** Next:** LQ Factorization
** Up:** Orthogonal Factorizations
and Linear ** Previous:** Orthogonal Factorizations and Linear **
Contents** **
Index**

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 non-singular. It is sometimes convenient to write
the factorization as

which reduces to

** ***A* = *Q*_{1} *R* ,

where *Q*_{1} consists of the first *n*
columns of *Q*, and *Q*_{2} the remaining
*m*-*n* columns.
If *m* < *n*, *R* is trapezoidal, and
the factorization can be written

where *R*_{1} is upper triangular and *R*_{2}
is rectangular.
The routine xGEQRF computes the *QR* factorization.
The matrix *Q* is not formed explicitly, but is represented as
a product of elementary reflectors,
as described in section 5.4. Users
need not be aware of the details of this representation, because associated
routines are provided to work with *Q*: xORGQR (or xUNGQR in the complex case) can generate all or part of *Q*,
while xORMQR (or xUNMQR) can pre- or post-multiply a given matrix
by *Q* or *Q*^{T} (*Q*^{H}
if complex).

The *QR* factorization can be used to solve the linear least
squares problem (2.1)
when
and *A* is of full rank, since

*c* can be computed by xORMQR (or xUNMQR ), and
*c*_{1} consists of its first *n* elements.
Then *x* is the solution of the upper triangular system

** ***Rx* = *c*_{1}

which can be computed by xTRTRS. The residual vector *r*
is given by

and may be computed using xORMQR (or
xUNMQR ). The residual sum of squares
**|r|**_{2}^{2} may be computed without forming *r*
explicitly, since

** |r|**_{2} = |b - *Ax*|_{2} = |c_{2}|_{2}.

** Next:** LQ Factorization
** Up:** Orthogonal Factorizations
and Linear ** Previous:** Orthogonal Factorizations and Linear **
Contents** **
Index**
*Susan Blackford*

*1999-10-01*