f08gbf computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix in packed storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
The routine may be called by the names f08gbf, nagf_lapackeig_dspevx or its LAPACK name dspevx.
The symmetric matrix is first reduced to tridiagonal form, using orthogonal similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.
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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput.11 873–912
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 , all eigenvalues will be found.
If , all eigenvalues in the half-open interval will be found.
On entry: if , the upper triangular part of is stored.
If , the lower triangular part of is stored.
4: – IntegerInput
On entry: , the order of the matrix .
5: – 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.
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 .
6: – Real (Kind=nag_wp)Input
7: – Real (Kind=nag_wp)Input
On entry: if , the lower and upper bounds of the interval to be searched for eigenvalues.
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval of width less than or equal to
where is the machine precision. If abstol is less than or equal to zero, then will be used in its place, where is the tridiagonal matrix obtained by reducing to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold , not zero. If this routine returns with , indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990).
11: – IntegerOutput
On exit: the total number of eigenvalues found. .
If , .
If , .
12: – Real (Kind=nag_wp) arrayOutput
On exit: the selected eigenvalues in ascending order.
13: – 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 , then
if , the first m columns of contain the orthonormal eigenvectors of the matrix corresponding to the selected eigenvalues, with the th column of holding the eigenvector associated with ;
if an eigenvector fails to converge (), then that column of contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
Background information to multithreading can be found in the Multithreading documentation.
f08gbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gbf 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 .