QR Factorization

The most common, and best known, of the factorizations is the QR factorization given by

$$A = Q\left( \begin{array}{c}R\\ 0\end{array}\right), \quad \mbox{if $m \ge n$,}$$

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

$$A = \left( \begin{array}{cc} Q_1 & Q_2\end{array} \right) \left( \begin{array}{c}R\\ 0\end{array}\right)$$

which reduces to

A = Q₁ R ,

where Q₁ consists of the first n columns of Q, and Q₂ the remaining m-n columns.

If m < n, R is trapezoidal, and the factorization can be written

$$A = Q\left( \begin{array}{cc}R_1 & R_2\end{array}\right), \quad \mbox{if $m < n$,}$$

where R₁ is upper triangular and R₂ 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 $m \geq n$ and A is of full rank, since

$$\Vert b - Ax\Vert _2 = \Vert Q^T b - Q^T A x\Vert _2 = \left... ...egin{array}{c} Q_1^T b \\ Q_2^T b \end{array} \right) = Q^T b;$$

c can be computed by xORMQR (or xUNMQR ), and c₁ consists of its first n elements. Then x is the solution of the upper triangular system

Rx = c₁

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

$$r = b - A x = Q \left( \begin{array}{c} 0 \\ c_2 \end{array} \right) ,$$

and may be computed using xORMQR (or xUNMQR ). The residual sum of squares |r|₂² may be computed without forming r explicitly, since

|r|₂ = |b - Ax|₂ = |c₂|₂.