User Tools

Site Tools


Math 753/853 Exam 1 topics

Floating point numbers

How do the number of bits dedicated to mantissa and exponents of floating point numbers determine

  • machine epsilon,
  • the number of digits of precision,
  • the smallest and largest numbers representable, and
  • the size of rounding error for a given number?

Solving nonlinear equations

We studied two methods for finding roots of the 1d equation $f(x) = 0$.

Bisection algorithm: understand

  • how it works
  • what it requires, in terms of the function $f(x)$ and the starting conditions for the iteration
  • the linear convergence rate

Newton method: understand

  • how it works, mathematically and geometrically
  • what it requires, in terms of the function $f(x)$ and the starting conditions for the iteration
  • the quadratic convergence rate and the assumptions necessary to get quadratic converegence

You should be able to write pseudo code or Julia code for either method. You're not expected to get the Julia syntax perfectly.

Solving systems of equations

We've studied two methods for solving systems of equations $Ax=b$.

LU decomposition: understand

  • how LU decomposition works
  • the formulae for the multipliers $\ell_{ij}$, why the formulae are what they are
  • the backsubstition and forward substitution algorithms (formulae and where they came from)
  • how you piece together LU decomp, forward substitution, and backward substitution to solve $Ax=b$

QR decomposition

  • how the Gram-Schimidt algorithm computes a QR factorization
  • the formulae for $q_j$, $r_{ij}$, and $r_{jj}$, and where the formulae came from
  • how to use a QR decomp to solve an $m \times m$ (square) $Ax=b$ problem
  • how to use a QR decomp to solve an $m \times n$ (oblong) least-squares $Ax=b$ problem

In addition we covered some basic linear algebra theory, including

  • transposes
  • inner product
  • the 2-norm, for both vectors and matrices
  • orthogonal matrices
  • how the inner product and 2-norm of both vectors and matrices are preserved under orthogonal transformation

About formulae: Rather than memorizing them, focus on understanding their meaning and being able to derive them or the first few instances of a general formula. You should be able to write pseudocode or Julia code (again, not necessarily perfect syntax) to implement a given formula.

gibson/teaching/fall-2016/math753/exam1topics.txt · Last modified: 2016/10/10 09:27 by gibson