Lecture 16: Review of the First 15 Lectures
===========================================

In the first 15 lectures we explored the extensive number system of Sage,
manipulated polynomials and general expressions.

Below is a first, preliminary list of questions to review.
Consider also the quizzes and homework assignments.

1. Explain the difference between ``1.0 + 10**(-32)`` and ``1 + 10**(-32)``.

2. The Gelfond-Schneider constant is \ :math:`2^{\sqrt{2}}`.
   Give all Sage commands

   1. to compute a rational approximation for \ :math:`2^{\sqrt{2}}`
      accurate with 5 decimal places;

   2. to show that the accuracy of this approximation 
      is indeed 5 decimal places;

   3. to compute a sequence of 10 consecutive rational approximations,
      starting with 5 decimal places and increasing in accuracy
      with one decimal place in each step of the sequence.

3. Explain the difference between ``x = polygen(QQ)`` and ``var('x')``.

4. Consider the polynomial \ :math:`p = 97 x^{45} - 62 x^{46} - 73 x^{31}`.
   What is the best way to evaluate \ :math:`p` 
   at hardware floating point numbers?
   Give the relevant Sage commands and compare between the straightforward
   evaluation of the expression for \ :math:`p`.

5. Define a function ``whatis`` which stores definitions.
   Below is a session with this function:

   1. ``whatis('computer algebra')`` returns

      ``call again with the definition for computer algebra``
         
         
   2. ``whatis('computer algebra', 'the study of algorithms in symbolic computation')`` 

      stores the definition for ``'computer algebra'``
            
            
   3. ``whatis('computer algebra')`` returns

      ``the study of algorithms in symbolic computation``

6. Consider the polynomial \ :math:`p = 25 x^3 + 12 x^2 + 26 x + 4`
   over a finite field of size 29, similar to working modulo 29.

   1. Does the polynomial factor over this field?

   2. Give the Sage commands to add sufficiently many formal roots
      to the field so \ :math:`p` has three roots over the extended 
      number field.

   3. Write the symbolic factorization of \ :math:`p` in linear factors below.

7. Write a Python function ``python_sum`` which takes on input 
   a positive integer \ :math:`n` and which returns the floating-point value of

   .. math::

      \frac{\pi}{n} \sum_{i=1}^{n-1}
      \cos\left( - \frac{\pi}{2} + i \frac{\pi}{n} \right).

   Write a more efficient version ``numpy_sum`` using vectorization.

   Time the two versions to illustrate the efficiency.

8. Explain how rational expressions are normalized.
   Illustrate with an example.

9. Type ``var('x,y'); q = (x^2 - y)/(y^2 - x)``
   and draw the expression three of ``q``.

10. Give the Sage commands to
    transform \ :math:`( x + (z^2 + 1))(y - (z^2+1))`
    into \ :math:`x y + (-z^2 - 1) x + (z^2 + 1) y - z^4 - 2 z^2 - 1`.