NAG Library Routine Document
F08JGF (DPTEQR) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite tridiagonal matrix, or of a real symmetric positive definite matrix which has been reduced to tridiagonal form.
||N, LDZ, INFO
||D(*), E(*), Z(LDZ,*), WORK(4*N)
The routine may be called by its
F08JGF (DPTEQR) 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
is a diagonal matrix whose diagonal elements are the eigenvalues
is the orthogonal matrix whose columns are the eigenvectors
The routine may also be used to compute all the eigenvalues and eigenvectors of a real symmetric positive definite matrix
which has been reduced to tridiagonal form
In this case, the matrix
must be formed explicitly and passed to F08JGF (DPTEQR), which must be called with
. The routines which must be called to perform the reduction to tridiagonal form and form
F08JGF (DPTEQR) first factorizes
is unit lower bidiagonal and
is diagonal. It forms the bidiagonal matrix
, and then calls F08MEF (DBDSQR)
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 factor
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
- 1: – CHARACTER(1)Input
: 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).
, or .
- 2: – INTEGERInput
On entry: , the order of the matrix .
- 3: – REAL (KIND=nag_wp) arrayInput/Output
the dimension of the array D
must be at least
On entry: the diagonal elements of the tridiagonal matrix .
eigenvalues in descending order, unless
, in which case D
- 4: – REAL (KIND=nag_wp) arrayInput/Output
the dimension of the array E
must be at least
On entry: the off-diagonal elements of the tridiagonal matrix .
- 5: – REAL (KIND=nag_wp) arrayInput/Output
the second dimension of the array Z
must be at least
and at least
must contain the orthogonal matrix
from the reduction to tridiagonal form.
need not be set.
required orthonormal eigenvectors stored as columns of
th column corresponds to the
th eigenvalue, where
is not referenced.
- 6: – INTEGERInput
: the first dimension of the array Z
as declared in the (sub)program from which F08JGF (DPTEQR) is called.
- if or , ;
- if , .
- 7: – REAL (KIND=nag_wp) arrayWorkspace
- 8: – INTEGEROutput
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.
If , the leading minor of order is not positive definite and the Cholesky factorization of could not be completed. Hence itself is not positive definite.
If , the algorithm to compute the singular values of the Cholesky factor failed to converge; off-diagonal elements did not converge to zero.
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
is diagonal with
is an exact eigenvalue of
is the corresponding computed value, then
is a modestly increasing function of
is the machine precision
is the condition number of
with respect to inversion defined by:
is the corresponding exact eigenvector of
is the corresponding computed eigenvector, then the angle
between them is bounded as follows:
is the relative gap between
and the other eigenvalues, defined by
8 Parallelism and Performance
F08JGF (DPTEQR) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08JGF (DPTEQR) 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 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 complex analogue of this routine is F08JUF (ZPTEQR)
This example computes all the eigenvalues and eigenvectors of the symmetric positive definite tridiagonal matrix
10.1 Program Text
Program Text (f08jgfe.f90)
10.2 Program Data
Program Data (f08jgfe.d)
10.3 Program Results
Program Results (f08jgfe.r)