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Linear least squares, condition number

  1. Aug 20, 2008 #1
    Hi,

    I am trying to learn some numerical algebra. Now I don't understand the following.

    I'm finding the solution to the Linear Least Squares problem [tex]min||A\lambda-y||_{2}[/tex], which turns out to be (1,1). I did this by doing a QR factorization using Givens rotations.

    with:

    [itex]
    A=
    \[ \left( \begin{array}{ccc}
    1 & 1\\
    1 & 1.0001\\
    1 & 1.0001\end{array} \right)\][/itex]
    and
    [itex]
    y=
    \[ \left( \begin{array}{ccc}
    2\\
    0.0001\\
    4.0001\end{array} \right)\]
    [/itex]

    Now, I have a Octave (matlab clone) program that does the same calculation. As the condition number of the matrix A is very large (4.2429e+004) (found by applying Octave's cond() function on A), I expect the solution to be at least not exact. Yet the Octave program gives the exact solution (1,1), at least, as far as I can see (6 digit accuracy I think), that is. Can someone explain this?

    Also, should one consider the condition number of the matrix A when considering the condition of the Linear Least Squares problem, or the condition number of the Matrix A|y?
    (The condition number of the latter is even bigger so my first question holds in any case).

    Thank you :)
     
  2. jcsd
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