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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 non-singular. It is sometimes convenient to write the factorization as 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 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 QT (QH 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 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 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.     Next: LQ Factorization Up: Orthogonal Factorizations and Linear Previous: Orthogonal Factorizations and Linear   Contents   Index
Susan Blackford
1999-10-01