% QUESTION 1
%    The operator * is the usual multiplication, which by default
%  applies to two matrices, e.g: A*B returns the product of two
%  matrices A and B.  With the dot, .* multiplies componentwise,
%  i.e.: when C = A*B, then C(i,j) = A(i,j)*B(i,j).
%    We use the operator * to compute the residual of a solution
%  to a linear system A*x = b, e.g.: b - A*x gives an idea how
%  accurate the solution is.
%    The operator .* is used to evaluate a formula, sampled for
%  plotting, e.g.: z = x.*y samples the surface z = x y.
% QUESTION 2
t = 0:0.1:25;
r = sqrt(t);
polar(t,r);
% QUESTION 3
r = -pi:pi/20:pi;
[x,y] = meshgrid(r,r);
z = y.^2.*cos(x+y);
mesh(x,y,z);
% QUESTION 4
% (a)
subplot(2,1,1);
t = -1:0.01:1;
x = t; y = t.^2; z = t.^3;
plot3(x,y,z);
% (b)
subplot(2,1,2);
r = -1:0.01:1;
[x1,y1] = meshgrid(r,r);
z1 = x1.^3;
mesh(x1,y1,z1);
view(3);
hold on
[x2,z2] = meshgrid(r,r);
y2 = x1.^2;
mesh(x2,y2,z2);
% QUESTION 5
%   The three commands we have seen to encounter approximate data
%   are polyfit, spline, and fft.  Their application domains are
%   the following:
%     polyfit: hypothesis testing, least squares fitting
%      (see answer to QUESTION 6);
%     spline: geometric design (see answer to QUESTION 6);
%     fft: signal processing, filtering of noise from signal by
%       the computation of the spectrum via discrete Fourier transform
%       implemented in fft.
% QUESTION 6
% (a) The command polyfit creates a polynomial of a given degree
%     which fits given data, using least squares.
%  The command spline approximates given data using a piecewise
%  function, where for each piece typically a cubic polynomial is used.
% (b) The command polyfit is useful to discover a relationship between
%  two variables x and y, to test a hypothesis on measured data,
%  e.g.: does y depend linearly or quadratically on y?
%  Since the degree is most important here, spline is inappropriate here.
% (c) The command spline is used in computer aided geometric design,
%  given some control points, we can draw a shape of an object.
%  Since in the design we want the adding of new points to have only 
%  a local effect, constructing a high degree polynomial as with polyfit
%  is not appropriate for this type of application.
% QUESTION 7
% (a) the file simpson.m:
function y = simpson(f,a,b)
% returns an approximation for the integral of f(x) over [a,b]
% using Simpson's rule
fa = feval(f,a);
fm = feval(f,(a+b)/2);
fb = feval(f,a);
y = (b-a)*(fa+4*fm+fb)/3;
% (b) 
simpson('cos',pi/4,pi/2)
% QUESTION 8
t = 0:1/30:1;
y = sin(2*pi*t);
plot(t,y);
axis([0 1 -2 2];
% Taking a step size of 1/30,
% -- or 30+1 smaples in [0,1[ --
% gives a good plot for sin(2*pi*t)
% 1. we need 3*30+1 = 91 samples for sin(2*pi*t) in [0,3]
%    and 10*91 = 910 samples for sin(2*pi*10*t) in [0,3]:
t = 0:1/900:3;
y = sin(2*pi*t);
hold on
plot(t,y);
% 2. for any k, w will need k*91 samples to have a good plot
%    of sin(2*pi*k*t) in [0,3], and use t = 0:1/(k*90):3;
% QUESTION 9
p3 = [ 0 0 1; 0 1 0; 1 0 0];
p3*[1:3]'
% the answer for any n, e.g.:
n = 31;
rows = [1:n];
cols = [n:-1:1];
p31 = sparse(rows,cols,1);
p31*[1:n]'
% QUESTION 10
[x y z] = sphere(30);
x1 = .1*x;
y1 = .1*y - 2;
z1 = .1*z + 2;
figure
hold on
surf(x,y,z);
surf(x1,y1,z1);
axis off
% QUESTION 11
f = [-23 -44];
A = [1 12; 24 30; -2 83; -1 0; 0 -1];
b = [123  912  134  0  0]';
s = linprog(f,A,b)
format bank
s
A*s - b
% the place of the zero elements in the answer 
% tell where the answer s reaches equality