Developer Reference for Intel® oneAPI Math Kernel Library for C
?stebz
Computes selected eigenvalues of a real symmetric tridiagonal matrix by bisection.
Syntax
lapack_intLAPACKE_sstebz ( charrange , charorder , lapack_intn , floatvl , floatvu , lapack_intil , lapack_intiu , floatabstol , constfloat*d , constfloat*e , lapack_int*m , lapack_int*nsplit , float*w , lapack_int*iblock , lapack_int*isplit );
lapack_intLAPACKE_dstebz ( charrange , charorder , lapack_intn , doublevl , doublevu , lapack_intil , lapack_intiu , doubleabstol , constdouble*d , constdouble*e , lapack_int*m , lapack_int*nsplit , double*w , lapack_int*iblock , lapack_int*isplit );
Include Files
mkl.h
Description
sstebz dstebz stebz
The routine computes some (or all) of the eigenvalues of a real symmetric tridiagonal matrix T by bisection. The routine searches for zero or negligible off-diagonal elements to see if T splits into block-diagonal form T = diag(T_{1}, T_{2}, ...) . Then it performs bisection on each of the blocks Ti and returns the block index of each computed eigenvalue, so that a subsequent call to stein can also take advantage of the block structure.
See also laebz .
Input Parameters
range
Must be ‘A’ or ‘V’ or ‘I’ .
If range = 'A' , the routine computes all eigenvalues.
If range = 'V' , the routine computes eigenvalues w [ i ] in the half-open interval: vl < w(i)[i]≤vu .
If range = 'I' , the routine computes eigenvalues with indices il to iu .
order
Must be ‘B’ or ‘E’ .
If order = 'B' , the eigenvalues are to be ordered from smallest to largest within each split-off block. If order = 'E' , the eigenvalues for the entire matrix are to be ordered from smallest to largest.
n
The order of the matrix T ( n≥ 0 ).
- vl , vu
-
REAL for sstebz DOUBLE PRECISION for dstebz .
If range = 'V' , the routine computes eigenvalues w [ i ] in the half-open interval:
vl < w(i)[i]) ≤vu .
If range = 'A' or ‘I’ , vl and vu are not referenced.
il , iu
Constraint: 1 ≤il≤iu≤n .
If range = 'I' , the routine computes eigenvalues w [ i ] such that il≤i≤iu (assuming that the eigenvalues w [ i ] are in ascending order).
If range = 'A' or ‘V’ , il and iu are not referenced.
- abstol
-
REAL for sstebz DOUBLE PRECISION for dstebz . 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 abstol . If abstol≤ 0.0 , then the tolerance is taken as eps*|T| , where eps is the machine precision, and |T| is the 1-norm of the matrix T .
- d , e , work
-
REAL for sstebz DOUBLE PRECISION for dstebz . Arrays:
d contains the diagonal elements of T .
The size of d must be at least max(1, n ).
e contains the off-diagonal elements of T .
The size of e must be at least max(1, n -1). work (*) is a workspace array. The dimension of work must be at least max(1, 4 n ).
iwork
Workspace.
Array, size at least max(1, 3 n ).
Output Parameters
m
The actual number of eigenvalues found.
nsplit
The number of diagonal blocks detected in T .
- w
-
REAL for sstebz DOUBLE PRECISION for dstebz .
Array, size at least max(1, n ). The computed eigenvalues, stored in w[0] to w[m - 1] .
iblock , isplit
Arrays, size at least max(1, n ).
A positive value iblock [i] is the block number of the eigenvalue stored in w [i] (see also info ).
The leading nsplit elements of isplit contain points at which T splits into blocks Ti as follows: the block T1 contains rows/columns 1 to isplit(1)[0] ; the block T2 contains rows/columns isplit(1)[0]+1 to isplit(2)[1] , and so on.
Return Values
This function returns a value info .
If info=0 , the execution is successful.
If info = 1 , for range = 'A' or ‘V’ , the algorithm failed to compute some of the required eigenvalues to the desired accuracy; iblock [ i ] < 0 indicates that the eigenvalue stored in w [ i ] failed to converge.
If info = 2 , for range = 'I' , the algorithm failed to compute some of the required eigenvalues. Try calling the routine again with range = 'A' .
If info = 3 :
for range = 'A' or ‘V’ , same as info = 1 ;
for range = 'I' , same as info = 2 .
If info = 4 , no eigenvalues have been computed. The floating-point arithmetic on the computer is not behaving as expected.
If info = -i , the i -th parameter had an illegal value.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
Application Notes
The eigenvalues of \(T\) 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 QR 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.