Re: 'economy size' qr ?

[Johan Kullstam]

"Pascal A. Dupuis" <address@hidden> writes:

> Hello,
> 
> I would like to solve a overdetermined set of equation Ax=B, where
> A is C m x n
> x is R n x 1
> B is C m x 1
> 
> In order for x to be real, the problem is augmented as:
> 
> [ A; conj(A) ] x = [b; conj(b)]
>
> The first step is to compute
> [qq, qr] =qr([A , b; conj(A) conj(b)]);

Why not make

AA = [real(A);
      imag(A)]
BB = [real(B);
      imag(B)]

and solve the least squares for
AA being 2*m by n
BB being 2*m by 1
and the same x as above?

The complex nature of A and B but having x be real just makes it a
twice as large problem in the reals, but hidden by the notation.

> qr is a n+1 x n+1 upper real triangular matrix
> 
> then the resolution is performed using various approaches on qr.
> 
> Given that
> 1) qq is not needed (2m x 2m complex matrix !)

You would have a 2*m by 2*m Q matrix with the real formulation but it
would at least be real valued.  There is an economy QR where Q is
truncated to the span dimensions of A and R is square -- just add a
2nd argument of 0 like SVD takes.  Unfortunately, the "help qr"
doesn't mention this (perhaps this is a documentation bug?).

   [qq, rr, pp] = qr(AA,0)

now qq is the same dimensions as AA which is savings.

> 2) although A and b are complex, qr is real
> 
> are there simpler approaches in order to obtain qr, skipping the qq
> computation ?

You have 2*m real constraints and n real unknowns.  Putting it in a
complex format doesn't change that and, in fact, seems to be a
hindrance in matrix size economy.

> Thanks in advance

Hope this helps.

-- 
Johan KULLSTAM



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