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### 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 nonsingular. 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 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).

The QR factorization can be used to solve the linear least squares problem (3.1)   when 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

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

Next: LQ Factorization Up: Orthogonal Factorizations and Linear Previous: Orthogonal Factorizations and Linear

Susan Blackford
Tue May 13 09:21:01 EDT 1997