Question 1
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function [lo,hi] = split(x)
% Sort 1-d array X and reutrn 2 arrays:
% LO contains all elements in X <= MEAN(X)
% HI contains all elements in X > MEAN(X)

% Sort array
   x = selectsort(x);

% Define LO<=mean
   loloc = (x <= mean(x));  % locations
   lo = x(loloc);           % values

% Define HI~=LO
   hi = x(~loloc);

%%%%%%%%%%%%%%%%%%%%%%%%%%

function y = selectsort(y)
% Sort and overwrite 1-d array Y in 
% ascending order

len = length(y);

for j = 1:len-1   % start of unsorted segment

   % Find min in unsorted segment
      [val,ind]=min(y(j:len));
      ind = ind+j-1;
   
   % Swap min into correct position
      y([j ind]) = y([ind j]);
end


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Question 2
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function newtonRalston(x)
% Find positive real root using Newton-Ralston
% method given a starting point X.
% Equation solved:
% f(x)= x^4 - 8.6x^3 - 35.51x^2 + 464.4x - 998.46

% Parameters:
% (use named constants--good programming style!)
  target = 0;       % stopping value
  it = 0;           % current no. of iterations
  maxit = 1000;     % maximum no. of iterations
  eps = 0.001;      % tolerance

% Initial function value:
  f=x^4 - 8.6*x^3 - 35.51*x^2 + 464.4*x - 998.46;

% Iterate for root:

  while ( abs(f-target)>eps & it<maxit )

    % Compute f'(x), f"(x), u(x), u'(x)
      fprime= 4*x^3 - 25.8*x^2 - 71.02*x + 464.4;
      f2prime = 12*x^2 - 51.6*x - 71.02;
      u = f/fprime;
      uprime = 1 - f*f2prime/fprime^2;

    % Compute new value of x
      x = x - u/uprime;

    % Compute new function value f(x)
      f=x^4 -8.6*x^3 -35.51*x^2 +464.4*x -998.46;

    % increment count of iterations
      it = it + 1;
  end

% Output

  if it<maxit
    fprintf('Positive real root %.2f found', x)
    fprintf(' after %d iterations\n', it)
  else
    fprintf('Stopped after %d iterations\n', maxit)
    fprintf('f(%.2f) = %.3f\n', x, f)
    fprintf('Restart with new x if necessary\n') 
  end


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Question 3
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Check your project notes/solutions for the bisection algorithm.
In fact, review the entire project!


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Question 4
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% get data from file elevat.mat
load elevat

% distances between measurements
  dx=25; dy=20;

% no. of measurements in x- and y-direction
  [rn,cn]=size(elevat);

% need at least 3 measurements in each direction
  if rn>=3 & cn>=3
     for i=2:rn-1
        for j=2:cn-1
           grad_x(i-1,j-1)=(elevat(i,j+1)...
                            -elevat(i,j-1))/(2*dx);
           grad_y(i-1,j-1)=(elevat(i+1,j)...
                            -elevat(i-1,j))/(2*dy);
        end
     end
  end

% save matrices in files
save grad_x
save grad_y


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