 
  
  
  
  
 
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
 consists of the first n columns of Q, and  the
remaining m-n columns.
 the
remaining m-n columns.
If m < n, R is trapezoidal, and the factorization can be written 

where  is upper triangular and
 is upper triangular and  is rectangular.
 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).
 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
 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
 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
 may be computed without forming r
explicitly, since

 
  
  
  
 