** 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*