NAG Library Routine Document
F08JJF (DSTEBZ) computes some (or all) of the eigenvalues of a real symmetric tridiagonal matrix, by bisection.
|SUBROUTINE F08JJF (
||RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
||N, IL, IU, M, NSPLIT, IBLOCK(N), ISPLIT(N), IWORK(3*N), INFO
||VL, VU, ABSTOL, D(*), E(*), W(N), WORK(4*N)
The routine may be called by its
F08JJF (DSTEBZ) uses bisection to compute some or all of the eigenvalues of a real symmetric tridiagonal matrix .
It searches for zero or negligible off-diagonal elements of
to see if the matrix splits into block diagonal form:
It performs bisection on each of the blocks
and returns the block index of each computed eigenvalue, so that a subsequent call to F08JKF (DSTEIN)
to compute eigenvectors can also take advantage of the block structure.
Kahan W (1966) Accurate eigenvalues of a symmetric tridiagonal matrix Report CS41 Stanford University
- 1: – CHARACTER(1)Input
: indicates which eigenvalues are required.
- All the eigenvalues are required.
- All the eigenvalues in the half-open interval (VL,VU] are required.
- Eigenvalues with indices IL to IU are required.
, or .
- 2: – CHARACTER(1)Input
: indicates the order in which the eigenvalues and their block numbers are to be stored.
- The eigenvalues are to be grouped by split-off block and ordered from smallest to largest within each block.
- The eigenvalues for the entire matrix are to be ordered from smallest to largest.
- 3: – INTEGERInput
On entry: , the order of the matrix .
- 4: – REAL (KIND=nag_wp)Input
- 5: – REAL (KIND=nag_wp)Input
, the lower and upper bounds, respectively, of the half-open interval
] within which the required eigenvalues lie.
is not referenced.
if , .
- 6: – INTEGERInput
- 7: – INTEGERInput
, the indices of the first and last eigenvalues, respectively, to be computed (assuming that the eigenvalues are in ascending order).
is not referenced.
if , .
- 8: – REAL (KIND=nag_wp)Input
On entry: the absolute tolerance to which each eigenvalue is required. An eigenvalue (or cluster) is considered to have converged if it lies in an interval of width . If , then the tolerance is taken as .
- 9: – REAL (KIND=nag_wp) arrayInput
the dimension of the array D
must be at least
On entry: the diagonal elements of the tridiagonal matrix .
- 10: – REAL (KIND=nag_wp) arrayInput
the dimension of the array E
must be at least
On entry: the off-diagonal elements of the tridiagonal matrix .
- 11: – INTEGEROutput
On exit: , the actual number of eigenvalues found.
- 12: – INTEGEROutput
On exit: the number of diagonal blocks which constitute the tridiagonal matrix .
- 13: – REAL (KIND=nag_wp) arrayOutput
On exit: the required eigenvalues of the tridiagonal matrix stored in to .
- 14: – INTEGER arrayOutput
: at each row/column
is zero or negligible,
is considered to split into a block diagonal matrix and
contains the block number of the eigenvalue stored in
. Note that
(see Section 6
- 15: – INTEGER arrayOutput
: the leading NSPLIT
elements contain the points at which
splits up into sub-matrices as follows. The first sub-matrix consists of rows/columns
, the second sub-matrix consists of rows/columns
, and the NSPLIT
(th) sub-matrix consists of rows/columns
- 16: – REAL (KIND=nag_wp) arrayWorkspace
- 17: – INTEGER arrayWorkspace
- 18: – 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 or , the algorithm failed to compute some (or all) of the required eigenvalues to the required accuracy. More precisely, indicates that eigenvalue (stored in ) failed to converge.
If , the algorithm failed to compute some (or all) of the required eigenvalues. Try calling the routine again with .
If , see the description above for .
If or , see the description above for .
No eigenvalues have been computed. The floating-point arithmetic on the computer is not behaving as expected.
If failures with
are causing persistent trouble and you have checked that the routine is being called correctly, please contact NAG
The eigenvalues of are computed to high relative accuracy which means that if they vary widely in magnitude, then any small eigenvalues will be computed more accurately than, for example, with the standard method. However, the reduction to tridiagonal form (prior to calling the routine) may exclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix if its eigenvalues vary widely in magnitude.
8 Parallelism and Performance
F08JJF (DSTEBZ) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.
There is no complex analogue of this routine.
See Section 10
in F08FGF (DORMTR).