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Linear Algebra (Sparse Matrix and Diff. Eq)

  1. Oct 11, 2012 #1
    1. The problem statement, all variables and given/known data

    25uphdd.jpg

    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. jcsd
  3. Oct 11, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    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?
     
  4. Oct 11, 2012 #3
    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.
     
  5. Oct 11, 2012 #4

    Ray Vickson

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    Homework Helper

    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
     
  6. Oct 11, 2012 #5
    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]

    ?
     
  7. Oct 11, 2012 #6

    Ray Vickson

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    Homework Helper

    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
     
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