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Other Factorizations

The QL and RQ factorizations are given by

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

and

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

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


next up previous contents index
Next: Generalized Orthogonal Factorizations and Up: Orthogonal Factorizations and Linear Previous: Complete Orthogonal Factorization   Contents   Index
Susan Blackford
1999-10-01