# Linear Algebra (Sparse Matrix and Diff. Eq)

1. Oct 11, 2012

### dreamspace

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

2. Relevant equations

Not sure.

3. The attempt at a solution

Have no idea, as I don't have any/much previous experience with Linear Algebra.
Can anyone help me with starting on this, hints/tips?

2. Oct 11, 2012

### HallsofIvy

Staff Emeritus
Surely you can solve $d^2y/dx^2= 1- x$? Do you know what LU factorization, Gauss-Seidel, etc. are? What is the matrix with n= 4?

3. Oct 11, 2012

### dreamspace

Hi

Yes, I can solve the Differential Equation by hand, and I have some limited knowledge/experience with LU factorization, Gauss-Seidel etc. And matrices in general, but it kinda stops there. I have in general problems understanding how to use everything for the problem, and some of the info included

For example "Xi = i/n is the interior, discrete spatial coordinates on [0,1] with steplength h = 1/n" ? I have no idea what that means, and googling it doesn't come up with a lot either. Neither does searching for Use of sparse matrices and differential equations.

4. Oct 11, 2012

### Ray Vickson

Do you actually understand what the question is about? It is about finding an approximate numerical solution to a DE by using a discrete approximation. So, you split up the interval [0,1] into n subintervals [0,1/n], [1/n,2/n],..., [(n-1)/n,1], then approximate d^2 u(x)/dx^2 by an appropriate finite-difference, etc. If you Google the appropriate topic you will find lots of relevant information. I'll leave that to you.

Anyway, you don't even need to know that to do the question: all you are asked to do is to perform some well-defined linear algebra tasks on a linear system that is given explicitly to you. You don't even need to know where the system comes from.

RGV

5. Oct 11, 2012

### dreamspace

Thanks. Yes, now I understand the problem. But alas, I'm still not sure how the set it up. I'm understanding (correctly?) that the U vector will be the approximate solutions for the Diff. EQ

Let's take the case of N=4 , would the Matrix equation look like this:

$n^{2} \begin{pmatrix} -2 & 1 & 0 & 0 & \cdots & 0\\ 1 & -2 & 1 & 0 & \cdots & 0\\ 0 & 1 & -2 & 1 & 0 & 0\\ \vdots & \vdots & 1 & \ddots & \\ \\ 0 & 0 & & & & 1\\ 0 & 0 & & & 1 & -2\\ \end{pmatrix} \begin{pmatrix} u_{1}\\ u_{2}\\ u_{3}\\ \vdots\\ u_{n-1}\\ \end{pmatrix} = \begin{pmatrix} f(x_{1})\\ f(x_{2})\\ f(x_{3})\\ \vdots\\ f(x_{n-1})\\ \end{pmatrix} \\ \\ \\ 4^{2} \begin{pmatrix} -2 & 1 & 0\\ 1 & -2 & 1\\ 0 & 1 & -2 \end{pmatrix} \begin{pmatrix} u_{1}\\ u_{2}\\ u_{3} \end{pmatrix} = \begin{pmatrix} f(x_{1})\\ f(x_{2})\\ f(x_{3}) \end{pmatrix} \\ \\ \\ \begin{pmatrix} -32 & 16 & 0\\ 16 & -32 & 16\\ 0 & 16 & -32 \end{pmatrix} \begin{pmatrix} u_{1}\\ u_{2}\\ u_{3} \end{pmatrix} = \begin{pmatrix} 1-\frac{1}{4}\\ 1-\frac{2}{4}\\ 1-\frac{3}{4} \end{pmatrix} \\ \\ \\ \begin{pmatrix} -32 & 16 & 0\\ 16 & -32 & 16\\ 0 & 16 & -32 \end{pmatrix} \begin{pmatrix} u_{1}\\ u_{2}\\ u_{3} \end{pmatrix} = \begin{pmatrix} \frac{3}{4}\\ \frac{1}{2}\\ \frac{1}{4} \end{pmatrix}$

?

6. Oct 11, 2012

### Ray Vickson

So now you are expected to solve this problem by a number of different methods. The first, Gaussian elimination (= LU factorization!) is familiar from beginning high-school algebra. The others are supposed to be what you are learning in the course, I think, judging from the wording of the problem.

RGV