The function may be called by the names: f08gfc, nag_lapackeig_dopgtr or nag_dopgtr.
f08gfc is intended to be used after a call to f08gec, which reduces a real symmetric matrix to symmetric tridiagonal form by an orthogonal similarity transformation: . f08gec represents the orthogonal matrix as a product of elementary reflectors.
This function may be used to generate explicitly as a square matrix.
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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.
2: – Nag_UploTypeInput
On entry: this must be the same argument uplo as supplied to f08gec.
3: – IntegerInput
On entry: , the order of the matrix .
4: – const doubleInput
Note: the dimension, dim, of the array ap
must be at least
On entry: details of the vectors which define the elementary reflectors, as returned by f08gec.
5: – 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 f08gec.
6: – doubleOutput
Note: the dimension, dim, of the array q
must be at least
The th element of the matrix is stored in
On exit: the orthogonal matrix .
7: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
8: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, . Constraint: .
On entry, and .
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.
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.
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.
f08gfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gfc 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.
The total number of floating-point operations is approximately .
This example computes all the eigenvalues and eigenvectors of the matrix , where
using packed storage. Here is symmetric and must first be reduced to tridiagonal form by f08gec. The program then calls f08gfc to form , and passes this matrix to f08jec which computes the eigenvalues and eigenvectors of .