The routine may be called by the names f08ahf, nagf_lapackeig_dgelqf or its LAPACK name dgelqf.
f08ahf forms the factorization of an arbitrary rectangular real matrix. No pivoting is performed.
If , the factorization is given by:
where is an lower 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 rows of , and the remaining rows.
If , is trapezoidal, and the factorization can be written
where is lower triangular and is rectangular.
The factorization of is essentially the same as the factorization of , since
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 in the first rows of the array a represents an factorization of the first rows of the original matrix .
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 elements above the diagonal are overwritten by details of the orthogonal matrix and the lower triangle is overwritten by the corresponding elements of the lower triangular matrix .
If , the strictly upper triangular part is overwritten by details of the orthogonal matrix and the remaining elements are overwritten by the corresponding elements of the lower trapezoidal matrix .
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08ahf is called.
5: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array tau
must be at least
On exit: further details of the orthogonal matrix .
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 f08ahf 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
Background information to multithreading can be found in the Multithreading documentation.
f08ahf 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 f08ahf may be followed by a call to f08ajf