# 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 $$r = \sin(8 t)$$. Make an interact to plot this curve.

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

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

1. The diagonal element of $$A$$ 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 $${\bf b}$$ of ones. Solve the system $$A {\bf x} = {\bf b}$$ and compute the residual.

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

1. What is $$a \star b$$?

2. What is the size of the group generated by $$a$$ and $$b$$?

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 $$(x+1) \sin(2 x)$$.

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

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

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

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