The function may be called by the names: f08fsc, nag_lapackeig_zhetrd or nag_zhetrd.
3Description
f08fsc reduces a complex Hermitian matrix to real symmetric tridiagonal form by a unitary 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).
4References
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: – Nag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of is stored.
The upper triangular part of is stored.
The lower triangular part of is stored.
Constraint:
or .
3: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
4: – ComplexInput/Output
Note: the dimension, dim, of the array a
must be at least
.
On entry: the Hermitian matrix .
If , is stored in .
If , is stored in .
If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: a is overwritten by the tridiagonal matrix and details of the unitary matrix as specified by uplo.
5: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
a.
Constraint:
.
6: – doubleOutput
Note: the dimension, dim, of the array d
must be at least
.
On exit: the diagonal elements of the tridiagonal matrix .
7: – doubleOutput
Note: the dimension, dim, of the array e
must be at least
.
On exit: the off-diagonal elements of the tridiagonal matrix .
8: – ComplexOutput
Note: the dimension, dim, of the array tau
must be at least
.
On exit: further details of the unitary matrix .
9: – 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: .
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 tridiagonal 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 and eigenvectors.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08fsc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fsc 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 real floating-point operations is approximately .
To form the unitary matrix f08fsc may be followed by a call to f08ftc
: