The QL and RQ factorizations are given by
and
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) 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.
--------------------------------------------------------------------------- Type of factorization Single precision Double precision and matrix Operation real complex real complex --------------------------------------------------------------------------- QR, general factorize with pivoting SGEQPF CGEQPF DGEQPF ZGEQPF 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 --------------------------------------------------------------------------- RQ, trapezoidal factorize, no pivoting STZRQF CTZRQF DTZRQF ZTZRQF ---------------------------------------------------------------------------Table 2.9: Computational routines for orthogonal factorizations