The routine may be called by the names f08abf, nagf_lapackeig_dgeqrt or its LAPACK name dgeqrt.
f08abf forms the factorization of an arbitrary rectangular real matrix. No pivoting is performed.
It differs from f08aef 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 factorization based on the algorithm of Elmroth and Gustavson (2000).
If , the factorization is given by:
where is an upper triangular matrix and is an orthogonal matrix. It is sometimes more convenient to write the factorization as
which reduces to
where consists of the first columns of , and the remaining columns.
If , is upper trapezoidal, and the factorization can be written
where is upper triangular and is rectangular.
The matrix is not formed explicitly but is represented as a product of elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with in this representation (see Section 9).
Note also that for any , the information returned represents a factorization of the first columns of the original matrix .
Elmroth E and Gustavson F (2000) Applying Recursion to Serial and Parallel 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
1: – IntegerInput
On entry: , the number of rows of the matrix .
2: – IntegerInput
On entry: , the number of columns of the matrix .
3: – IntegerInput
On entry: the explicitly chosen block size to be used in computing the factorization. See Section 9 for details.
if , .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the matrix .
On exit: if , the elements below the diagonal are overwritten by details of the orthogonal matrix and the upper triangle is overwritten by the corresponding elements of the upper triangular matrix .
If , the strictly lower triangular part is overwritten by details of the orthogonal matrix and the remaining elements are overwritten by the corresponding elements of the upper trapezoidal matrix .
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08abf is called.
6: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array t
must be at least
On exit: further details of the orthogonal matrix . The number of blocks is , where and each block is of order nb except for the last block, which is of order . For each of the blocks, an upper triangular block reflector factor is computed: . These are stored in the matrix as .
7: – IntegerInput
On entry: the first dimension of the array t as declared in the (sub)program from which f08abf is called.
8: – Real (Kind=nag_wp) arrayWorkspace
9: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The computed factorization is the exact factorization of a nearby matrix , where
and is the machine precision.
8Parallelism and Performance
f08abf 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.
The total number of floating-point operations is approximately if or if .
To apply to an arbitrary real rectangular matrix , f08abf may be followed by a call to f08acf
. For example,
To form the orthogonal matrix explicitly, simply initialize the matrix to the identity matrix and form using f08acf.
The block size, nb, used by f08abf 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 is likely to achieve good efficiency and it is unlikely that an optimal value would exceed .
To compute a factorization with column pivoting, use f08bbforf08bff.