabel('x');
ylabel('y');
zlabel('z');

% add text
text(x,y,'text');

Euler's Method:

Given

1. first order ordinary differential equation: $\frac{dy}{dx}$ as dxdt
2. initial condition $y_0 = y (x_0)$

We determine $y_i = y(x_0 + ih)$ for $i = 1,2,...$ by using Euler's Method. $y_0 = y(x_0)$ $y_{i+1}=y_i+h* \dot f(x_i,x_y)$ for $i=1,2,...$

for i = 1:max_steps
    y(i+1) = y(i) + dydt(t(i), y(i))*(h);
end

Runge-Kutta Method

These methods acheive the accurracy of the taylor series approach without requiring the calculations of higher derivatives.

$y_{i+1} = y_i + \psi (x_i,y_i,h)h$ $\psi = a_1 k_1 + a_2 k_2 + ... + a_n k_n$ <-- Increment Function

$a$'s are constants

$k_1 = f(x_i, y_i)$ $k_2 = f(x_i + p_i h, y_i + q_{11} k_1 h)$ <-- $p$'s and $q$'s are constants $k_3 = f(x_i + p_3 h, y_i + q_{21} k_2 h + q_{22} k_2 h)$ ... $k_n = f(x_i + p_{n-1} h, y_i + q_{n-1} k_1 h + q_{n-1,2} k_2 h + ... + q_{n-1,n-1} k_{n-1} h)$

The order is chosen as $n$ and the values of a, p, q are evalulated by setting the general equation equal to terms in a Taylor series expansion.

Fourth Order Runge Kutta (RK4): $K_1 = f(x_i, y_i)$ $K_2 = f(x_i + 1/2h, y_i + 1/2 k_1 h)$ $K_3 = f(x_i + 1/2 h, y_i + 1/2 K_2 h)$ $K_4 = f(x_i + h, y_i + K_3 h)$ $y_{i+1} = y_i + \frac{h}{6} (K_1 + 2K_2 + 2 K_3 + K_4)$

You need the following input

• dydt the name of the M-file that evalues that ODE
• tspan consisting of ti and tf that corresponds to the initial and final times.
• y0 is the initial values of the dependent variable
• h is the stetpsize

Before we start:

1. check that nargin >= 4 (if you are accepting optional inputs)
2. check that tspan is span some time
3. populate the independent vector (create time t)

The computation:

for i=1:(length(x)-1) % calculation loop
   k1 = dydt(x(i),y(i));
   k2 = dydt(x(i)+0.5*h,y(i)+0.5*h*k1);
   k3 = dydt((x(i)+0.5*h),(y(i)+0.5*h*k2));
   k4 = dydt((x(i)+h),(y(i)+k3*h));
   
   y(i+1) = y(i) + (1/6)*(k1+2*k2+2*k3+k4)*h; % main equation
end

Solving system of equations

If A is the matrix and B is the column of solutions. Solve with one of the following ways

% inverse 
x = inv(A)*b

% rref
r = rref([a b]);
x = r(:, 4);

Solve an equation

syms x
eqn = sin(x) == x^2 - 1;
S = solve(eqn,x)

Polynomials

Fit a set of points to a polynomial with n degrees with polyfit:

p = polyfit(x,y,1)

p contains two coefficients that describe a linear curve fit.

Use polyval to evaluate the a polynomial at the given points

p = polyfit(x,y,1);
x = [25 30 35 40 45];
y = polyfit(p, x);

Mean Squared Error (MSE)

Matlab function: err = immse(x,y).

Doing it manually: $MSE = \frac{1}{n} * \Sigma (actual-forcast)^2$

• where n is sample size

Newton's Method

Approximates the real zeros of a function by using tangent lines to approximate the graph of the functions near its x-intercepts.

Consider function f that is continuous and differentiable on the interval [a b]. Choose a starting point x1.

The equation of the tangent line is $y = f'(x_1) (x-x_1) + f(x_1)$

If we let $y=0$ and solve for $x$ we find the new estimate: $x = x_1 - \frac{f(x_1)}{f'(x_1)}$

We can improve on the value by calculating another estimate using the old estimate $x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}$

Generally: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

The code look something like

function [xr, iter, err, results] = newtons_method(f, fder, xi, iter_max, tol) 
    xr = xi;
    for iter=1:iter_max
        xrold = xr;
        xr = newtons_update(f, fder, xrold);
        err(iter) = abs((xrold - xr)/xrold);
        results(iter) = xr;
        if (err(iter) < tol) 
            break;
        end
    end
end

where newtons_update is:

function [xr] = newtons_update(f,fder, xi)
    xr = xi - f(xi)/fder(xi);
end

Least Squares Approximation

Do it with matlab's built in function: polyfit.

It's goal is to minimize error betweent the fit line and the data poins. $Minimize\{Error = \Sigma [y_i - (ax_i + b)]^2\}$

• Error can also be written as $\Sigma [y_i - f(x_i)]^2$

Taking the derivative of this to minimize involves quite a bit of work... refer to lab10

Reading Input

Use fopen to open the file. Use fscapf(fileID, '%f') to read numbers

• or use %c to read as strings

You can choose a whole line with tline = fgetl(fileID) and iterate through the whole file until it reaches the end. Here is an example of reading and displaying each line of a text file:

fid = fopen('badpoem.txt');

while ~feof(fid)
	tline = fgetl(fid);
	disp(tline)
end

fclose(fid);