f08jef uses the implicitly shifted algorithm, switching between the and variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that , but are determined only to within a factor .
If only the eigenvalues of are required, it is more efficient to call f08jff instead. If is positive definite, small eigenvalues can be computed more accurately by f08jgf.
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville https://www.netlib.org/lapack/lawnspdf/lawn17.pdf
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia
1: – Character(1)Input
On entry: 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
Note: the dimension of the array d
must be at least
On entry: the diagonal elements of the tridiagonal matrix .
On exit: the eigenvalues in ascending order, unless (in which case see Section 6).
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array e
must be at least
On entry: the off-diagonal elements of the tridiagonal matrix .
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 has failed to find all the eigenvalues after a total of iterations. In this case, d and e contain on exit the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix orthogonally similar to . off-diagonal elements have not converged to zero.
The computed eigenvalues and eigenvectors are exact for a nearby matrix , where
and is the machine precision.
If is an exact eigenvalue and is the corresponding computed value, then
where is a modestly increasing function of .
If is the corresponding exact eigenvector, and is the corresponding computed eigenvector, then the angle between them is bounded as follows:
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
8Parallelism and Performance
f08jef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jef 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.