The routine may be called by the names f08faf, nagf_lapackeig_dsyev or its LAPACK name dsyev.
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 https://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
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
2: – Character(1)Input
On entry: if , the upper triangular part of is stored.
If , the lower triangular part of is stored.
3: – IntegerInput
On entry: , the order of the matrix .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the symmetric matrix .
If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if , a contains the orthonormal eigenvectors of the matrix .
If then on exit the lower triangle (if ) or the upper triangle (if ) of a, including the diagonal, is overwritten.
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08faf is called.
6: – Real (Kind=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
7: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
8: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08faf is called.
If , a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
for optimal performance, , where is the optimal block size for f08fef.
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.
The computed eigenvalues and eigenvectors are exact for a nearby matrix , where
f08faf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08faf 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 .