The function may be called by the names: f08cfc, nag_lapackeig_dorgql or nag_dorgql.
3Description
f08cfc is intended to be used after a call to f08cec, which performs a factorization of a real matrix . The orthogonal matrix is represented as a product of elementary reflectors.
This function may be used to generate explicitly as a square matrix, or to form only its trailing columns.
Usually is determined from the factorization of an matrix with . The whole of may be computed by
:
nag_lapackeig_dorgql(order,m,m,p,a,pda,tau,&fail)
(note that the array a must have at least columns)
or its trailing columns by
:
nag_lapackeig_dorgql(order,m,p,p,a,pda,tau,&fail)
The columns of returned by the last call form an orthonormal basis for the space spanned by the columns of ; thus f08cec followed by f08cfc can be used to orthogonalize the columns of .
The information returned by f08cec also yields the factorization of the trailing columns of , where . The orthogonal matrix arising from this factorization can be computed by
:
nag_lapackeig_dorgql(order,m,m,k,a,pda,tau,&fail)
or its trailing columns by
:
nag_lapackeig_dorgql(order,m,k,k,a,pda,tau,&fail)
4References
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
5Arguments
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
4: – IntegerInput
On entry: , the number of elementary reflectors whose product defines the matrix .
Constraint:
.
5: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
when
;
when
.
On entry: details of the vectors which define the elementary reflectors, as returned by f08cec.
On exit: the matrix .
6: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
if ,
;
if , .
7: – const doubleInput
Note: the dimension, dim, of the array tau
must be at least
.
On entry: further details of the elementary reflectors, as returned by f08cec.
8: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_INT
On entry, .
Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
7Accuracy
The computed matrix differs from an exactly orthogonal matrix by a matrix such that
where is the machine precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08cfc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The total number of floating-point operations is approximately ; when , the number is approximately .