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

📦 version 3.12.0, in opensuse-leap 16.0

🌸 `/home/abui...x_2stage.f(3)` 🌸

okay so welcome to this page from the Library Functions Manual, bestie — it's part 1 of 2, and we're getting into it 💅✨

✨ NAME ✨

the full file path is right here, no cap:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/chbevx_2stage.f

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

subroutine CHBEVX_2STAGE — the main character of this page, fr fr. the signature is exactly this:

subroutine CHBEVX_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)

so what does it do? CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 👑 like yes, the page really says OTHER in all caps — iconic, no notes 😭🫶

💫 Function/Subroutine Documentation 💫

   subroutine CHBEVX_2STAGE (character jobz, character range, character uplo,
       integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab,
       complex, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer
       il, integer iu, real abstol, integer m, real, dimension( * ) w, complex,
       dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work,
       integer lwork, real, dimension( * ) rwork, integer, dimension( * )
       iwork, integer, dimension( * ) ifail, integer info)

okay so CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. That’s the whole one-liner, no cap. Main character energy for a band-matrix eigensolver, honestly 💅

🎯 Purpose

This routine computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. You can pick which eigenvalues/eigenvectors you want by specifying either a range of values or a range of indices for the desired eigenvalues. It’s giving “choose your own eigenvalue adventure,” bestie ✨

🧾 Parameters

Every single parameter is here. I’m not skipping a crumb, fr fr.

`JOBZ`

JOBZ is CHARACTER*1. It tells the routine what to compute:

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.

Not available in this release. So if you were about to get hype for 'V', babe, it’s not available here — periodt 😭

`RANGE`

RANGE is CHARACTER*1. It picks the selection mode:

= '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. It says which triangle of A is stored:

= '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 COMPLEX 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 yeah, she gets changed on the way out — noted, bestie 💜

`LDAB`

LDAB is INTEGER.

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

`Q`

Q is COMPLEX 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 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.

The accuracy note is real, bestie — setting ABSTOL to zero is not the move here. 2*SLAMCH('S') is the slay choice when you want max accuracy 💅

`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. Organized, iconic, exactly what we want 🌸

`Z`

Z is COMPLEX 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. Ma’am, please allocate enough columns — this is not a drill 🫶

`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 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.

The workspace-query life is lowkey a whole vibe: LWORK = -1 doesn’t make a big array, it just asks the routine to spill the optimal sizes. Bestie behavior, honestly 😌

`RWORK`

RWORK is REAL 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.

🌈 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 323 of file chbevx_2stage.f.

🌸 Author

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Catch you in the next doc, bestie — this page is LAPACK Version 3.12.0, sourced from /home/abui...x_2stage.f(3). Stay slay, stay convergent 💅✨🫶

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