matlab lecture 4 solving mathematical problems.ppt

MATLAB

1. MATLAB
Lecture Two (Part II)
Thursday/Friday,
3 -4 July 2003

2. Chapter 5
Applications

3. Linear Algebra
Solving a linear system
5x = 3y -2z + 10
8y + 4z = 3x +20
2x + 4y -9z = 9
Cast in standard matrix form
A x = b

4. Linear Equations
A = [5 -3 2; -3 8 4; 2 4 -9];
b = [10; 20; 9];
x = A \b
Check solution
c = A*x
c should be equal to b.

5. Gaussian Elimination
C = [ A b]; % augmented matrix
row reduced echelon form
Cr = rref(C);

6. Eigenvalues and
Eigenvectors
The problem A v = v
With pencil-paper, we must
solve for
det (A -) = 0
then solve the linear equation
MATLAB way
[V, D] = eig(A)

7. Matrix Factorizations
LU decomposition
[L, U] = lu(A);
such that L*U = A, L is a lower
triangular matrix, U is an upper
triangular matrix.

8. Matrix Factorization
QR factorization
[Q, R] = qr(A);
such that Q*R = A; Q is
orthogonal matrix and R is
upper triangular matrix.

9. Matrix factorization
Singular Value Decomposition
[U, D, V] = svd(A);
such that UDV = A, where U and
V are orthogonal and D is
diagonal matrix.

10. Sparse Matrix
See
help sparfun
for detail

11. Curve Fitting
Polynomial curve fitting
a=polyfit(x,y,n)
do a least-square fit with
polynomial of degree n. x and
y are data vectors, and a is
coefficients of the polynomial,
a = [a
n, a
n-1, …, a
1, a
0]

12. Curve Fitting
y = polyval(a, x)
compute the value of
polynomial at value x
y = a(1) x
n
+ a(2) x
n-1
+ …
+ a(n) x + a(n+1)
x can be a vector

13. Example-1:
Straight-line fit:
Problem: Fit the set of data,
and make a plot of it.
Step-1: find the coefficients
Step-2: Evaluate y at finer scale
Step-3: Plot and see

14. Interpolation
Find a curve that pass through
all the points
ynew = interp1(x,y,xnew,method)
methodis a string. Possible
choices are 'nearest', 'linear',
'cubic', or 'spline'.

15. Data Analysis and
Statistics
mean average value
median half way
std standard deviation
max largest value
min smallest value
sum sum total
prod product

16. Numerical Integration
Quad Adaptive Simpson's rule
Quad8 Adaptive Newton-Cotes
Use
quad('your_func', a, b);
or quad('your_func',a, b, opt…)

17. Ordinary Differential
Equations
General first-order ODE
dx/dt = f(x, t)
where x and f are vectors
MATLAB ODE solvers
ode23 2nd/3rd Runge-Kutta
ode45 4/5th Runge-Kutta

18. Examples of ODE
Solve the first order differential
equation
dx/dt = x + t
with the initial condition x(0)=0.

19. dx/dt = x + t, Example
function xdot = simpode(t,x)
% SIMPODE: computes
% xdot = x + t.
xdot = x + t;

20. dx/dt = x + t example
tspan = [0 2]; x0;
[t, x] = ode23('simpode', tspan, x0);
plot(t, x)
xlabel('t'), ylabel('x')

21. Second Order Equations
Nonlinear pendulum
d
2
/dt
2
+ 
2
sin= 0
Convert into set of first-order
ODE with z
1 = , z
2 = d/dt,
dz
1/dt = z
2,
dz
2/dt = -
2
sin(z
1)

22. Pendulum Function File
Function zdot = pend(t, z)
%Inputs : t = time
% z = [z(1); z(2)]
%Outputs: zdot = [z(2); -w^2 sin (z1)]
wsq = 1.56; % omega value
zdot = [z(2); -wsq*sin(z(1))];

23. Nonlinear Algebraic
Equations
Finding zeros of equation
f(x) = 0
MATLAB function
x_sol = fzero('your_func', x0, tol,
trace);
where tol and trace are optional
arguments

24. Examples of Algebraic
Equations
Solve transcendental equation
sin x = exp(x) -5
Define function
f(x) = sin(x) -exp(x) + 5

25. Chapter 6
Graphics

26. Simple Plotting
plot(x,y,'r')
plot(x,y,':', x2,y2, '+')
plot(x,y,'b--')
See Table on Chapter 6 for style
options.

27. Other Plotting
Commands
xlabel, ylabellabels on axis
title title text
text text in graphics
legend legend
axis axis limits/scale
gtext text with local
 located by mouse

28. Specialized 2D Plots
semilogx log x vs y
semilogy x vs log y
loglog log x vs log y
polar polar plot
bar bar chart
hist histogram
pie pie plot

29. 3D Plots
plot3(x, y, z, 'style-option')
Example:
t=linspace(0, 6*pi, 100);
x=cos(t); y=sin(t); z = t;
plot3(x,y,z)

30. Advanced Features
"Handle graphics" gives a better
control of graphic output. It is
a lower level graphics.

31. Chapter 8, What Else is
There?
Symbolic Math and other
Toolboxes
External Interface: Mex-files
Graphics User Interface