The most
common, and best known, of the factorizations
is the QR factorization
given by
where R is an n-by-n upper triangular matrix and Q is an m-by-m
orthogonal (or unitary) matrix. If A is of full rank n, then R is
nonsingular.
It is sometimes convenient to write the factorization as
which reduces to
where 
consists of the first n columns of Q, and 
the
remaining m-n columns.
If m < n, R is trapezoidal, and the factorization can be written
where 
is upper triangular and 
is rectangular.
The routine PxGEQRF
computes the QR factorization . The matrix Q is not
formed explicitly, but is represented as a product of elementary reflectors,
as described in section 3.4.
Users need not be aware of the details of this representation,
because associated routines are provided to work with Q:
PxORGQR (or PxUNGQR
in the complex case) can generate all or part of Q,
while PxORMQR (or PxUNMQR) can pre- or post-multiply
a given matrix by Q or 
(
if complex).
The QR factorization can be used to solve the linear least squares
problem (3.1) when 
and
A is of full rank, since
c can be computed by PxORMQR (or PxUNMQR ), and 
consists of its first
n elements. Then
x is the solution of the upper triangular system
which can be computed by PxTRTRS .
The residual vector r is given by
and may be computed using PxORMQR (or PxUNMQR ).
The residual sum of squares 
may be computed without forming r
explicitly, since
