ESMF_DSYEVR Subroutine

subroutine ESMF_DSYEVR(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

\brief ESMF_DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices \htmlonly Download ESMF_DSYEVR + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_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.

ESMF_DSYEVR first reduces the matrix A to tridiagonal form T with a call to ESMF_DSYTRD. Then, whenever possible, ESMF_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). Gram-Schmidt 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) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general. (b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d). (c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy. (d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter ABSTOL.

For more details, see DSTEMR’s documentation and: - Inderjit S. Dhillon and Beresford N. Parlett: “Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices,” Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. - Inderjit Dhillon and Beresford Parlett: “Orthogonal Eigenvectors and Relative Gaps,” SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154. - Inderjit Dhillon: “A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem”, Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1 : ESMF_DSYEVR calls DSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. ESMF_DSYEVR calls DSTEBZ and DSTEIN on non-ieee 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. \endverbatim \param[in] JOBZ \verbatim JOBZ is CHARACTER*1 = ‘N’: Compute eigenvalues only; = ‘V’: Compute eigenvalues and eigenvectors. \endverbatim

\param[in] RANGE \verbatim 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, DSTEBZ and DSTEIN are called \endverbatim

\param[in] UPLO \verbatim UPLO is CHARACTER*1 = ‘U’: Upper triangle of A is stored; = ‘L’: Lower triangle of A is stored. \endverbatim

\param[in] N \verbatim N is INTEGER The order of the matrix A. N >= 0. \endverbatim

\param[in,out] A \verbatim A is DOUBLE PRECISION array, dimension (LDA, N) On entry, the symmetric matrix A. If UPLO = ‘U’, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = ‘L’, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO=’L’) or the upper triangle (if UPLO=’U’) of A, including the diagonal, is destroyed. \endverbatim

\param[in] LDA \verbatim LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). \endverbatim

\param[in] VL \verbatim VL is DOUBLE PRECISION If RANGE=’V’, the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = ‘A’ or ‘I’. \endverbatim

\param[in] VU \verbatim VU is DOUBLE PRECISION If RANGE=’V’, the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = ‘A’ or ‘I’. \endverbatim

\param[in] IL \verbatim 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’. \endverbatim

\param[in] IU \verbatim 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’. \endverbatim

\param[in] ABSTOL \verbatim ABSTOL is 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 1-norm 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
     ESMF_DLAMCH( '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.

\endverbatim

\param[out] M \verbatim 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. \endverbatim

\param[out] W \verbatim W is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. \endverbatim

\param[out] Z \verbatim Z is DOUBLE PRECISION 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). If JOBZ = ‘N’, 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 = ‘V’, the exact value of M is not known in advance and an upper bound must be used. Supplying N columns is always safe. \endverbatim

\param[in] LDZ \verbatim LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = ‘V’, LDZ >= max(1,N). \endverbatim

\param[out] ISUPPZ \verbatim ISUPPZ is INTEGER array, dimension ( 2max(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 ) through ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the orthogonal transformations applied by ESMF_DORMTR. Implemented only for RANGE = ‘A’ or ‘I’ and IU - IL = N - 1 \endverbatim

\param[out] WORK \verbatim WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. \endverbatim

\param[in] LWORK \verbatim LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,26N). For optimal efficiency, LWORK >= (NB+6)N, where NB is the max of the blocksize for ESMF_DSYTRD and ESMF_DORMTR returned by ESMF_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 ESMF_XERBLA.

\endverbatim

\param[out] IWORK \verbatim IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. \endverbatim

\param[in] LIWORK \verbatim LIWORK is 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 ESMF_XERBLA.

\endverbatim

\param[out] INFO \verbatim INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: Internal error \endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date June 2016 \ingroup doubleSYeigen \par Contributors:

Inderjit Dhillon, IBM Almaden, USA \n
Osni Marques, LBNL/NERSC, USA \n
Ken Stanley, Computer Science Division, University of
  California at Berkeley, USA \n
Jason Riedy, Computer Science Division, University of
  California at Berkeley, USA \n

Arguments