The routine may be called by the names f08cjf, nagf_lapackeig_dorgrq or its LAPACK name dorgrq.
f08cjf is intended to be used following a call to f08chf, which performs an factorization of a real matrix and represents the orthogonal matrix as a product of elementary reflectors of order .
This routine may be used to generate explicitly as a square matrix, or to form only its trailing rows.
Usually is determined from the factorization of a matrix with . The whole of may be computed by
(note that the matrix must have at least rows)
or its trailing rows by
The rows of returned by the last call form an orthonormal basis for the space spanned by the rows of ; thus f08chf followed by f08cjf can be used to orthogonalize the rows of .
The information returned by f08chf also yields the factorization of the trailing rows of , where . The orthogonal matrix arising from this factorization can be computed by
or its leading columns by
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: – IntegerInput
On entry: , the number of elementary reflectors whose product defines the matrix .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: details of the vectors which define the elementary reflectors, as returned by f08chf.
On exit: the matrix .
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08cjf is called.
6: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array tau
must be at least
On entry: must contain the scalar factor of the elementary reflector , as returned by f08chf.
7: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
8: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08cjf 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.
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 matrix differs from an exactly orthogonal matrix by a matrix such that
and is the machine precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08cjf 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 ; when this becomes .