NAG CL Interface
f08jsc (zsteqr)
1
Purpose
f08jsc computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix which has been reduced to tridiagonal form.
2
Specification
void |
f08jsc (Nag_OrderType order,
Nag_ComputeZType compz,
Integer n,
double d[],
double e[],
Complex z[],
Integer pdz,
NagError *fail) |
|
The function may be called by the names: f08jsc, nag_lapackeig_zsteqr or nag_zsteqr.
3
Description
f08jsc computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix
.
In other words, it can compute the spectral factorization of
as
where
is a diagonal matrix whose diagonal elements are the eigenvalues
, and
is the orthogonal matrix whose columns are the eigenvectors
. Thus
The function stores the real orthogonal matrix
in a complex array, so that it may also be used to compute all the eigenvalues and eigenvectors of a complex Hermitian matrix
which has been reduced to tridiagonal form
:
In this case, the matrix
must be formed explicitly and passed to
f08jsc, which must be called with
. The functions which must be called to perform the reduction to tridiagonal form and form
are:
f08jsc uses the implicitly shifted
algorithm, switching between the
and
variants in order to handle graded matrices effectively (see
Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that
, but are determined only to within a complex factor of absolute value
.
If only the eigenvalues of
are required, it is more efficient to call
f08jfc instead. If
is positive definite, small eigenvalues can be computed more accurately by
f08juc.
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem
LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville
https://www.netlib.org/lapack/lawnspdf/lawn17.pdf
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia
5
Arguments
-
1:
– Nag_OrderType
Input
-
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_ComputeZType
Input
-
On entry: indicates whether the eigenvectors are to be computed.
- Only the eigenvalues are computed (and the array z is not referenced).
- The eigenvalues and eigenvectors of are computed (and the array z must contain the matrix on entry).
- The eigenvalues and eigenvectors of are computed (and the array z is initialized by the function).
Constraint:
, or .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– double
Input/Output
-
Note: the dimension,
dim, of the array
d
must be at least
.
On entry: the diagonal elements of the tridiagonal matrix .
On exit: the
eigenvalues in ascending order, unless
NE_CONVERGENCE (in which case see
Section 6).
-
5:
– double
Input/Output
-
Note: the dimension,
dim, of the array
e
must be at least
.
On entry: the off-diagonal elements of the tridiagonal matrix .
On exit:
e is overwritten.
-
6:
– Complex
Input/Output
-
Note: the dimension,
dim, of the array
z
must be at least
- when
or ;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: if
,
z must contain the unitary matrix
from the reduction to tridiagonal form.
If
,
z need not be set.
On exit: if
or
, the
required orthonormal eigenvectors stored as columns of
; the
th column corresponds to the
th eigenvalue, where
, unless
NE_CONVERGENCE.
If
,
z is not referenced.
-
7:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if or , ;
- if , .
-
8:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error 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_CONVERGENCE
-
The algorithm has failed to find all the eigenvalues after a total of
iterations. In this case,
d and
e contain on exit the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix unitarily similar to
.
off-diagonal elements have not converged to zero.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if or , ;
if , .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
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.
7
Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrix
, where
and
is the
machine precision.
If
is an exact eigenvalue and
is the corresponding computed value, then
where
is a modestly increasing function of
.
If
is the corresponding exact eigenvector, and
is the corresponding computed eigenvector, then the angle
between them is bounded as follows:
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
8
Parallelism and Performance
f08jsc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jsc 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 real floating-point operations is typically about if and about if or , but depends on how rapidly the algorithm converges. When , the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when or can be vectorized and on some machines may be performed much faster.
The real analogue of this function is
f08jec.
10
Example
See Section 10 in
f08ftc,
f08gtc or
f08hsc, which illustrate the use of this function to compute the eigenvalues and eigenvectors of a full or band Hermitian matrix.