/home/abui...x_2stage.f(3)

📦 version 3.12.0, in opensuse-leap 16.0

okay bestie, welcome to /home/abui...x_2stage.f(3) from the Library Functions Manual — manual name folded in near the top, no cap. This is a LAPACK moment, so let’s get into it. 💖

✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ssbevx_2stage.f

That’s the whole name, literally that whole path, periodt. 🫶

🦋 SYNOPSIS 🦋

🎀 Functions/Subroutines 🎀

Here’s the main-character signature, copied exactly so nobody comes for me:

       subroutine SSBEVX_2STAGE (jobz, range, uplo, n, kd, ab, ldab, q, ldq,
           vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail,
           info)

And what does this icon do? SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. 💅✨

🎀 Function/Subroutine Documentation 🎀

okay bestie, this is the actual subroutine, like the whole entire thing, right here:

subroutine SSBEVX_2STAGE (character jobz, character range, character uplo,
    integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab,
    real, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il,
    integer iu, real abstol, integer m, real, dimension( * ) w, real,
    dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer
    lwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail,
    integer info)

and like, what does it even do? SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. that's the whole vibe. ✨

🌸 Purpose 🌸

so listen 💖 — SSBEVX_2STAGE computes selected eigenvalues and, if you ask nicely, eigenvectors of a real symmetric band matrix A, using the 2stage technique for the reduction to tridiagonal. and because it's a selected situation, you get to pick which eigenvalues you want by specifying either a range of values or a range of indices. choose your own adventure, eigenvalue edition. 👑

🌈 Parameters 🌈

buckle up bestie, there's a lot of them, but every single one is important, no skipping. 🫶

💅 JOBZ

JOBZ is CHARACTER*1

  • = 'N': Compute eigenvalues only;
  • = 'V': Compute eigenvalues and eigenvectors. Not available in this release.
ngl, 'V' is lowkey not available in this release, so don't come for me when it doesn't do the thing. 😭

🦋 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.

⭐ UPLO

UPLO is CHARACTER*1

  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.

🎀 N

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

🍓 KD

KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.

💜 AB

AB is REAL array, dimension (LDAB, N)

On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.

she gets overwritten on exit, bestie. your original data is gone-gone. 💀

🧚 LDAB

LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.

🌸 Q

Q is REAL array, dimension (LDQ, N)

  • If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction to tridiagonal form.
  • If JOBZ = 'N', the array Q is not referenced.

🌈 LDQ

LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).

✨ 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'.

✨ 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'.

🎀 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'.

🎀 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'.

🔥 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 AB to tridiagonal form.

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

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

ok so the iconic accurate move is 2*SLAMCH('S') and not zero. zero is giving nothing. do it right. 💅

💖 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.

⭐ W

W is REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.

🦋 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).
  • 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. like, you have to guess big enough, bestie. no cap. 🙈

🧚 LDZ

LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

🌸 WORK

WORK is REAL array, dimension (LWORK)

🔥 LWORK

LWORK is 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, 7*N, dimension) where
                         dimension = (2KD+1)*N + KD*NTHREADS + 2*N
                         where KD is the size of the band.
                         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.

the -1 thing is the workspace-query slay ✨ — you ask, it tells you how much room it needs, and XERBLA stays quiet about it. love that for us.

🎀 IWORK

IWORK is INTEGER array, dimension (5*N)

💜 IFAIL

IFAIL is 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

INFO is 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: 🧚

ok so if you're obsessed (and honestly same) and you want all the details about the 2stage techniques, they're all available in these absolute bangers 📚:

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

✨ and that's the whole thing — Definition at line 319 of file ssbevx_2stage.f. ✨

💫 Author 💫

Generated automatically by Doxygen for LAPACK from the source code.

and that's a wrap bestie, we did it 🫶💖 this whole glow-up is brought to you by LAPACK Version 3.12.0, served fresh from /home/abui...x_2stage.f(3) — go forth and compute, you absolute icon. 👑✨

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