The routine may be called by the names f08fsf, nagf_lapackeig_zhetrd or its LAPACK name zhetrd.
3Description
f08fsf 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). Routines 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: – Character(1)Input
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 .
2: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
3: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On entry: the Hermitian matrix .
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.
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08fsf is called.
Constraint:
.
5: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array d
must be at least
.
On exit: the diagonal elements of the tridiagonal matrix .
6: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array e
must be at least
.
On exit: the off-diagonal elements of the tridiagonal matrix .
7: – Complex (Kind=nag_wp) arrayOutput
Note: the dimension of the array tau
must be at least
.
On exit: further details of the unitary matrix .
8: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
9: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08fsf 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.
Suggested value:
for optimal performance, , where is the optimal block size.
Constraint:
or .
10: – 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.
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.
f08fsf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fsf 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.
9Further Comments
The total number of real floating-point operations is approximately .
To form the unitary matrix f08fsf may be followed by a call to f08ftf
:
Call zungtr(uplo,n,a,lda,tau,work,lwork,info)
To apply to an complex matrix f08fsf may be followed by a call to f08fuf
. For example,