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Let us suppose we have a, say, 10 dimensional real space with 3 orthogonal unit vectors given. How do I complete this orthonormal system with 7 additional vectors into a complete ONS in a way that is numerically stable?

An approach I could think of is to take a random vector, then perform a step in the Gram-Schmidt method to obtain a random unit vector that is orthogonal to all the previous ones. But I am concerned that choosing such a vector arbitrarily can introduce numerical errors which continue growing until I add the last vector to the system.

Is there a way I can choose the additional vectors in a way that I minimize numerical error?

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Don't use Gram-Schmidt; simply take a QR factorization of your matrix $U$. If your starting matrix was exactly orthogonal, then the top $3\times 3$ block $R_1$ of $R$ will be orthogonal as well (hence also diagonal, since it's upper triangolar -- if it's not the case and you know the initial data was correct, then you can fix the numerical errors by changing the small off-diagonal elements to zero). Then just multiply $Q$ by $\operatorname{diag}(R_1,I_7)$ to get an orthogonal matrix whose first rows are $U$.

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  • $\begingroup$ Thanks for pointing me to the right direction. I did not realize that I actually needed to compute a good QR decomposition of a rectangular matrix.(After I was reading about QR, I realize it is essentially same as Gram-Schmidt for square matrices, so I think "don't use G-S" is a bit misleading.) $\endgroup$
    – MrX
    Commented Dec 4, 2013 at 9:39
  • $\begingroup$ It's a chicken-and-egg issue. The way I see it, the QR factorization is the primitive concept, and G-S is an algorithm to compute it (not the best one, often). I guess you were taught G-S but not QR, so you find the former more natural, but arguably (at least for a numerical analyst) it's the other way round. Anyway, sorry if the first sentence was misleading for you! $\endgroup$
    – Federico Poloni
    Commented Dec 4, 2013 at 10:05

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