The routine may be called by the names f08nef, nagf_lapackeig_dgehrd or its LAPACK name dgehrd.
f08nef 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). Routines are provided to work with in this representation (see Section 9).
The routine can take advantage of a previous call to f08nhf, 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 f08nhf can be supplied to the routine directly. If f08nhf 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: – IntegerInput
On entry: , the order of the matrix .
2: – IntegerInput
3: – IntegerInput
On entry: if has been output by f08nhf, ilo and ihimust contain the values returned by that routine. Otherwise, ilo must be set to and ihi to n.
if , ;
if , and .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the general matrix .
On exit: a is overwritten by the upper Hessenberg matrix and details of the orthogonal matrix .
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08nef is called.
6: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array tau
must be at least
On exit: further details of the orthogonal matrix .
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 f08nef 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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
An explanatory message is output, and execution of the program is terminated.
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.
f08nef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nef 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 , where ; if and , the number is approximately .
To form the orthogonal matrix f08nef may be followed by a call to f08nff
To apply to an real matrix f08nef may be followed by a call to f08ngf.