The routine may be called by the names f08apf, nagf_lapackeig_zgeqrt or its LAPACK name zgeqrt.
3Description
f08apf forms the $QR$ factorization of an arbitrary rectangular complex $m\times n$ matrix. No pivoting is performed.
It differs from f08asf in that it: requires an explicit block size; stores reflector factors that are upper triangular matrices of the chosen block size (rather than scalars); and recursively computes the $QR$ factorization based on the algorithm of Elmroth and Gustavson (2000).
where $R$ is an $n\times n$ upper triangular matrix (with real diagonal elements) and $Q$ is an $m\times m$ unitary matrix. It is sometimes more convenient to write the factorization as
where ${R}_{1}$ is upper triangular and ${R}_{2}$ is rectangular.
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 $k<n$, the information returned represents a $QR$ factorization of the first $k$ columns of the original matrix $A$.
4References
Elmroth E and Gustavson F (2000) Applying Recursion to Serial and Parallel $QR$ Factorization Leads to Better Performance IBM Journal of Research and Development. (Volume 44)4 605–624
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore
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 $A$.
Constraint:
${\mathbf{n}}\ge 0$.
3: $\mathbf{nb}$ – IntegerInput
On entry: the explicitly chosen block size to be used in computing the $QR$ factorization. See Section 9 for details.
Constraint:
if $\mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{m}},{\mathbf{n}})>0$, $1\le {\mathbf{nb}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{m}},{\mathbf{n}})$.
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 $m\ge n$, the elements below the diagonal are overwritten by details of the unitary matrix $Q$ and the upper triangle is overwritten by the corresponding elements of the $n\times n$ upper triangular matrix $R$.
If $m<n$, the strictly lower triangular part is overwritten by details of the unitary matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m\times n$ upper trapezoidal matrix $R$.
The diagonal elements of $R$ are real.
5: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08apf is called.
Note: the second dimension of the array t
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 $Q$. The number of blocks is $b=\lceil \frac{k}{{\mathbf{nb}}}\rceil $, where $k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(m,n)$ and each block is of order nb except for the last block, which is of order $k-(b-1)\times {\mathbf{nb}}$. For each of the blocks, an upper triangular block reflector factor is computed: ${\mathit{T}}_{1},{\mathit{T}}_{2},\dots ,{\mathit{T}}_{b}$. These are stored in the ${\mathbf{nb}}\times n$ matrix $T$ as $\mathit{T}=\left[{\mathit{T}}_{1}\right|{\mathit{T}}_{2}|\cdots |{\mathit{T}}_{b}]$.
7: $\mathbf{ldt}$ – IntegerInput
On entry: the first dimension of the array t as declared in the (sub)program from which f08apf is called.
f08apf 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 $\frac{8}{3}{n}^{2}(3m-n)$ if $m\ge n$ or $\frac{8}{3}{m}^{2}(3n-m)$ if $m<n$.
To apply $Q$ to an arbitrary $m\times p$ complex rectangular matrix $C$, f08apf may be followed by a call to f08aqf
. For example,
To form the unitary matrix $Q$ explicitly, simply initialize the $m\times m$ matrix $C$ to the identity matrix and form $C=QC$ using f08aqf as above.
The block size, nb, used by f08apf is supplied explicitly through the interface. For moderate and large sizes of matrix, the block size can have a marked effect on the efficiency of the algorithm with the optimal value being dependent on problem size and platform. A value of ${\mathbf{nb}}=64\ll \mathrm{min}\phantom{\rule{0.125em}{0ex}}(m,n)$ is likely to achieve good efficiency and it is unlikely that an optimal value would exceed $340$.
To compute a $QR$ factorization with column pivoting, use f08bpforf08btf.