The routine may be called by the names f08jjf, nagf_lapackeig_dstebz or its LAPACK name dstebz.
f08jjf 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 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
On entry: indicates which eigenvalues are required.
All the eigenvalues are required.
All the eigenvalues in the half-open interval (vl,vu] are required.
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 , the tolerance is taken as .
9: – Real (Kind=nag_wp) arrayInput
Note: 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
Note: 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
On exit: at each row/column where is zero or negligible, is considered to split into a block diagonal matrix and
contains the block number of the eigenvalue stored in , for . Note that for some whenever or (see Section 6) and or .
15: – Integer arrayOutput
On exit: the leading nsplit elements contain the points at which splits up into sub-matrices as follows. The first sub-matrix consists of rows/columns to , the second sub-matrix consists of rows/columns to , , and the nsplit(th) sub-matrix consists of rows/columns to ().
16: – Real (Kind=nag_wp) arrayWorkspace
17: – Integer arrayWorkspace
18: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If failures with are causing persistent trouble and you have checked that the routine is being called correctly, please contact NAG.
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.
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08jjf 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.