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Warm-up

  • Code an arbitrary precision (integer) addition code. The start of the code should be where the "numbers" are defined. For example: 
    N1=[1 9 5 4 3 8 4 8 5 6 0 3 4 0 5 3 2 4 5 6 ]; % to mean  19543848560340532456
N2=[2 3 4 3 2 3 4 5 4 8 6 4 7 8 9 7 6 5 3 2 1 4 6 7 9 ]; % stands for
    2343234548647897653214679

    These numbers are to big to be represented exactly in MATLAB®. Your task is to write code that follows the rules of addition (adding from smallest to largest) and gets the precise answer, in the same format (that is an array of integers). There are various limitations that need to be discussed:

    • First try doing this while allowing yourself only to add small 20 numbers using MATLAB +. You will still need to be crafty about how you find the "ones'' and "tens'' of a given number. Since MATLAB is good with lists, think how you can use a list to do this. You may not use high level functions like mod, rem etc. (But you are encouraged to familiarize yourself with them.)
    • Can you do it with a matrix that represents the addition table for numbers between 0 and 11? Once you create this matrix, you can solve this exercise with +1 as the only "allowed'' arithmetic operation (because MATLAB's index of lists starts from 1 and not 0).
    • Redo this exercise for a different definition of addition, for example, for Z16.
  • Modify the Newton code you wrote for the basin of attraction to look at the basins for another function. How about \(f(x,y)=(x^4-y, y^4-x)^T\)? You will have to look for zeros (manually or otherwise) and then change your code and recalculate the Jacobian matrix.
  • What is \(\sqrt{-1}\)?
  • What is \((1+i)^4\)?
  • Can you find the roots of \(x^2+2x+4\)?
  • Where are the roots of \(x^5+1\)?
  • To operate NOT in a linear algebra way on vectors, you need to put a point (.) in front of the operator: thus 1./[1 2 4] is [1 .5 .25]. Similarly with .*. Calculate \(\pi\) by this really (really) bad way: For a large \(N\) calculate \(\sum_{n=1}^N\frac{1}{n^2}\). This converges to \(\pi^2/6\) as \(N\rightarrow\infty\). Calculate \(\pi\) using this and find out how large \(N\) must be for the resulting value of \(\pi\) to be within 1e-6 of the MATLAB value pi.