LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

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Functions/Subroutines  
subroutine  sstevr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) 
SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 
subroutine sstevr  (  character  JOBZ, 
character  RANGE,  
integer  N,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real  VL,  
real  VU,  
integer  IL,  
integer  IU,  
real  ABSTOL,  
integer  M,  
real, dimension( * )  W,  
real, dimension( ldz, * )  Z,  
integer  LDZ,  
integer, dimension( * )  ISUPPZ,  
real, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
integer  LIWORK,  
integer  INFO  
) 
SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
Download SSTEVR + dependencies [TGZ] [ZIP] [TXT]SSTEVR 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, SSTEVR calls SSTEMR to compute the eigenspectrum using Relatively Robust Representations. SSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various "good" L D L^T representations (also known as Relatively Robust Representations). GramSchmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the ith unreduced block of T, (a) Compute T  sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation, (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm, (c) If there is a cluster of close eigenvalues, "choose" sigma_i close to the cluster, and go to step (a), (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rankrevealing twisted factorization. The desired accuracy of the output can be specified by the input parameter ABSTOL. For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD97971, UC Berkeley, May 1997. Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested on machines which conform to the ieee754 floating point standard. SSTEVR calls SSTEBZ and SSTEIN on nonieee machines and when partial spectrum requests are made. Normal execution of SSTEMR may create NaNs 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.
[in]  JOBZ  JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. 
[in]  RANGE  RANGE is CHARACTER*1 = 'A': all eigenvalues will be found. = 'V': all eigenvalues in the halfopen interval (VL,VU] will be found. = 'I': the ILth through IUth eigenvalues will be found. For RANGE = 'V' or 'I' and IU  IL < N  1, SSTEBZ and SSTEIN are called 
[in]  N  N is INTEGER The order of the matrix. N >= 0. 
[in,out]  D  D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, D may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues. 
[in,out]  E  E is REAL array, dimension (max(1,N1)) On entry, the (n1) subdiagonal elements of the tridiagonal matrix A in elements 1 to N1 of E. On exit, E may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues. 
[in]  VL  VL is REAL 
[in]  VU  VU is REAL If RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. 
[in]  IL  IL is INTEGER 
[in]  IU  IU is INTEGER If RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. 
[in]  ABSTOL  ABSTOL is REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( a,b ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*T will be used in its place, where T is the 1norm of the tridiagonal matrix obtained by reducing A to tridiagonal form. See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy. 
[out]  M  M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IUIL+1. 
[out]  W  W is REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. 
[out]  Z  Z is REAL array, dimension (LDZ, max(1,M) ) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with W(i). Note: the user 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. 
[in]  LDZ  LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). 
[out]  ISUPPZ  ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The ith eigenvector is nonzero only in elements ISUPPZ( 2*i1 ) through ISUPPZ( 2*i ). Implemented only for RANGE = 'A' or 'I' and IU  IL = N  1 
[out]  WORK  WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal (and minimal) LWORK. 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK. LWORK >= 20*N. If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal 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. 
[out]  IWORK  IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK. 
[in]  LIWORK  LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= 10*N. If LIWORK = 1, then a workspace query is assumed; the routine only calculates the optimal 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. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: Internal error 
Definition at line 298 of file sstevr.f.