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Question 1:  iteration, swap, function, logical array

Given a one-dimensional numeric array, create two arrays sorted in
ascending order:  one contains all the values at or below the mean;
the other contains the values above the mean.  Write two functions: 

  $selectsort$	accepts a numeric array as input, sorts it in ascending 
		order in-place (i.e., without creating a new array), 
		and returns the sorted array.  You must use the 
		Selection Sort method described below.

  $split$	accepts a numeric array as input, calls $selectsort$ to 
		sort the array, and returns two arrays:  one contains 
		all the values at or below the mean; the other contains 
		the values above the mean.  Loops are not allowed in 
		this function.

Selection Sort (in ascending order):
------------------------------------
- Start with a numeric array of length N and assume that it is unsorted.
- Select the smallest value and put it in the right place, now N-1 elements 
  remain unsorted.
- Select the second smallest value and put it in the right place, now N-2
  elements remain unsorted.
- Continue until all elements are sorted.

Matlab reminders:
-----------------
Function $mean(x)$ returns the mean of the values in (1-d) array $x$.
Function $min(x)$ for a 1-d array $x$ can return both the minimum
value and its position (first occurrence).  E.g.,

	>> [value,index]=min([4 3 5 2 2 5])
	value =
	     2
	index =
	     4
	>>

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Question 2:  Newton's method with a twist

The Newton-Ralston method is a modified Newton's method for dealing
with multiple roots.  The central idea is to apply the Newton's
method to a function of the function whose roots we wish
determine--the second derivative is needed in addition to the first
dervative.  Write a function $newtonRalston$ to find the positive
real roots of the function

  f(x) = x^4 - 8.6x^3 - 35.51x^2 + 464.4x - 998.46 

using the Newton-Ralston method, described below.

Newton-Ralston Method (for root finding):
-----------------------------------------
Suppose we wish to find the roots of a function f(x).  Define a new function

  u(x) = f(x)/f'(x)		Eq. 1

Function u(x) has the same roots as f(x) does, since u(x) becomes zero
everywhere that f(x) is zero.  Apply Newton's method to u(x) instead
of f(x): 

  delta = x - x(0) = -u(x)/u'(x)

Eq. 1 can be differentiated to obtain u'(x):

  u'(x) = 1 - (f(x)*f"(x))/(f'(x))^2

Algorithm for the Newton-Ralston method:
1.  Pick an initial value x
2.  Calculate f(x)
3.  While tolerance is not met, iterate as follows:
	- compute f'(x), f"(x), u(x), u'(x)
	- compute the new value of x, denoted xn
	- compute the new function value f(xn)
For this question, do not worry about the number of iterations.

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Question 3:  Bisection method

Write out the algorithm for the bisection method.  Try to do this
without looking at your project notes.

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Question 4:  2d-array, nested for-loops

Suppose someone has measured the elevations at a soccer field as shown in 
following picture.  At all the crosses a measurement of the elevation was 
taken.  The distances between the measurements in the x-direction 
(dx) and y-direction (dy) are 25m and 20m, respectively.  The surveyer knew 
about Matlab and stored the measurements in a 5-by-6 matrix that he saved 
in a Matlab file called elevat.mat.

   
     +------+------+------+------+------+
     |      |      |      |      |      |
     |      |      |      |      |      |
     +------+------+------+------+------+
     |      |      |      |      |      |
     |      |      |      |      |      |     Y
     +------+------+------+------+------+     ^
     |      |      |      |      |      |     |
     |      |      |      |      |      |     |
     +------+------+------+------+------+     -----> X
     |      |      |      |      |      |
     |      |      |      |      |      |
     +------+------+------+------+------+


You are asked to write a Matlab script file to determine the gradients 
in the x and y directions at all interior points (3*4=12 points), i.e., 
all points excluding the points on the boundary.  The gradients should be 
stored in two 3 by 4 matrices and written to files called grad_x.mat and 
grad_y.mat.

This is how you determine the gradient for an interior point where i
refers to the row and j to the column of matrix elevat, respectively:

     grad_x(i,j)=(elevat(i,j+1)-elevat(i,j-1))/(2*dx)   
     grad_y(i,j)=(elevat(i+1,j)-elevat(i-1,j))/(2*dy)

Which values of the matrix elevat did you not use for your
calculations?
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