The routine may be called by the names f08cef, nagf_lapackeig_dgeqlf or its LAPACK name dgeqlf.
f08cef forms the factorization of an arbitrary rectangular real matrix.
If , the factorization is given by:
where is an lower triangular matrix and is an orthogonal matrix. If the factorization is given by
where is an lower trapezoidal matrix and is again an orthogonal matrix. In the case where the factorization can be expressed as
where consists of the first columns of , and the remaining columns.
The matrix is not formed explicitly but is represented as a product of elementary reflectors (see Section 3.3.6 in 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 in the last columns of the array a represents a factorization of the last columns of the original matrix .
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd 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: – 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 lower triangle of the subarray contains the lower triangular matrix .
If , the elements on and below the th superdiagonal contain the lower trapezoidal matrix . The remaining elements, with the array tau, represent the orthogonal matrix as a product of elementary reflectors (see Section 3.3.6 in the F08 Chapter Introduction).
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08cef is called.
5: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array tau
must be at least
On exit: the scalar factors of the elementary reflectors (see Section 9).
6: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
7: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08cef is called.
If , 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.
for optimal performance, , where is the optimal block size.
8: – 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
f08cef 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 form the orthogonal matrix f08cef may be followed by a call to f08cff