Lecture 46: Fifth ReviewΒΆ

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.