# Linear least squares, condition number

#### azay

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 $$min||A\lambda-y||_{2}$$, which turns out to be (1,1). I did this by doing a QR factorization using Givens rotations.

with:

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

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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#### fresh_42

Mentor
2018 Award
The condition numbers don't tell you that something is definitely not accurate. They tell you what at least can be expected. There are a couple of possible reasons: the input is sufficiently precise for the data memory and algorithmic steps used, the algorithms are identical, or just by chance.

#### marcusl

Gold Member
Matlab computes with internal numerical precision of approximately 1 part in 10^15 or 10^16, I forget which (unless you set it for greater precision). It therefore easily inverts a matrix of condition number 10^4. (In fact, it should work with condition numbers nearing 10^15.) Presumably Octave is similar.

"Linear least squares, condition number"

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