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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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| 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
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!> !> 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 L D L^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, !> (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, 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 !> rank-revealing twisted factorization. !> The desired accuracy of the output can be specified by the input !> parameter ABSTOL. !> !> For more details, see , by Inderjit Dhillon, !> Computer Science Division Technical Report No. UCB//CSD-97-971, !> UC Berkeley, May 1997. !> !> !> Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested !> on machines which conform to the ieee-754 floating point standard. !> SSTEVR calls SSTEBZ and SSTEIN on non-ieee 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 half-open interval (VL,VU] !> will be found. !> = 'I': the IL-th through IU-th 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,N-1)) !> On entry, the (n-1) subdiagonal elements of the tridiagonal !> matrix A in elements 1 to N-1 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 !> If RANGE='V', the lower bound of the interval to !> be searched for eigenvalues. VL < VU. !> Not referenced if RANGE = 'A' or 'I'. !> |
| [in] | VU | !> VU is REAL !> If RANGE='V', the upper bound of the interval to !> be searched for eigenvalues. VL < VU. !> Not referenced if RANGE = 'A' or 'I'. !> |
| [in] | IL | !> IL is INTEGER !> If RANGE='I', the index of the !> smallest eigenvalue 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] | IU | !> IU is INTEGER !> If RANGE='I', the index of the !> largest eigenvalue 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 1-norm of the tridiagonal matrix obtained !> by reducing A to tridiagonal form. !> !> See 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, !> , 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 = IU-IL+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 i-th !> 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 i-th eigenvector !> is nonzero only in elements ISUPPZ( 2*i-1 ) 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 i-th argument had an illegal value !> > 0: Internal error !> |
Definition at line 301 of file sstevr.f.
1.11.0