/home/abui...d_2stage.f(3)

💅 NAME 💅

so bestie, this whole page is literally just the source path of one Fortran file:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ssyevd_2stage.f

like, that's the whole name. it's giving "my name is my address" energy and honestly? iconic. it lives in the LAPACK Library Functions Manual (Version 3.12.0), and we're about to read it together, mwah 🫶

✨ SYNOPSIS ✨

Functions/Subroutines 🦋

subroutine SSYEVD_2STAGE (jobz, uplo, n, a, lda, w, work, lwork, iwork, liwork, info) — SSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

🔥 Function/Subroutine Documentation 🔥

`subroutine SSYEVD_2STAGE (character jobz, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) w, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)`

SSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices ✨

🎀 Purpose:

!>
!> SSYEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a
!> real symmetric matrix A using the 2stage technique for
!> the reduction to tridiagonal. If eigenvectors are desired, it uses a
!> divide and conquer algorithm.
!>
!>

okay so real talk — it finds all the eigenvalues (and eigenvectors if you ask nicely) of a real symmetric matrix A, but it does the tridiagonal reduction in two stages (hence the name, no cap). if you want eigenvectors, it pulls out the divide and conquer algorithm. fancy 💅

💖 Parameters 💖

JOBZ

!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors.
!>                  Not available in this release.
!>

'N' — eigenvalues only, minimal effort queen
'V' — eigenvalues and eigenvectors… except it says "Not available in this release." so like, look but don't touch bestie 😭

UPLO

!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored.
!>

'U' — you're an upper triangle girlie
'L' — you're a lower triangle girlie

pick a side, we only read half the matrix anyway 💀

N

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!>

the order of matrix A, and it must be N >= 0. zero is allowed, don't come for her 🌸

A

!>          A is REAL 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, if JOBZ = 'V', then if INFO = 0, A contains the
!>          orthonormal eigenvectors of the matrix A.
!>          If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
!>          or the upper triangle (if UPLO='U') of A, including the
!>          diagonal, is destroyed.
!>

so it's a REAL array, dimension (LDA, N). on entry it's your symmetric matrix — UPLO = 'U' means the leading N-by-N upper triangular part holds it, UPLO = 'L' means the leading N-by-N lower triangular part does.

on exit: if JOBZ = 'V' and INFO = 0, A holds the orthonormal eigenvectors, slay.

⚠️ but if JOBZ = 'N', on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A — including the diagonal — is destroyed. gone. reduced to atoms. do not expect your matrix back, bestie 😭

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>

leading dimension of A, and LDA >= max(1,N). that's the whole vibe.

W

!>          W is REAL array, dimension (N)
!>          If INFO = 0, the eigenvalues in ascending order.
!>

REAL array, dimension (N). if INFO = 0, you get the eigenvalues in ascending order — organized, sorted, main character behavior 👑

WORK

!>          WORK is REAL array,
!>                                         dimension (LWORK)
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

REAL array, dimension (LWORK). on exit, if INFO = 0, WORK(1) gives you the optimal LWORK — like a little receipt for how much workspace you should've brought 🧾

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK.
!>          If N <= 1,               LWORK must be at least 1.
!>          If JOBZ = 'N' and N > 1, LWORK must be queried.
!>                                   LWORK = MAX(1, dimension) where
!>                                   dimension = max(stage1,stage2) + (KD+1)*N + 2*N+1
!>                                             = N*KD + N*max(KD+1,FACTOPTNB)
!>                                               + max(2*KD*KD, KD*NTHREADS)
!>                                               + (KD+1)*N + 2*N+1
!>                                   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 at least
!>                                                1 + 6*N + 2*N**2.
!>
!>          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.
!>

okay this is the big one, buckle up 🎢

dimension of the array WORK
if N <= 1, LWORK must be at least 1
if JOBZ = 'N' and N > 1, LWORK must be queried (yes, really, go ask):

LWORK = MAX(1, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 2*N+1
          = N*KD + N*max(KD+1,FACTOPTNB)
            + max(2*KD*KD, KD*NTHREADS)
            + (KD+1)*N + 2*N+1

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 at least 1 + 6*N + 2*N**2
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. so you can probe with -1 and nobody yells at you 🫶

IWORK

!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
!>

INTEGER array, dimension (MAX(1,LIWORK)). on exit, if INFO = 0, IWORK(1) hands you the optimal LIWORK.

LIWORK

!>          LIWORK is INTEGER
!>          The dimension of the array IWORK.
!>          If N <= 1,                LIWORK must be at least 1.
!>          If JOBZ  = 'N' and N > 1, LIWORK must be at least 1.
!>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*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.
!>

the integer twin of LWORK 🧚

dimension of the array IWORK
if N <= 1, LIWORK must be at least 1
if JOBZ = 'N' and N > 1, LIWORK must be at least 1
if JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N
if LIWORK = -1, a workspace query is assumed — again it only computes the optimal sizes of WORK and IWORK, returns them in the first entries, and no error message related to LWORK or LIWORK is issued by XERBLA

INFO

!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  if INFO = i and JOBZ = 'N', then the algorithm failed
!>                to converge; i off-diagonal elements of an intermediate
!>                tridiagonal form did not converge to zero;
!>                if INFO = i and JOBZ = 'V', then the algorithm failed
!>                to compute an eigenvalue while working on the submatrix
!>                lying in rows and columns INFO/(N+1) through
!>                mod(INFO,N+1).
!>

the vibe check at the end 🚦

= 0 — successful exit, we ate 💅
< 0 — if INFO = -i, the i-th argument had an illegal value (you passed something cursed)
> 0 — if INFO = i and JOBZ = 'N', the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero 😭
- if INFO = i and JOBZ = 'V', the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1)

💀 so basically a positive INFO means "it did not cook." read it carefully, it's telling you exactly which part flopped.

👑 Author 👑

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

the whole squad showed up, fr fr 🫶

🌈 Contributors: 🌈

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified description of INFO. Sven, 16 Feb 05.

shoutout Sven for fixing the INFO description, that's the real hero arc ⭐

🦋 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
!>
!>

three whole papers if you wanna go full academia mode 📚 — the 2stage lore is right there, iykyk.

🍓 Definition at line 219 of file `ssyevd_2stage.f`. 🍓

💜 Author (the other one) 💜

Generated automatically by Doxygen for LAPACK from the source code. yes, a robot wrote this part, and it ate.

okay that's the whole page bestie, you're officially educated ✨ signing off with the footer deets: this is LAPACK, Version 3.12.0, and the whole thing is filed under /home/abui...d_2stage.f(3). go forth and compute your eigenvalues, ily, bye 💖🎀🦋

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