The function may be called by the names: f08nec, nag_lapackeig_dgehrd or nag_dgehrd.
f08nec reduces a real general matrix to upper Hessenberg form by an orthogonal similarity transformation: .
The matrix is not formed explicitly, but is represented as a product of elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with in this representation (see Section 9).
The function can take advantage of a previous call to f08nhc, which may produce a matrix with the structure:
where and are upper triangular. If so, only the central diagonal block , in rows and columns to , needs to be reduced to Hessenberg form (the blocks and will also be affected by the reduction). Therefore, the values of and determined by f08nhc can be supplied to the function directly. If f08nhc has not previously been called however, then must be set to and to .
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: – IntegerInput
On entry: , the order of the matrix .
3: – IntegerInput
4: – IntegerInput
On entry: if has been output by f08nhc, ilo and ihimust contain the values returned by that function. Otherwise, ilo must be set to and ihi to n.
if , ;
if , and .
5: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
The th element of the matrix is stored in
On entry: the general matrix .
On exit: a is overwritten by the upper Hessenberg matrix and details of the orthogonal matrix .
6: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
7: – doubleOutput
Note: the dimension, dim, of the array tau
must be at least
On exit: further details of the orthogonal matrix .
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 .
On entry, , and .
Constraint: if , ;
if , 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 Hessenberg matrix is exactly similar to a nearby matrix , where
is a modestly increasing function of , and is the machine precision.
The elements of themselves may be sensitive to small perturbations in or to rounding errors in the computation, but this does not affect the stability of the eigenvalues, eigenvectors or Schur factorization.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08nec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nec 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 , where ; if and , the number is approximately .
To form the orthogonal matrix f08nec may be followed by a call to f08nfc