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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 :)

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# Linear least squares, condition number

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