Re: [ublas] lapack geqrf matrix Q

From: Dima Sorkin (dsorkin_at_[hidden])
Date: 2005-12-22 03:13:44

Quoting Eddie Vedder :
> The following is from the documentation of geqrf:
> 
> The matrix Q is represented as a product of elementary reflectors
>   Q = H(1) H(2) . . . H(k), where k = min(m,n).
> Each H(i) has the form
>   H(i) = I - tau * v * v'
> where tau is a real scalar, and v is a real vector with
> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
> and tau in TAU(i).
> 
> Does anybody known:
> Is H a matrix, vector or scalar?
H is a matrix, but it is not stored explicitly.

> Does v*v' result in a scalar or a matrix?
It results in a matrix,but it is usually not computed explicitly.

> What does the : in v(1:i-1) mean?
It is Matlab notation. it means subarray of "v", starting
from index 1 upto index i-1 (including "i-1").

> How is the vector v constructed?
> Is there a good paper about elementary reflectors?
It is Householder reflectors. At any book about numerical methods
of linear algebra. Probably it will be in
"G. Golub and C. Van Loan, Matrix Computations"
There are also lots of material in the internet.

> Did anyone else ever tried to extract matrix Q?
I didn't.

Regards,
 Dima.