Numerical Analysis Blog

This week's material

September 29, 2017 by Instructor

The material covered this week may be structured as follows:

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Lecture 1 (Curs 1)

1. Vector Norms (Norme de vectori in ${ℝ}^n$ ). We discuss about the most commonly used vector norms and their equivalence.
2. Matrix norms in general (norme de matrici) and operator (matrix) norms (norme de operator). We give the most important operator norms: ${\left|\right|A\left|\right|}_1,{\left|\right|A\left|\right|}_{\infty },{\left|\right|A\left|\right|}_2$ where $A\in {ℳ}_{m,n}\left(ℝ\right)$.
3. Conditioning numbers (numere de conditionare) and how these numbers affect the numerical solution of linear systems.
4. Direct methods for solving linear systems (Metode directe de rezolvare a sistemelor liniare).

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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.