SRC/dsyevx_2stage.f(3)

📦

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okay so bestie, we're in the LAPACK Library Functions Manual, and the page is SRC/dsyevx_2stage.f(3) — yes, the _2stage one. It's giving “selected symmetric eigenvalues with optional eigenvectors” energy, and we're gonna get through every single arg together. 💅✨

🎀 NAME 🎀

SRC/dsyevx_2stage.f

✨ SYNOPSIS ✨

🌸 Functions/Subroutines 🌸

       subroutine dsyevx_2stage (jobz, range, uplo, n, a, lda, vl, vu, il, iu,
           abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)
            DSYEVX_2STAGE computes the eigenvalues and, optionally, the left
           and/or right eigenvectors for SY matrices

💖 Function/Subroutine Documentation 💖

   subroutine dsyevx_2stage (character jobz, character range, character uplo,
       integer n, double precision, dimension( lda, * ) a, integer lda, double
       precision vl, double precision vu, integer il, integer iu, double
       precision abstol, integer m, double precision, dimension( * ) w, double
       precision, dimension( ldz, * ) z, integer ldz, double precision,
       dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
       integer, dimension( * ) ifail, integer info)
        DSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or
       right eigenvectors for SY matrices

🌟 Purpose 🌟

okay so the glow-up here: DSYEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A using the 2stage technique for the reduction to tridiagonal. You can pick which eigenvalues you want by specifying either a range of values or a range of indices for the desired eigenvalues. That's the whole personality. 🦋

💅 Parameters 💅

JOBZ — CHARACTER*1. = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.
RANGE — 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.
UPLO — CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
N — INTEGER. The order of the matrix A. N >= 0.
A — 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. 🫠
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
VL — 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'.
VU — 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'.
IL — 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'.
IU — 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'.
ABSTOL — 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.

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S').

See by Demmel and Kahan, LAPACK Working Note #3.

M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W — DOUBLE PRECISION array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order.
Z — 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. 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.
LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — INTEGER. The length of the array WORK. LWORK >= 1, when N <= 1; otherwise:
- If JOBZ = 'N' and N > 1, LWORK must be queried.

LWORK = MAX(1, 8*N, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 3*N
          = N*KD + N*max(KD+1,FACTOPTNB)
            + max(2*KD*KD, KD*NTHREADS)
            + (KD+1)*N + 3*N
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.
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 — INTEGER array, dimension (5*N).
IFAIL — INTEGER array, dimension (N). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.
INFO — INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.

👑 Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌈 Further Details 🌈

All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

Definition at line 297 of file dsyevx_2stage.f.

🫶 Author 🫶

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okay bestie, that's the whole page. The footer bundles it up as LAPACK Version 3.12.0, source SRC/dsyevx_2stage.f(3). You ate that. 💖✨

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