# Matrix numerical analysis: Need explanations

1. Apr 30, 2009

### eddiechai2003

1. The problem statement, all variables and given/known data

Given that $$\Omega \subset \mathbb{R}^2$$ and $$(x,y) \in [0,1] \times [0,1]$$.

The goal is to find $$u:(x,y) \in \Omega \longmapsto u(x,y) \in \mathbb{R}$$.

2. Relevant equations

Details:

$$(P1) \left\{ \begin{array}{rlc} -\frac{\partial}{\partial x^2} u(x,y) - \frac{\partial}{\partial y^2} u(x,y) &=\quad f(x,y), & \forall (x,y) \in \Omega \\ u(x=0,y)&=\quad 0 \\ u(x,y=0)&=\quad 0 \\ u(x=1,y)&=\quad y^2 \\ u(x,y=1)&=\quad x^2 \end{array}\right$$

Where $$f$$ is given as:

$$f(x,y)=-2\sin(4\pi x)-2x^2-2y^2+(4\pi)^2y(y-1)\sin(4\pi x)$$

We are, supposedly, using finite difference method to approach numerically the solution of $$(P1)$$.

As advised by the author, one begins by discretizing $$\Omega$$ with $$N=n^2$$ points attributed regularly and numbered accordingly, like in the example below:

http://www.chokleong.com/ESIEE/images/exemple_de_discretisation.png [Broken]

Notice that the point $$1$$ has $$(\frac{1}{n+1},\frac{1}{n+1})$$ as its coordinate, etc...

He, then, searches one equation (can be solved numerically) satisfied by approximated values of $$u$$ at points $$1,2,3,$$ etc... He gathers these values in a vector with the size of $$N=n^2$$:

$$U=\begin{pmatrix} u_1\\ u_2\\ \vdots\\ u_N\end{pmatrix} \in \mathbb{R}^N$$

He introduces the $$N \times N$$ matrix below:

$$A_N= \begin{pmatrix} A & B & 0 & \cdots & 0 \\ B^T & \ddots & \ddots & \ddots & \vdots\\ 0 & \ddots & A & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & B\\ 0 & \cdots & 0 & B^T & A \end {pmatrix} \in \mathbb{R}^{N\times N}$$

Where $$A$$ and $$B$$ are both $$n \times n$$ matrices given as:

$$A= \begin{pmatrix} 4 & -1 & 0 & \cdots & 0\\ -1& \ddots & \ddots & \ddots & \vdots\\ 0 & \ddots & 4 & \ddots & 0\\ \vdots & \ddots & \ddots & \ddots & -1\\ 0 & \cdots & 0 & -1 & 4\end{pmatrix} \qquad,\quad B= \begin{pmatrix} -1 & 0 & \cdots & \cdots & 0\\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & -1 & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0\\ -1 & 0 & \cdots & 0 & -1\end{pmatrix}$$

Questions:
1) Explain why $$(P1)$$ can be approached by $$(P2)$$ below:

$$(P2)$$ Find $$U \in \mathbb{R}^N$$ such that $$\frac{1}{(n+1)^2}A_N U = b$$ where the form of the vector $$b \in \mathbb{R}^N$$ will be given by the student.
Hint: Begin with $$n$$ small $$( n = 2, 3,\quad etc...)$$.

2) By introducing $$X \in \mathbb{R}^N$$, a $$N$$ vector, explain why $$(P2)$$ is equivalent to the least squares problem below:

$$(P3)$$ Find $$X \in \mathbb{R}^N$$ minimum of $$J(X)=\frac{1}{2}X^TA_N X-(n+1)^2X^Tb$$.

That's it.

3. The attempt at a solution

There are a total of 5 questions to answer, but I am having problems with the above 2 questions which I don't have a clue of.

(Update: My biggest problem is that I don't know how to write the vectors $$U$$ and $$b$$ explicitly which prevent me from solving the problem. If someones know this matter better than me, I would be glad that you can shed some light for me on this matter.
My biggest concern is what represent the vector $$b \in \mathbb{R}^N$$ actually? What could it possibly be? Any idea?)