NAG Library Routine Document
generates the complex unitary matrix
, which was determined by f08gsf (zhptrd)
when reducing a Hermitian matrix to tridiagonal form.
|Integer, Intent (In)||:: ||n, ldq|
|Integer, Intent (Out)||:: ||info|
|Complex (Kind=nag_wp), Intent (In)||:: ||ap(*), tau(*)|
|Complex (Kind=nag_wp), Intent (Inout)||:: ||q(ldq,*)|
|Complex (Kind=nag_wp), Intent (Out)||:: ||work(n-1)|
|Character (1), Intent (In)||:: ||uplo|C Header Interface
f08gtf_ (const char *uplo, const Integer *n, const Complex ap, const Complex tau, Complex q, const Integer *ldq, Complex work, Integer *info, const Charlen length_uplo)|
The routine may be called by its
is intended to be used after a call to f08gsf (zhptrd)
, which reduces a complex Hermitian matrix
to real symmetric tridiagonal form
by a unitary similarity transformation:
. f08gsf (zhptrd)
represents the unitary matrix
as a product of
This routine 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: – Character(1)Input
: this must
be the same argument uplo
as supplied to f08gsf (zhptrd)
- 2: – IntegerInput
On entry: , the order of the matrix .
- 3: – Complex (Kind=nag_wp) arrayInput
the dimension of the array ap
must be at least
: details of the vectors which define the elementary reflectors, as returned by f08gsf (zhptrd)
- 4: – Complex (Kind=nag_wp) arrayInput
the dimension of the array tau
must be at least
: further details of the elementary reflectors, as returned by f08gsf (zhptrd)
- 5: – Complex (Kind=nag_wp) arrayOutput
the second dimension of the array q
must be at least
On exit: the by unitary matrix .
- 6: – IntegerInput
: the first dimension of the array q
as declared in the (sub)program from which f08gtf (zupgtr)
- 7: – Complex (Kind=nag_wp) arrayWorkspace
- 8: – IntegerOutput
unless the routine detects an error (see Section 6
Error 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 unitary matrix by a matrix
is the machine precision
Parallelism and Performance
f08gtf (zupgtr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gtf (zupgtr) 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 real floating-point operations is approximately .
The real analogue of this routine is f08gff (dopgtr)
This example computes all the eigenvalues and eigenvectors of the matrix
using packed storage. Here
is Hermitian and must first be reduced to tridiagonal form by f08gsf (zhptrd)
. The program then calls f08gtf (zupgtr)
, and passes this matrix to f08jsf (zsteqr)
which computes the eigenvalues and eigenvectors of
Program Text (f08gtfe.f90)
Program Data (f08gtfe.d)
Program Results (f08gtfe.r)