SRC/dsbev_2stage.f(3)

📦

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💅 SRC/dsbev_2stage.f(3) — like, the Library Functions Manual, bestie

Okay so this is a page straight out of the LAPACK library docs (the Library Functions Manual vibe, yk), and it's all about one lil subroutine called dsbev_2stage. Buckle up 🎀

✨ NAME ✨

SRC/dsbev_2stage.f — that's the file, that's the whole name section, no notes 💅

🌸 SYNOPSIS 🌸

Functions/Subroutines

   subroutine dsbev_2stage (jobz, uplo, n, kd, ab, ldab, w, z, ldz, work,
       lwork, info)

DSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 👑

🦋 Function/Subroutine Documentation 🦋

   subroutine dsbev_2stage (character jobz, character uplo, integer n, integer
       kd, double precision, dimension( ldab, * ) ab, integer ldab, double
       precision, dimension( * ) w, double precision, dimension( ldz, * ) z,
       integer ldz, double precision, dimension( * ) work, integer lwork,
       integer info)

DSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices ✨ (yes, the description really does say "OTHER matrices", that's canon, don't come for me 😭)

🎀 Purpose 🎀

!> !> DSBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of !> a real symmetric band matrix A using the 2stage technique for !> the reduction to tridiagonal. !>

So like — real symmetric band matrix A, all the eigenvalues (and eigenvectors if you want them, which, spoiler, you can't have them in this release 👀), and it gets there via the 2stage technique for the reduction to tridiagonal. iconic behavior honestly 💅

🌈 Parameters 🌈

`JOBZ`

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

Bestie, 'V' is not available in this release — so rn it's basically 'N' or nothing 🙈 don't say I didn't warn you.

`UPLO`

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

Pick a triangle, any triangle 💖

`N`

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

The order of A, and it's gotta be N >= 0 — no negative orders, that's not a vibe.

`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. !>

Superdiagonals if UPLO = 'U', subdiagonals if UPLO = 'L'. And KD >= 0, periodt.

`AB`

!> AB is DOUBLE PRECISION 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. !>

Okay so here's where it gets real, and also where the indexing lives, so I'm giving it to you verbatim because the math facts do NOT get to be vibed away 🫶

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 entry: the upper-or-lower triangle of the symmetric band matrix A, chilling in the first KD+1 rows of the array.

⚠️ On exit, AB gets overwritten — the values generated during the reduction to tridiagonal form take its place. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T come back in rows KD and KD+1 of AB; if UPLO = 'L', the diagonal and first subdiagonal of T come back in the first two rows of AB. So like, don't get attached to your input 😭 it's giving transformation arc.

`LDAB`

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

Leading dimension of AB, and it's LDAB >= KD + 1 — non-negotiable 💅

`W`

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

If INFO = 0, congrats, you get the eigenvalues in ascending order ⭐

`Z`

!> Z is DOUBLE PRECISION array, dimension (LDZ, N) !> If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal !> eigenvectors of the matrix A, with the i-th column of Z !> holding the eigenvector associated with W(i). !> If JOBZ = 'N', then Z is not referenced. !>

If JOBZ = 'V' (which, again, not available in this release 🙈) and INFO = 0, then Z holds the orthonormal eigenvectors of A, with the i-th column of Z holding the eigenvector that goes with W(i). If JOBZ = 'N', then Z is literally not referenced — she's off duty, don't even look at her.

`LDZ`

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

Leading dimension of Z. Base requirement: LDZ >= 1. And if JOBZ = 'V', then LDZ >= max(1,N) 👑

`WORK`

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

On exit, if INFO = 0, WORK(1) hands you the optimal LWORK. Iconic little self-report ✨

`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 + 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. !>

Length of the WORK array. LWORK >= 1 when N <= 1. Otherwise:

LWORK = MAX(1, dimension) where
dimension = (2KD+1)*N + KD*NTHREADS + N

where KD is the size of the band, and NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1 🧵

If JOBZ = 'N' and N > 1, LWORK must be queried. If JOBZ = 'V' and N > 1, LWORK must be queried too — but that's Not yet available 💀

And the big one:

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. Sneaky but legal 😌

`INFO`

!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, the algorithm failed to converge; i !> off-diagonal elements of an intermediate tridiagonal !> form did not converge to zero. !>

= 0: successful exit, we ate 💅
< 0: if INFO = -i, the i-th argument had an illegal value 🙈
> 0: if INFO = i, the algorithm failed to converge — i off-diagonal elements of an intermediate tridiagonal form did not converge to zero 😭

💜 Author 💜

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

(the whole squad 🫶)

⭐ 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, bestie — if you wanna go deep on 2stage techniques, the receipts are RIGHT there 📚✨

🧚 Definition location 🧚

Definition at line 202 of file dsbev_2stage.f. That's where she lives 📍

🌟 Author 🌟

Generated automatically by Doxygen for LAPACK from the source code. (Not a human, just a very diligent little robot 🤖💖)

mwah 💋 that's the whole page — LAPACK, Version 3.12.0, and the file timestamp is SRC/dsbev_2stage.f(3). Go forth and compute your eigenvalues, you gorgeous nerd 🦋💜🎀

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