====== Math 445 HW1 Solutions ======


1. Assign the value 0.00014 to the variable //x//, using compact scientific notation.

<code>
x = 1.4e-04;
</code>


2. Set the variable //x// to a vector whose components are the even numbers between 14 and 36.

<code>
x = 14:2:36;
</code>

3. Set the variable //x// to 200 evenly spaced points between 0 and 2.

<code>
x = linspace(0,2,200);
</code>

4. Produce a plot of //y = x^2 - 2x + 3// for the //x// of the previous problem. Label the axes.

<code>
plot(x, x.^2 - 2*x + 3)
xlabel('x')
ylabel('y = x^2 - 2*x + 3)')
</code>


5. Plot //tan(x)// versus //x// for 200 evenly spaced points between 0 and 1.57 ≈ pi/2, using
a logarithmic scale on the //y// axis. 

<code> 
x = linspace(0, 1.57, 200); 
semilogy(x, tan(x));
xlabel('x')
ylabel('y = tan(x)')
</code>

6. Plot //sin(x)// and //cos(x)// for 200 evenly spaced points between -π and π, on the same plot,
using red for //sin(x)// and blue for //cos(x)//. Add a legend that indicates which function is which color.

<code> 
x = linspace(-pi, pi, 200); 
plot(x, sin(x), 'r-', x, cos(x), 'b-', );
xlabel('x')
legend('sin(x)', 'cos(x)');
</code>

7. Produce a vector whose components are random integers between 0 and 10, inclusive.

<code>
x = randi([0,10], 13,1);  % The problem didn't specify the length of the vector, choose 13
</code>

8. Produce a vector whose components are random real numbers between 0 and 10, inclusive.

<code>
x = 10*rand(13,1);        % The problem didn't specify the length of the vector, choose 13
</code>

9. Show how you would solve the following system of equations with Matlab

<code>
  x + 2y -  z =   5
 3x +  y + 6z =  37
-3x +  y + 2z = -11
</code>

<code>
A = [1 2 -1; 3 1 6; -3 1 2];
b = [5 37 -11]';
x = A\b
</code>

10. Write a conditional expression that is true if scalar variables //x// and //y// are both nonzero 
and false otherwise.

<code>
(x ~= 0) && (y ~= 0)
</code>

11. Set variable //A// to a 3 x 5 matrix of zeros.

<code>
A = zeros(3,5);
</code>

12. Set variable //A// to a 4 x 7 matrix of random real numbers, using a guassian (normal) distribution.

<code>
A = randn(4,7);
</code>

13. Write a conditional expression that is true if a matrix //A// is square and false otherwise.

<code>
size(A,1) == size(A,2)
</code>

14. Write a conditional expression that is true if either //x// or //y// is an integer.

<code>
 x == round(x) || y == round(y)
</code>


Write a Matlab function that

15. returns 1 (true) if its argument is divisible by 3 and 0 (false) if it's not. 

<code>
function r = divisible_by_3(x)
  r = (x == 3*round(x/3));
end
</code>

or better, use the ''mod'' function

<code>
function r = divisible_by_3(x)
  r = (mod(x,3) == 0);
end
</code>

16. takes a vector //x// as input and returns 1 if the components of //x// are sorted
in ascending order, 0 if not.

<code>
function s = is_sorted(x)

  s = 1; % start assuming x is sorted
  for n=1:length(s)-1;
     if s(n) > s(n+1)
       s = 0;
       break;
     end
  end
end  
</code>

17. finds a zero of another function using the bisection search algorithm.

<code>
function xcenter = bisectsearch(f, xleft, xright)
% function xcenter = bisectsearch(f, xleft, xright)
% find a zero of f between xleft and xright using bisection search algorithm

  eps = 1e-13; % stopping condition: abs(f(xcenter)) < eps

  fleft = f(xleft);
  fright = f(xright);

  % should put in check that fleft and fright have opposite signs

  xcenter = 0.5*(xleft + xright);
  fcenter = f(xcenter);

  % repeat until f(xcenter) is close enough to zero
  while (abs(fcenter) > eps)

    if (sign(fleft) == sign(fcenter))
      % replace the left stuff with the center stuff;
      fleft = fcenter;
      xleft = xcenter;
    else
      % replace the right stuff with the center stuff;
      fright = fcenter;
      xright = xcenter;  
    end

    xcenter = 0.5*(xleft + xright);
    fcenter = f(xcenter);
  end
  
end  
</code>

18. computes the product //y = Ax// of an //m x n// matrix //A// and an //n x 1// vector //x//, according the formula
<latex>
y_i = \sum_{j=1}^n A_{ij} x_j
</latex>

<code>
function y = matvecmult(A,x);
% returns y = A*x;

  [M,N] = size(A); % assume x is N x 1

  y = zeros(M,1);

  for i = 1:M
    for j = 1:N
      y(i) = y(i) + A(i,j)*x(j);
    end
  end
end
</code>