Developer Reference for Intel® oneAPI Math Kernel Library for C
?stevr
Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix using the Relatively Robust Representations.
Syntax
lapack_intLAPACKE_sstevr ( intmatrix_layout , charjobz , charrange , lapack_intn , float*d , float*e , floatvl , floatvu , lapack_intil , lapack_intiu , floatabstol , lapack_int*m , float*w , float*z , lapack_intldz , lapack_int*isuppz );
lapack_intLAPACKE_dstevr ( intmatrix_layout , charjobz , charrange , lapack_intn , double*d , double*e , doublevl , doublevu , lapack_intil , lapack_intiu , doubleabstol , lapack_int*m , double*w , double*z , lapack_intldz , lapack_int*isuppz );
Include Files
mkl.h
Description
sstevr dstevr stevr
The routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T . Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
Whenever possible, the routine calls stemr to compute the eigenspectrum using Relatively Robust Representations. stegr computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various “good” L*D*L:code:`T` representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of T :
Compute T - σ_{i} = L_{i}*D_{i}*L_{i}^{T} , such that Li*Di*Li:code:`T` is a relatively robust representation.
Compute the eigenvalues, λ j , of Li*Di*Li:code:`T` to high relative accuracy by the dqds algorithm.
If there is a cluster of close eigenvalues, “choose” σ i close to the cluster, and go to Step (a).
Given the approximate eigenvalue λ j of Li*Di*Li:code:`T` , compute the corresponding eigenvector by forming a rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the input parameter abstol .
The routine ?stevr calls stemr when the full spectrum is requested on machines which conform to the IEEE-754 floating point standard. ?stevr calls stebz and stein on non-IEEE machines and when partial spectrum requests are made.
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
jobz
Must be ‘N’ or ‘V’ .
If jobz = 'N' , then only eigenvalues are computed. If jobz = 'V' , then eigenvalues and eigenvectors are computed.
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)w[i] in the half-open interval: vl<w(i)w[i]≤vu . If range = 'I' , the routine computes eigenvalues with indices il to iu . For range = 'V' or ‘I’ and iu - il < n - 1, sstebz / dstebz and sstein / dstein are called.
n
The order of the matrix T ( n≥ 0 ).
- d , e , work
-
REAL for sstevr DOUBLE PRECISION for dstevr . Arrays: d(*) contains the n diagonal elements of the tridiagonal matrix T . The dimension of d must be at least max(1, n ). e(*) contains the n -1 subdiagonal elements of A . The dimension of e must be at least max(1, n -1). The n -th element of this array is used as workspace. work is a workspace array, its dimension max(1, lwork) .
- vl , vu
-
REAL for sstevr DOUBLE PRECISION for dstevr . If range = 'V' , the lower and upper bounds of the interval to be searched for eigenvalues. Constraint: vl< vu . If range = 'A' or ‘I’ , vl and vu are not referenced.
il , iu
If range = 'I' , the indices in ascending order of the smallest and largest eigenvalues to be returned. Constraint: 1 ≤il≤iu≤n , if n > 0 ; il=1 and iu=0 if n = 0 . If range = 'A' or ‘V’ , il and iu are not referenced.
- abstol
-
REAL for sstevr DOUBLE PRECISION for dstevr . The absolute error tolerance to which each eigenvalue/eigenvector is required. If jobz = 'V' , the eigenvalues and eigenvectors output have residual norms bounded by abstol , and the dot products between different eigenvectors are bounded by abstol . If abstol < n *eps*||T|| , then n *eps*||T|| will be used in its place, where eps is the machine precision, and ||T|| is the 1-norm of the matrix T . The eigenvalues are computed to an accuracy of eps*||T|| irrespective of abstol . If high relative accuracy is important, set abstol to ?lamch (‘S’).
ldz
The leading dimension of the output array z .
Constraints: ldz≥ 1 if jobz = 'N' ;
ldz≥ max(1, n) for column major layout and ldz ≥ max(1, m ) for row major layout if jobz = 'V' .
lwork
The dimension of the array work . Constraint: lwork≥ max(1, 20*n) . If lwork = -1 , then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla . See Application Notes for details.
iwork
Workspace array, its dimension max(1, liwork) .
liwork
The dimension of the array iwork , lwork≥ max(1, 10*n) . If liwork = -1 , then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla . See Application Notes for details.
Output Parameters
m
The total number of eigenvalues found,
0 ≤m≤n . If range = 'A' , m = n , if range = 'I' , m = iu-il+1 , and if range = 'V' the exact value of m is unknown..
- w , z
-
REAL for sstevr DOUBLE PRECISION for dstevr . Arrays: w(*) , size at least max(1, n ). The first m elements of w contain the selected eigenvalues of the matrix T in ascending order.
z (size at least max(1, ldz * m ) for column major layout and max(1, ldz * n ) for row major layout) .
The second dimension of z must be at least max(1, m ). If jobz = 'V' , then if info = 0 , the first m columns of z contain the orthonormal eigenvectors of the matrix T corresponding to the selected eigenvalues, with the i -th column of z holding the eigenvector associated with w(i)w[i - 1] . If jobz = 'N' , then z is not referenced. Note: you must ensure that at least max(1, m ) columns are supplied in the array z ; if range = 'V' , the exact value of m is not known in advance and an upper bound must be used.
- d , e
-
On exit, these arrays may be multiplied by a constant factor chosen to avoid overflow or underflow in computing the eigenvalues.
isuppz
Array, size at least 2 *max(1, m ). The support of the eigenvectors in z , i.e., the indices indicating the nonzero elements in z . The i -th eigenvector is nonzero only in elements isuppz( 2i-1)isuppz[2i - 2] through isuppz(2i)isuppz[2i - 1] . Implemented only for range = 'A' or ‘I’ and iu-il = n-1 .
- work(1)
-
On exit, if info = 0 , then work(1) returns the required minimal size of lwork .
- iwork(1)
-
On exit, if info = 0 , then iwork(1) returns the required minimal size of liwork .
Return Values
This function returns a value info .
If info=0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
If info = i , an internal error has occurred.
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
Normal execution of the routine ?stegr may create NaN s and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the IEEE standard default manner.