Linear Algebra (Sparse Matrix and Diff. Eq)

[dreamspace]

Homework Statement

Homework Equations

Not sure.

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?

 

[HallsofIvy]

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

 

[dreamspace]

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

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.

 

[Ray Vickson]

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.

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

 

[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:

[itex]

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}

[/itex]

?

 

[Ray Vickson]

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:

[itex]

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}

[/itex]

?

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