Projection of a vector into subspace defined by non-orthogonal basis

[dmitrey]

Hi all,

suppose I run an algorithm that returns sequentially normalized (but
not necessarily orthogonal) vectors a1, a2, a3, ..., ak, ... with
(same) typical size n = 1'000-100'000.
In each i-th iteration, for all i>m, I would like to find a projection
(or, at least, its length) of a_i to a subspace defined by a_i-1,
a_i2, ..., a_i-m (m is constant, typical value is 5..15);

So, what is the best (fastest, computationally) way to perform this?

First of all I would like to get solution for all-dense vectors, but
some may be (very) sparse, so it would be nice to get benefits of this
if possible.

Thank you in advance, D.

[Peter Spellucci]

In article <f8f52b0e-2fc8-4d6c...@k17g2000yqh.googlegroups.com>,

remove column/add column for QR-decomposition, use QR for least squares solving.
with this relation of number of rows (your n0 and the number of columns (m)
this should be best.
http://www.netlib.org/toms/686
hth
peter