heevx_2stage(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

👑 heevx_2stage — {he,sy}evx_2stage: eig, bisection 💅

okay so bestie, welcome to heevx_2stage(3) from the Library Functions Manual. this is the ✨two-stage✨ glow-up of the classic ?evx eigenvalue hunt, and we're gonna get through it together. grab a boba, we got a lot of parameters. 🧋

✨ NAME ✨

heevx_2stage — {he,sy}evx_2stage: eig, bisection

like, eigenvalues by bisection, but make it 2stage. periodt. 💜

🌸 SYNOPSIS 🌸

🎀 Functions 🎀

  • subroutine cheevx_2stage (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info) — CHEEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
  • subroutine dsyevx_2stage (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info) — DSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine ssyevx_2stage (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info) — SSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine zheevx_2stage (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info) — ZHEEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

bestie notice the pattern: C = complex HE, Z = complex HE, D = double SY, S = single SY. the real ones don't get an rwork and nobody's mad about it. 💅

🦋 Detailed Description 🦋

(this is where the actual lore lives — keep scrolling, the good stuff's right below 🫶)

⭐ Function Documentation ⭐

💖 subroutine cheevx_2stage (character jobz, character range, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, 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) 💖

CHEEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

🌈 Purpose: 🌈

!>
!> CHEEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
!> of a complex Hermitian 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.
!>

so basically: you hand it a complex Hermitian matrix A, it does the 2stage reduction to tridiagonal, and then it picks out exactly the eigenvalues you asked for — either by value range or by index range. iconic behavior honestly. 👑

🍓 Parameters 🍓

JOBZ

!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors.
!>                  Not available in this release.
!>
⚠️ heads up bestie: 'V' is Not available in this release. so it's giving "menu item that's out of stock". don't come for me, that's what the page 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.
!>

note the half-open interval (VL,VU] — square-ish bracket on the right, open on the left. math girls will get it. ✨

UPLO

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

which triangle of A you actually put data in. pick a side and commit, no cap. 💅

N

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

A

!>          A is COMPLEX array, dimension (LDA, N)
!>          On entry, the Hermitian 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, the lower triangle (if UPLO='L') or the upper
!>          triangle (if UPLO='U') of A, including the diagonal, is
!>          destroyed.
!>
💀 the way that A gets destroyed — on exit, whichever triangle you fed it (including the diagonal) is gone. like fully obliterated. don't count on reading A back afterwards, bestie, it's not coming home. 🙈

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= 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'.
!>

IL/IU are the index-range squad — only relevant when RANGE='I'. and yes, if N = 0 it's IL = 1 and IU = 0. quirky but that's the rule. 🦋

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 A 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 convergence vibe check: an approximate eigenvalue counts as converged when it lives in an interval [a,b] with width ≤

ABSTOL + EPS *   max( |a|,|b| )

where EPS is the machine precision. if ABSTOL is ≤ zero, then EPS*|T| slides in to take its place, where |T| is the 1-norm of the tridiagonal matrix you got from reducing A to tridiagonal form.

🔥 the accuracy tea: eigenvalues come out most accurate when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S') — not zero. if this routine comes back with INFO>0 (meaning some eigenvectors didn't converge), try setting ABSTOL to 2*SLAMCH('S'). like, that's the fix, bestie. 📌

and yeah — See by Demmel and Kahan, LAPACK Working Note #3. there's literally a blank where the title should be in the source and i'm not gonna invent one for you. the Demmel & Kahan, LAPACK Working Note #3 part is real though. 🫶

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)
!>          On normal exit, the first M elements contain the selected
!>          eigenvalues in ascending order.
!>

the first M slots of W hold your chosen eigenvalues, in ascending order. sorted for you, no notes. 🎀

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.
!>
⚠️ real talk: you (the user) have to make sure at least max(1,M) columns are supplied in Z. and if RANGE = 'V', the exact M isn't known ahead of time, so you gotta use an upper bound. plan accordingly bestie, don't get caught lacking. 😭

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 (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal 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, 8*N, dimension) where
!>                                   dimension = max(stage1,stage2) + (KD+1)*N + N
!>                                             = N*KD + N*max(KD+1,FACTOPTNB)
!>                                               + max(2*KD*KD, KD*NTHREADS)
!>                                               + (KD+1)*N + N
!>                                   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 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.
!>

okay this is the main character of the parameter list honestly. let me break it down:

  • LWORK >= 1, when N <= 1; otherwise…
  • if JOBZ = 'N' and N > 1, LWORK must be queried:
LWORK = MAX(1, 8*N, dimension) where
          dimension = max(stage1,stage2) + (KD+1)*N + N
                    = N*KD + N*max(KD+1,FACTOPTNB)
                      + max(2*KD*KD, KD*NTHREADS)
                      + (KD+1)*N + N
  • 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.

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

🫶 the lil shortcut: 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. it's giving "just asking, not committing". 💅

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.
!>

the receipts array — it tells you which eigenvectors ghosted you. 🫶

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.
!>

the classic INFO trio: 0 = slay, < 0 = the i-th argument was illegal, > 0 = i eigenvectors failed to converge and their indices are chilling in IFAIL. 🔥

🌟 Author 🌟

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

(academic royalty, all of them 👑)

💜 Further Details: 💜

(...and this is where part 1 leaves off bestie — the details continue in the next part, don't go anywhere 💖)

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 303 of file cheevx_2stage.f.

subroutine dsyevx_2stage 💖

Signature 🎀

   subroutine dsyevx_2stage (character jobz, character range, character uplo,
       integer n, double precision, dimension( lda, * ) a, integer lda, double
       precision vl, double precision vu, integer il, integer iu, double
       precision abstol, integer m, double precision, dimension( * ) w, double
       precision, dimension( ldz, * ) z, integer ldz, double precision,
       dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
       integer, dimension( * ) ifail, integer info)
        DSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or
       right eigenvectors for SY matrices

Purpose 🎯

DSYEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric 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. So yes, bestie, you get to pick your faves — by value range or by index range. 💅

Parameters 💅

  • JOBZ — JOBZ is CHARACTER*1. It decides what you actually get back:
    • 'N': Compute eigenvalues only;
    • 'V': Compute eigenvalues and eigenvectors. Not available in this release. 🚨
  • RANGE — RANGE is CHARACTER*1 and tells the routine which eigenvalues you’re hunting for:
    • '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.
  • A — A is DOUBLE PRECISION 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, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. Destroyed means GONE, bestie — keep a copy if you need A later. 💀
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= 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'.
  • 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 A 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. iykyk, the accuracy lore is real. 🧚

  • 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). On normal exit, the first M elements contain the selected eigenvalues in ascending order. Sorted and cute, as they should be. ✨
  • Z — Z is DOUBLE PRECISION 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. That “upper bound” note is not a drill — give Z enough columns, bestie. 🫶
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal 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, 8*N, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 3*N
          = N*KD + N*max(KD+1,FACTOPTNB)
            + max(2*KD*KD, KD*NTHREADS)
            + (KD+1)*N + 3*N

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 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. Workspace-query era, we love to see it. 🔥
  • 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:

  1. 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
  2. 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
  3. 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 297 of file dsyevx_2stage.f.

ssyevx_2stage 💖 (the selected-eigenvalue era)

subroutine ssyevx_2stage (character jobz, character range, character uplo,
    integer n, real, dimension( lda, * ) a, integer lda, 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)

SSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose ✨

Okay so SSYEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric 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. Very choose-your-own-adventure, very main character energy 💅.

Parameters 🎀

  • JOBZ — CHARACTER*1
    • 'N': Compute eigenvalues only;
    • 'V': Compute eigenvalues and eigenvectors. Not available in this release.
  • RANGE — 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 — CHARACTER*1
    • 'U': Upper triangle of A is stored;
    • 'L': Lower triangle of A is stored.
  • N — INTEGER The order of the matrix A. N >= 0.
  • A — 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, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. Yes, destroyed — like your weekend plans 💀.
  • LDA — INTEGER The leading dimension of the array A. LDA >= max(1,N).
  • VL — 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 — 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 — 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 — 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 — 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 A 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.

  • M — INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W — REAL array, dimension (N) On normal exit, the first M elements contain the selected eigenvalues in ascending order.
  • Z — 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.
  • LDZ — INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — 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, 8*N, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 3*N
          = N*KD + N*max(KD+1,FACTOPTNB)
            + max(2*KD*KD, KD*NTHREADS)
            + (KD+1)*N + 3*N

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 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.

  • IWORK — INTEGER array, dimension (5*N)
  • IFAIL — 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 — 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 🌈

Ngl, if you actually want to understand the 2stage tea, these papers are the whole syllabus 👑 read them, cite them, live them:

  • 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 🔥
Bestie, bookmark these, seriously. Also lol, the page says Definition at line 297 of file ssyevx_2stage.f — yes, the s file, we're in the z section, it's giving crossover episode 🙈 iconic anyway.

🎀 subroutine zheevx_2stage 🎀

subroutine zheevx_2stage (character jobz, character range, character uplo,
    integer n, complex*16, dimension( lda, * ) a, integer lda, double
    precision vl, double precision vu, integer il, integer iu, double
    precision abstol, integer m, double precision, dimension( * ) w,
    complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( *
    ) work, integer lwork, double precision, dimension( * ) rwork, integer,
    dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHEEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices 💜

✨ Purpose ✨

Okay so real talk: ZHEEVX_2STAGE computes selected eigenvalues and, if you want them, eigenvectors of a complex Hermitian matrix A — and it does it using the 2stage technique for the reduction to tridiagonal. You get to pick which eigenvalues you want by giving either a range of values or a range of indices. Choose your own adventure, bestie 🦋

🌸 Parameters 🌸

  • JOBZ — CHARACTER*1, the "how much do you want back" switch:
    • 'N': compute eigenvalues only;
    • 'V': compute eigenvalues and eigenvectors. 🚨 Not available in this release.
  • RANGE — CHARACTER*1, tells us which eigenvalues to go hunting for:
    • '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 — CHARACTER*1, which triangle is doing the work:
    • 'U': upper triangle of A is stored;
    • 'L': lower triangle of A is stored.
  • N — INTEGER, the order of the matrix A. N >= 0.
  • A — COMPLEX*16 array, dimension (LDA, N). On entry, the Hermitian 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, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed 💀 like, gone, don't come looking for it.
  • LDA — INTEGER, the leading dimension of the array A. LDA >= max(1,N).
  • VL — 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 — 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'.
  • IL — 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 — 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 — 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 A 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.

  • M — INTEGER, the total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W — DOUBLE PRECISION array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order ✨
  • Z — 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. Like, you gotta over-provision, babe, no cap.
  • LDZ — INTEGER, the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — 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, 8*N, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + N
          = N*KD + N*max(KD+1,FACTOPTNB)
            + max(2*KD*KD, KD*NTHREADS)
            + (KD+1)*N + N
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 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 classic "just tell me how much room I need" move 💅

  • RWORK — DOUBLE PRECISION array, dimension (7*N).
  • IWORK — INTEGER array, dimension (5*N).
  • IFAIL — 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 — INTEGER, the vibe check at the end:
    • = 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

okay so if you’re obsessed with the 2stage techniques and want all the details, bestie, they’re fully available in these papers — like, the whole lore, no cap:

  • 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 303 of file zheevx_2stage.f. yes, line 303, periodt 💅

👑 AUTHOR 👑

This whole page? Generated automatically by Doxygen for LAPACK from the source code. literally built by the docs machine, and honestly? iconic. ✨🤖

💖 and that’s a wrap on heevx_2stage(3), bestie — LAPACK Version 3.12.0, straight from the source code. mwah, slay, bye 🎀

📦 where this page comes from