NAG Library Routine Document
F08GAF (DSPEV) computes all the eigenvalues and, optionally, all the eigenvectors of a real by symmetric matrix in packed storage.
||N, LDZ, INFO
||AP(*), W(N), Z(LDZ,*), WORK(3*N)
The routine may be called by its
The symmetric matrix is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide
(3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
- 1: – CHARACTER(1)Input
: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
- 2: – CHARACTER(1)Input
, the upper triangular part of
If , the lower triangular part of is stored.
- 3: – INTEGERInput
On entry: , the order of the matrix .
- 4: – REAL (KIND=nag_wp) arrayInput/Output
the dimension of the array AP
must be at least
: the upper or lower triangle of the
, packed by columns.
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of
- 5: – REAL (KIND=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
- 6: – REAL (KIND=nag_wp) arrayOutput
the second dimension of the array Z
must be at least
, and at least
contains the orthonormal eigenvectors of the matrix
, with the
th column of
holding the eigenvector associated with
is not referenced.
- 7: – INTEGERInput
: the first dimension of the array Z
as declared in the (sub)program from which F08GAF (DSPEV) is called.
- if , ;
- otherwise .
- 8: – REAL (KIND=nag_wp) arrayWorkspace
- 9: – 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 algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
The computed eigenvalues and eigenvectors are exact for a nearby matrix
is the machine precision
. See Section 4.7 of Anderson et al. (1999)
for further details.
8 Parallelism and Performance
F08GAF (DSPEV) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08GAF (DSPEV) 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 proportional to .
The complex analogue of this routine is F08GNF (ZHPEV)
This example finds all the eigenvalues of the symmetric matrix
together with approximate error bounds for the computed eigenvalues.
10.1 Program Text
Program Text (f08gafe.f90)
10.2 Program Data
Program Data (f08gafe.d)
10.3 Program Results
Program Results (f08gafe.r)