where $L$ is an $m\times m$ lower triangular matrix (with real diagonal elements) and $Q$ is an $n\times n$ unitary matrix. It is sometimes more convenient to write the factorization as
The matrix $Q$ is not formed explicitly but is represented as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}(m,n)$ elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with $Q$ in this representation (see Section 9).
Note also that for any $k<m$, the information returned in the first $k$ rows of the array a represents an $LQ$ factorization of the first $k$ rows of the original matrix $A$.
4References
None.
5Arguments
1: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint:
${\mathbf{m}}\ge 0$.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Note: the second dimension of the array a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: the $m\times n$ matrix $A$.
On exit: if $m\le n$, the elements above the diagonal are overwritten by details of the unitary matrix $Q$ and the lower triangle is overwritten by the corresponding elements of the $m\times m$ lower triangular matrix $L$.
If $m>n$, the strictly upper triangular part is overwritten by details of the unitary matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m\times n$ lower trapezoidal matrix $L$.
The diagonal elements of $L$ are real.
4: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08avf is called.
Note: the dimension of the array tau
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{m}},{\mathbf{n}}))$.
On exit: further details of the unitary matrix $Q$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
7: $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08avf is called.
If ${\mathbf{lwork}}=\mathrm{-1}$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value:
for optimal performance, ${\mathbf{lwork}}\ge {\mathbf{m}}\times \mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint:
${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{m}})$ or ${\mathbf{lwork}}=\mathrm{-1}$.
8: $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
7Accuracy
The computed factorization is the exact factorization of a nearby matrix $(A+E)$, where
f08avf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The total number of real floating-point operations is approximately $\frac{8}{3}{m}^{2}(3n-m)$ if $m\le n$ or $\frac{8}{3}{n}^{2}(3m-n)$ if $m>n$.
To form the unitary matrix $Q$f08avf may be followed by a call to f08awf
: