dsyevr (l)  Linux Manuals
dsyevr: computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A
NAME
DSYEVR  computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix ASYNOPSIS
 SUBROUTINE DSYEVR(
 JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO )
 CHARACTER JOBZ, RANGE, UPLO
 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
 DOUBLE PRECISION ABSTOL, VL, VU
 INTEGER ISUPPZ( * ), IWORK( * )
 DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
PURPOSE
DSYEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.DSYEVR first reduces the matrix A to tridiagonal form T with a call to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute the eigenspectrum using Relatively Robust Representations. DSTEMR 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 each unreduced block (submatrix) of T,
(a)
(b)
(c)
(d)
For more details, see DSTEMRaqs documentation and:
 Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
 Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested on machines which conform to the ieee754 floating point standard. DSYEVR calls DSTEBZ and SSTEIN on nonieee machines and
when partial spectrum requests are made.
Normal execution of DSTEMR 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.
ARGUMENTS
 JOBZ (input) CHARACTER*1

= aqNaq: Compute eigenvalues only;
= aqVaq: Compute eigenvalues and eigenvectors.  RANGE (input) CHARACTER*1

= aqAaq: all eigenvalues will be found.
= aqVaq: all eigenvalues in the halfopen interval (VL,VU] will be found. = aqIaq: the ILth through IUth eigenvalues will be found.  UPLO (input) CHARACTER*1

= aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored.  N (input) INTEGER
 The order of the matrix A. N >= 0.
 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
 On entry, the symmetric matrix A. If UPLO = aqUaq, the leading NbyN upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = aqLaq, the leading NbyN lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO=aqLaq) or the upper triangle (if UPLO=aqUaq) of A, including the diagonal, is destroyed.
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,N).
 VL (input) DOUBLE PRECISION
 VU (input) DOUBLE PRECISION If RANGE=aqVaq, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = aqAaq or aqIaq.
 IL (input) INTEGER
 IU (input) INTEGER If RANGE=aqIaq, 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 = aqAaq or aqVaq.
 ABSTOL (input) DOUBLE PRECISION
 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 DLAMCH( aqSafe minimumaq ). 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.
 M (output) INTEGER
 The total number of eigenvalues found. 0 <= M <= N. If RANGE = aqAaq, M = N, and if RANGE = aqIaq, M = IUIL+1.
 W (output) DOUBLE PRECISION array, dimension (N)
 The first M elements contain the selected eigenvalues in ascending order.
 Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
 If JOBZ = aqVaq, 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). If JOBZ = aqNaq, then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = aqVaq, the exact value of M is not known in advance and an upper bound must be used. Supplying N columns is always safe.
 LDZ (input) INTEGER
 The leading dimension of the array Z. LDZ >= 1, and if JOBZ = aqVaq, LDZ >= max(1,N).
 ISUPPZ (output) 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 ).
 WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
 LWORK (input) INTEGER
 The dimension of the array WORK. LWORK >= max(1,26*N). For optimal efficiency, LWORK >= (NB+6)*N, where NB is the max of the blocksize for DSYTRD and DORMTR returned by ILAENV. If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
 On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
 LIWORK (input) INTEGER
 The dimension of the array IWORK. LIWORK >= max(1,10*N). If LIWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.
 INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
> 0: Internal error
FURTHER DETAILS
Based on contributions byInderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of
Jason Riedy, Computer Science Division, University of