The routine may be called by the names f08gaf, nagf_lapackeig_dspev or its LAPACK name dspev.
3Description
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.
4References
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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: – Character(1)Input
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint:
or .
2: – Character(1)Input
On entry: if , the upper triangular part of is stored.
If , the lower triangular part of is stored.
Constraint:
or .
3: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap
must be at least
.
On entry: the upper or lower triangle of the symmetric matrix , packed by columns.
More precisely,
if , the upper triangle of must be stored with element in for ;
if , the lower triangle of must be stored with element in for .
On exit: ap 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
Note: the second dimension of the array z
must be at least
if , and at least otherwise.
On exit: if , z contains the orthonormal eigenvectors of the matrix , with the th column of holding the eigenvector associated with .
On entry: the first dimension of the array z as declared in the (sub)program from which f08gaf is called.
Constraints:
if , ;
otherwise .
8: – Real (Kind=nag_wp) arrayWorkspace
9: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
7Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrix , where
and is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08gaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gaf 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.
9Further Comments
The total number of floating-point operations is proportional to .