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Vandermonde matrix

  1. Jul 9, 2009 #1
    In polynomial interpolation:

    I see some connection between:

    The Vandermonde matrix, the monomial basis and the fact that 'the monomial basis is not a good basis because it's components are not very orthogonal'.

    Now, I still don't really grasp sufficiently the reason why exactly a Vandermonde matrix is often ill-conditioned. Also, I don't feel I understand why an orthogonal basis in general leads to better conditioned problems, how self-evident it may look from a certain point of view.

    Any insights?
  2. jcsd
  3. Jul 9, 2009 #2
    Here is a rapid insight,
    Obtain, the vector below, at each cases
    [itex]\left[\begin{array}{cc}1 &0\\0 &1\end{array}\right]\left(\begin{array}{c}x\\y\end{array}\right)= \left(\begin{array}{c}3\\5\end{array}\right)[/itex]

    [itex]\left[\begin{array}{cc}1 &-1\\1 &1\end{array}\right]\left(\begin{array}{c}x\\y\end{array}\right)= \left(\begin{array}{c}3\\5\end{array}\right)[/itex]

    Please repeat it for the vector [tex]\left(\begin{array}{c}3.1\\5\end{array}\right)[/tex]

    You can relate orthogonality to being able to isolate arbitrary effects to one subset of orthogonal elements. Usually, complicated things are just for simplicity...
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