NAG Library Routine Document
F08JUF (ZPTEQR)
1 Purpose
F08JUF (ZPTEQR) 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
| INTEGER |
N, LDZ, INFO |
| REAL (KIND=nag_wp) |
D(*), E(*), WORK(4*N) |
| COMPLEX (KIND=nag_wp) |
Z(LDZ,*) |
| CHARACTER(1) |
COMPZ |
|
The routine may be called by its
LAPACK
name zpteqr.
3 Description
F08JUF (ZPTEQR) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite tridiagonal matrix
T.
In other words, it can compute the spectral factorization of
T as
where
Λ is a diagonal matrix whose diagonal elements are the eigenvalues
λi, and
Z is the orthogonal matrix whose columns are the eigenvectors
zi. Thus
The routine stores the real orthogonal matrix
Z in a complex array, so that it may be used to compute all the eigenvalues and eigenvectors of a complex Hermitian positive definite matrix
A which has been reduced to tridiagonal form
T:
In this case, the matrix
Q must be formed explicitly and passed to F08JUF (ZPTEQR), which must be called with
COMPZ='V'. The routines which must be called to perform the reduction to tridiagonal form and form
Q are:
F08JUF (ZPTEQR) first factorizes
T as
LDLH where
L is unit lower bidiagonal and
D is diagonal. It forms the bidiagonal matrix
B=LD12, and then calls
F08MSF (ZBDSQR) to compute the singular values of
B which are the same as the eigenvalues of
T. The method used by the routine allows high relative accuracy to be achieved in the small eigenvalues of
T. The eigenvectors are normalized so that
zi2=1, but are determined only to within a complex factor of absolute value
1.
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 Parameters
- 1: COMPZ – CHARACTER(1)Input
-
On entry: indicates whether the
eigenvectors
are to be computed.
- COMPZ='N'
- Only the eigenvalues
are computed (and the array Z is not referenced).
- COMPZ='I'
- The
eigenvalues and eigenvectors of T are computed (and the array Z is initialized by the routine).
- COMPZ='V'
- The
eigenvalues and eigenvectors
of A are computed (and the array Z must contain the matrix Q on entry).
Constraint:
COMPZ='N', 'V' or 'I'.
- 2: N – INTEGERInput
-
On entry:
n, the order of the matrix T.
Constraint:
N≥0.
- 3: D(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
D
must be at least
max1,N.
On entry: the diagonal elements of the tridiagonal matrix T.
On exit: the
n eigenvalues in descending order, unless
INFO>0, in which case
D is overwritten.
- 4: E(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
E
must be at least
max1,N-1.
On entry: the off-diagonal elements of the tridiagonal matrix T.
On exit:
E is overwritten.
- 5: Z(LDZ,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
Z
must be at least
max1,N if
COMPZ='V' or
'I' and at least
1 if
COMPZ='N'.
On entry: if
COMPZ='V',
Z must contain the
unitary
matrix
Q from the reduction to tridiagonal form.
If
COMPZ='I',
Z need not be set.
On exit: if
COMPZ='I' or
'V', the
n required orthonormal eigenvectors stored as columns of
Z; the
ith column corresponds to the
ith eigenvalue, where
i=1,2,…,n, unless
INFO>0.
If
COMPZ='N',
Z is not referenced.
- 6: LDZ – INTEGERInput
-
On entry: the first dimension of the array
Z as declared in the (sub)program from which F08JUF (ZPTEQR) is called.
Constraints:
- if COMPZ='V' or 'I', LDZ≥max1,N;
- if COMPZ='N', LDZ≥1.
- 7: WORK(4×N) – REAL (KIND=nag_wp) arrayWorkspace
- 8: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0
If INFO=i, the leading minor of order i is not positive definite and the Cholesky factorization of T could not be completed. Hence T itself is not positive definite.
If INFO=N+i, the algorithm to compute the singular values of the Cholesky factor B failed to converge; i off-diagonal elements did not converge to zero.
7 Accuracy
The eigenvalues and eigenvectors of T 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 QR 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
H be the tridiagonal matrix defined by
H=DTD, where
D is diagonal with
dii
=
t
ii
-12
, and
hii
=
1
for all
i. If
λi is an exact eigenvalue of
T and
λ~i is the corresponding computed value, then
where
cn is a modestly increasing function of
n,
ε is the
machine precision, and
κ2H is the condition number of
H with respect to inversion defined by:
κ2H=H·H-1.
If
zi is the corresponding exact eigenvector of
T, and
z~i is the corresponding computed eigenvector, then the angle
θz~i,zi between them is bounded as follows:
where
relgapi is the relative gap between
λi and the other eigenvalues, defined by
8 Further Comments
The total number of real floating point operations is typically about 30n2 if COMPZ='N' and about 12n3 if COMPZ='V' or 'I', but depends on how rapidly the algorithm converges. When COMPZ='N', the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when COMPZ='V' or 'I' can be vectorized and on some machines may be performed much faster.
The real analogue of this routine is
F08JGF (DPTEQR).
9 Example
This example computes all the eigenvalues and eigenvectors of the complex Hermitian positive definite matrix
A, where
9.1 Program Text
Program Text (f08jufe.f90)
9.2 Program Data
Program Data (f08jufe.d)
9.3 Program Results
Program Results (f08jufe.r)