Numerical Analysis Blog

Week 2

October 7, 2017 by Instructor

The material covered this week may be structured as follows:

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Lecture 2 (Curs 2)

1. Iterative methods for linear systems (Metode iterative pentru rezolvarea sistemelor liniare):
   1. Jacobi's Method,
   2. Gauss-Seidel Method,
   3. $SOR\left(\omega \right)$ .
2. Convergence of iterative methods.

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Laboratory, Weeks: 1-2 (Laborator, sapt. 1-2).

1. A quick overview of MatLab/Octave.
2. Floating point representation of real numbers, round off errors (reprezentarea numerelor in virgula mobila, erori de rotunjire). Some examples of disasters caused by numerical errors can be found here.

   Some exercises will involve the concept of epsilon_machine . In MatLab/Octave the value for epsilon_machine is given by the constant eps. Some sample MatLab code related to this is given below:

   >> 1+eps/4+eps/4+eps/4+eps/4 == 1+(eps/4+eps/4+eps/4+eps/4)
                           

3. Vector norms and matrix norms (both operator and non-operator norms). Here we discuss the the MatLab/Mathematica commands for computing these norms as well as solve some problems without the direct use of computer.

   Sample Problems

   1. Desenati bilele unitate din ${ℝ}^2$ pentru ${\left|\right|\cdot \left|\right|}_2$ , ${\left|\right|\cdot \left|\right|}_{\infty }$ si ${\left|\right|\cdot \left|\right|}_1$ .
   2. Fie $A\in {ℳ}_{n,n}\left(ℝ\right)$ , astfel incat: $\begin{array}{cc}& A=\left(\begin{array}{ccccccc}\hfill 4\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill ...\hfill & \hfill 0\hfill \\ \hfill ...\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill ...\hfill \\ \hfill 0\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill ...\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 4\hfill \end{array}\right)\hfill \end{array}$ Sa se arate ca $r_{\sigma }\left(A\right)\le 6$ si $r_{\sigma }\left(A^{-1}\right)\le 1/2$ , unde $r_{\sigma }\left(A\right)$ este raza spectrala.

4. Direct methods for linear systems.