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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):
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- 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:
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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):
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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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