Lecture 46: Fifth Review
========================

For the last part of the course took,
we can divide the topics roughly as:

1. building interactive web pages;

2. the numpy, scipy, and sympy stack;

3. introduction to Julia

4. GAP, PARI/GP, Singular and R.

Below are some additional, preliminary questions.

1. Consider the curve defined by :math:`r = \sin(8 t)`.
   Make an interact to plot this curve.

   The range for :math:`t` always starts at zero.
   The end of the range for :math:`t` is controlled by a slider.
   The initial value for the end is :math:`\pi/2`.
   The increment for the end value is :math:`\pi/40`.

2. Use numpy to solve a 5-by-5 tridiagonal system
   :math:`A {\bf x} = {\bf b}`.

   1. The diagonal element of :math:`A` is 5, the elements just above 
      and below the diagonal are one.
      Everywhere else the matrix is zero.

   2. Define a 5-dimensional right hand side vector :math:`{\bf b}` of ones.
      Solve the system :math:`A {\bf x} = {\bf b}` and compute the residual.

3. Consider the permutations 
   :math:`a = (1, 4)(2, 3)` and :math:`b = (4, 5)(3, 6)`.

   1. What is :math:`a \star b`?
   
   2. What is the size of the group generated 
      by :math:`a` and :math:`b`?

4.  Use the Cauchy integral formula to compute the number of
    complex roots in a disk centered at 0 and with radius 1.1 
    of :math:`(x+1) \sin(2 x)`.

    Give the number of roots in that disk of the complex plane.

5. Consider the polynomial system defined by the polynomials
   :math:`p = x^2 y - 2 x + 3` and :math:`q = x y^2 - 2 y + 3`.

   1. Bring the system in triangular form.  
      Use this triangular form to determine the number of complex solutions.

   2. If :math:`N` is the number of solutions, compute the companion matrix 
      of the system.  The rows of this matrix are the reductions of
      the products of :math:`y` with :math:`y^k` 
      for :math:`k` ranging between 0 and :math:`N-1`.
      Show that the *y*-coordinates of the solutions are the eigenvalues 
      of this companion matrix.