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

💖 NAME 💖

Okay so the file for this manual page is literally:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zhbevx_2stage.f

and the manual-name energy up top is /home/abui...x_2stage.f(3) — yes bestie, the title is a whole path, and it's a Library Functions Manual situation. No cap, iconic. 💅

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

Here's the one and only star of this part, with every argument kept exactly as-is:

subroutine ZHBEVX_2STAGE (jobz, range, uplo, n, kd, ab, ldab, q, ldq,
    vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork,
    ifail, info)

The description is short and sweet: ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. That's the whole vibe for this page part — eigenvalues first, eigenvectors if you ask nicely, and OTHER matrices doing main character work. 🦋

🦋 Function/Subroutine Documentation 🦋

okay bestie, this is the part where we get into the actual tea — what the routine does, what every single argument wants from you, and who wrote it. buckle up 💅

🎯 Purpose

ZHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. ✨

it's giving "I'm a banded Hermitian matrix and I'm about to get reduced" and honestly? iconic.

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🎀 Parameters

JOBZ

JOBZ is CHARACTER*1

• = 'N' — compute eigenvalues only;
• = 'V' — compute eigenvalues and eigenvectors. Not available in this release. 👀

lowkey important: that 'V' option is literally marked not-available in this release. don't come for me, that's what the source says 🫶

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.

pick a side, bestie 🌸 (just one, and tell us which)

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 COMPLEX*16 array, dimension (LDAB, N)

on entry, the upper or lower triangle of the Hermitian 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. 💀

so fr, if you needed your original AB — copy it first. she's not coming back.

LDAB

LDAB is INTEGER

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

Q

Q is COMPLEX*16 array, dimension (LDQ, N)

• if JOBZ = 'V', the N-by-N unitary 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 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

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'.

yes bestie, VL strictly less than VU. an empty interval is not the vibe 🙈

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 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 AB 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.

yep, the reference title got eaten in the source — that's genuinely how it appears. not my fault, I don't make the rules 💀

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 DOUBLE PRECISION array, dimension (N)

the first M elements contain the selected eigenvalues in ascending order. 🔥 sorted and everything, we love a tidy queen.

Z

Z is COMPLEX*16 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. 😭 plan ahead bestie, that's the whole game here.

LDZ

LDZ is INTEGER

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

WORK

WORK is COMPLEX*16 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, dimension) where
dimension = (2KD+1)*N + KD*NTHREADS
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 sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. ✨

workspace query my beloved — she tells you how much room she needs instead of just yelling at you. main character energy fr.

RWORK

RWORK is DOUBLE PRECISION array, dimension (7*N)

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.

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👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

the academic avengers assembled, we love to see it 🫶

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🧠 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

if you actually wanna understand the 2stage pipeline, these three papers are the assigned reading. no skipping, bestie, this is the syllabus 📚

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Definition at line 323 of file zhbevx_2stage.f.

🎀 Author

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole page, bestie, you made it to the end 🎉💖 this one's from LAPACK Version 3.12.0, and if you ever forget where you found it, the file it lives in is /home/abui...x_2stage.f(3) — go forth and compute those selected eigenvalues, you absolute icon 👑✨