NAG FL Interface
f08juf (zpteqr)
1
Purpose
f08juf computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian positive definite matrix which has been reduced to tridiagonal form.
2
Specification
Fortran Interface
Integer, Intent (In) |
:: |
n, ldz |
Integer, Intent (Out) |
:: |
info |
Real (Kind=nag_wp), Intent (Inout) |
:: |
d(*), e(*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
work(4*n) |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
z(ldz,*) |
Character (1), Intent (In) |
:: |
compz |
|
C Header Interface
#include <nag.h>
void |
f08juf_ (const char *compz, const Integer *n, double d[], double e[], Complex z[], const Integer *ldz, double work[], Integer *info, const Charlen length_compz) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08juf_ (const char *compz, const Integer &n, double d[], double e[], Complex z[], const Integer &ldz, double work[], Integer &info, const Charlen length_compz) |
}
|
The routine may be called by the names f08juf, nagf_lapackeig_zpteqr or its LAPACK name zpteqr.
3
Description
f08juf computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite 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 routine stores the real orthogonal matrix
in a complex array, so that it may be used to compute all the eigenvalues and eigenvectors of a complex Hermitian positive definite matrix
which has been reduced to tridiagonal form
:
In this case, the matrix
must be formed explicitly and passed to
f08juf, which must be called with
. The routines which must be called to perform the reduction to tridiagonal form and form
are:
f08juf first factorizes
as
where
is unit lower bidiagonal and
is diagonal. It forms the bidiagonal matrix
, and then calls
f08msf to compute the singular values of
which are the same as the eigenvalues of
. The method used by the routine allows high relative accuracy to be achieved in the small eigenvalues of
. The eigenvectors are normalized so that
, but are determined only to within a complex factor of absolute value
.
4
References
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
5
Arguments
-
1:
– Character(1)
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 routine).
Constraint:
, or .
-
2:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
3:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
d
must be at least
.
On entry: the diagonal elements of the tridiagonal matrix .
On exit: the
eigenvalues in descending order, unless
, in which case
d is overwritten.
-
4:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
e
must be at least
.
On entry: the off-diagonal elements of the tridiagonal matrix .
On exit:
e is overwritten.
-
5:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
z
must be at least
if
or
and at least
if
.
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
.
If
,
z is not referenced.
-
6:
– Integer
Input
-
On entry: the first dimension of the array
z as declared in the (sub)program from which
f08juf is called.
Constraints:
- if or , ;
- if , .
-
7:
– Real (Kind=nag_wp) array
Workspace
-
-
8:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
-
The leading minor of order is not positive definite and the Cholesky factorization of could not be completed. Hence itself is not positive definite.
-
The algorithm to compute the singular values of the Cholesky factor failed to converge; off-diagonal elements did not converge to zero.
7
Accuracy
The eigenvalues and eigenvectors of are computed to high relative accuracy which means that if they vary widely in magnitude, then any small eigenvalues (and corresponding eigenvectors) will be computed more accurately than, for example, with the standard method. However, the reduction to tridiagonal form (prior to calling the routine) may exclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix if its eigenvalues vary widely in magnitude.
To be more precise, let
be the tridiagonal matrix defined by
, where
is diagonal with
, and
for all
. If
is an exact eigenvalue of
and
is the corresponding computed value, then
where
is a modestly increasing function of
,
is the
machine precision, and
is the condition number of
with respect to inversion defined by:
.
If
is the corresponding exact eigenvector of
, and
is the corresponding computed eigenvector, then the angle
between them is bounded as follows:
where
is the relative gap between
and the other eigenvalues, defined by
8
Parallelism and Performance
f08juf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08juf 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 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 routine is
f08jgf.
10
Example
This example computes all the eigenvalues and eigenvectors of the complex Hermitian positive definite matrix
, where
10.1
Program Text
10.2
Program Data
10.3
Program Results