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    "# MA934 - Week 6 (unassessed!) Problem Sheet\n",
    "\n",
    "The problems below give you a chance to practice some of the concepts we learned about in week 1 and also provide a chance to hand in work that would receive feedback ahead of any assessed submission. This is an opportunity to double-check both if mathematical concepts are covered sufficiently thoroughly and whether implementation elements (commenting, plot creation) are likely to result in full marks.\n",
    "\n",
    "\n",
    "## Task 1 [10 marks]\n",
    "\n",
    "Create a notebook and implement code that reads in parameters $\\alpha$ and $n$ and produces a log plot of some samples of the function $$f(x) = x^\\alpha \\, \\log(x)$$\n",
    "at values of $x$ that increase in powers of 2 from 1 to $2^n$."
   ]
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    "## Task 2 [40 marks]\n",
    "\n",
    "Following a similar structure (including a function - or more!) to the Notebook presented in class for matrix-matrix multiplication, add to the respective code such that it can also:\n",
    " * execute the Strassen multiplication algorithm\n",
    " * plot the runtime on the same figure as the standard matrix-matrix multiplication results and compare the two\n",
    " \n",
    "Note: you may benefit from reducing the size of the problem while you do so (to avoid unnecessarily long execution while you debug)."
   ]
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    "## Task 3 [30 marks]\n",
    "\n",
    "Write a new module containing functions that compute the $n^{th}$ term, $a_n$, in the Fibonacci sequence:\n",
    " * Iteratively\n",
    " * Recursively\n",
    " * Using memoization\n",
    " \n",
    "The functions should work starting from any given values of $a_1$ and $a_2$.\n",
    "\n",
    "Measure the run-time of each of these functions over a range of values of $n$ and produce a plot to illustrate your results.\n",
    "\n",
    "The recursive function implementation gets slow very quickly - I could only do up to $n=48$ on my laptop. Write a recursion relation for the computational complexity of the recursive version of the task. Solve it to prove that the computational complexity grows exponentially with $n$."
   ]
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    "## Task 4 [20 marks]\n",
    "\n",
    "The computational complexity of the naive divide-and-conquer algorithm for matrix multiplication satisfies the functional equation $$F(n) = 8 F(n/2) + 4 (n/2)^2$$ with $F(1)=1$.\n",
    "\n",
    "The corresponding equation for Strassen multiplication is $$F(n) = 7 F(n/2) + 18 (n/2)^2$$ with $F(1)=1$.\n",
    "\n",
    "Solve these recurrence relations explicitly to prove that the computational complexity of the two algorithms are $O(n^3)$ and $O(n^{\\log_2(7)})$ respectively. \n",
    "\n",
    "It is helpful to adopt the change of variables $n=2^p$ with $a_p = F(2^p)$ to obtain linear (albeit inhomogeneous) recursion relations. \n",
    "\n",
    "Some helpful online notes about solving recursion relations can be found at:\n",
    "\n",
    "https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematics_recurrence_relation.htm"
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