<var>QR</var> Factorization

next up previous contents index
Next: Factorization Up: Orthogonal Factorizations and Previous: Orthogonal Factorizations and

QR Factorization

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

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 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 R, while xORMQR   (or xUNMQR  ) 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 (2.1)  when m > = n and A is of full rank, since

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

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 may be computed without forming r explicitly, since

Tue Nov 29 14:03:33 EST 1994