====== Math 445 mid-term exam with solutions ======

**1.** Write one line of Matlab code that returns the 4th column of 
the matrix $A$.

  A(:,4)
  
**2.** Write Matlab code that sets all entries in the 3rd row of the  
matrix $A$ to zero. Its possible to do this in one line, but you can use several.

  A(3,:) = 0;
  
or

  A(3,:) = zeros(1,size(A,2));
  
or

  [M,N] = size(A);
  A(3,:) = zeros(1,N);
  
**3.**  Write one line of Matlab code for an anonymous function that 
computes the value of the polynomial $3x^2 - 2x - 7$ for an input argument $x$.

  f = @(x) 3*x^2 - 2*x - 7;
  
**4.** How would you use Matlab and the anonymous function from 
problem 3 to find a numerical solution to the equation  $3x^2 - 2x - 7 = 0$? 
One line of code should do it. 

  x = newtonsearch(f,2); 
  
or 

  x = fzero(f,2);
  
Note: 2 is an initial guess for the solution, chosen because f(2) = 1 (this is relatively close to zero).

**5.** Write one line of Matlab code that evaluates to 1 (true) if $x$ 
is negative and $y$ is positive, and 0 (false) otherwise.

  x < 0 && y > 0
  
**6.**  Write one line of Matlab code that evaluates to 1 (true) if
both $x$ and $y$ are positive or if both are negative, and 0 (false) otherwise.

  (x < 0 && y < 0) || (x > 0 && y > 0)
  
**7.** Write one line of Matlab code that counts how many components
of the vector $v$ are exactly zero.
  
  sum(v==0)
  
**8.**  Show how to solve the system of equations with three lines of Matlab code.

<latex>
\begin{align*}
3x + y + 2z - 6 &= 0 \\
9z - x - 8       &= 0 \\
5y - 4x - 1 &= 0
\end{align*}
</latex>

  A = [ 3 1 2; -1 0 9; -4 5 0];
  b = [6; 8; 1];
  x = A\b
  
**9.**  Write a Matlab function that computes the mean (i.e. average) of 
the components of a vector $x$ according to the formula 

<latex>
  \text{mean}(x) = (1/N) \sum_{i=1}^{N} x_i
</latex>

where $N$ is the length of the vector. Your function should evaluate this 
sum directly instead of using the Matlab %%sum%% or %%mean%% functions.

  function m = mean(x)
  % compute mean of vector x
  
    m = 0;
    N = length(x);
    
    for i=1:N
      m = m + x(i);
    end
    
    m = m/N;
  end
  
**10.**  Write a Matlab function that takes an $M \times M$  transition 
matrix $T$ for a network of $M$ web pages and returns the page rank vector $p$ 
of the steady-state distribution of visitors to each page. The page rank is 
given by $p = T^n e$, where $e$ is an arbitrary $M$-vector whose components 
sum to 1, and $n$ is large number. You can set $n$ to 100.
    
  function p = pagerank(T);
  % compute that page rank vector for transition matrix T
  
    [M,M] = size(T);
  
    e = zeros(M,1);
    e(1) = 1;
    
    n=100;
    
    p = T^n * e;
    
  end