Other Factorizations

The QL and RQ factorizations are given by

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

and

$$A = \left( \begin{array}{cc} 0 & R \end{array} \right) Q, \quad \mbox{if $m \leq n$.}$$

These factorizations are computed by xGEQLF and xGERQF, respectively; they are less commonly used than either the QR or LQ factorizations described above, but have applications in, for example, the computation of generalized QR factorizations [2].

All the factorization routines discussed here (except xTZRQF and xTZRZF) allow arbitrary m and n, so that in some cases the matrices R or L are trapezoidal rather than triangular. A routine that performs pivoting is provided only for the QR factorization.

Table 2.9: Computational routines for orthogonal factorizations

Type of factorization	Operation	Single precision		Double precision
and matrix		real	complex	real	complex
QR, general	factorize with pivoting	SGEQP3	CGEQP3	DGEQP3	ZGEQP3
	factorize, no pivoting	SGEQRF	CGEQRF	DGEQRF	ZGEQRF
	generate Q	SORGQR	CUNGQR	DORGQR	ZUNGQR
	multiply matrix by Q	SORMQR	CUNMQR	DORMQR	ZUNMQR
LQ, general	factorize, no pivoting	SGELQF	CGELQF	DGELQF	ZGELQF
	generate Q	SORGLQ	CUNGLQ	DORGLQ	ZUNGLQ
	multiply matrix by Q	SORMLQ	CUNMLQ	DORMLQ	ZUNMLQ
QL, general	factorize, no pivoting	SGEQLF	CGEQLF	DGEQLF	ZGEQLF
	generate Q	SORGQL	CUNGQL	DORGQL	ZUNGQL
	multiply matrix by Q	SORMQL	CUNMQL	DORMQL	ZUNMQL
RQ, general	factorize, no pivoting	SGERQF	CGERQF	DGERQF	ZGERQF
	generate Q	SORGRQ	CUNGRQ	DORGRQ	ZUNGRQ
	multiply matrix by Q	SORMRQ	CUNMRQ	DORMRQ	ZUNMRQ
RZ, trapezoidal	factorize, no pivoting	STZRZF	CTZRZF	DTZRZF	ZTZRZF
	(blocked algorithm)
	multiply matrix by Q	SORMRZ	CUNMRZ	DORMRZ	ZUNMRZ