SRC/cheevx_2stage.f(3)

📦

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💖 SRC/cheevx_2stage.f 💖

okay bestie, welcome to the LAPACK Library Functions Manual, and strap in, because this one is a baddie. It's a Fortran subroutine that does eigenvalues two-stage style and honestly? she's iconic. 🎀

✨ NAME ✨

SRC/cheevx_2stage.f — that's the file, that's the vibe, no notes. 💅

🌸 SYNOPSIS 🌸

Functions/Subroutines

So the whole function signature is long, like, unreasonably long, but we're not skipping a single character of it because every argument matters:

subroutine cheevx_2stage (jobz, range, uplo, n, a, lda, vl, vu, il, iu,
    abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)

and what she does is:

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

like, that's the whole personality. 💜

🦋 Function/Subroutine Documentation 🦋

Here's the full signature with all the types spelled out, character for character:

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 yeah — she computes selected eigenvalues (not necessarily all of them, you get to pick 😌) plus optionally the eigenvectors, of a complex Hermitian matrix A, and she does it with the 2stage technique for the reduction to tridiagonal. Obsessed. ✨

👑 Parameters 👑

JOBZ

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

⚠️ heads up: 'V' (eigenvalues and eigenvectors) is not available in this release. so like, don't come for her when it doesn't work, she told you. 💀

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.
!>

three modes, pick your fighter:

'A' → all eigenvalues, the whole buffet 🍓
'V' → everything in the half-open interval (VL,VU] (note the half-open, bestie, that bracket is doing WORK)
'I' → the IL-th through IU-th eigenvalues

UPLO

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

which triangle of A you're storing. upper or lower, choose wisely. 🦋

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.
!>

😭 warning: on exit, that triangle of A — including the diagonal — is destroyed. gone. reduced to atoms. back up your matrix if you needed it, fr fr.

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'.
!>

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.
!>

so the convergence test is literally the interval width ABSTOL + EPS * max( |a|,|b| ) where EPS is machine precision. if ABSTOL is <= 0, then EPS*|T| steps in instead, where |T| is the 1-norm of the tridiagonal matrix from reducing A to tridiagonal form.

💡 patch notes from the devs: eigenvalues come out most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. and if INFO>0 (some eigenvectors didn't converge), try setting ABSTOL to 2*SLAMCH('S'). iykyk. 🧚

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.
!>

ascending order, always, she's organized like that. 🌸

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.
!>

📌 the note is important, bestie: you have to make sure at least max(1,M) columns exist in Z. and if RANGE = 'V' you can't even know M in advance, so use an upper bound. main character responsibility. 🫶

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 big one so let's break it down:

LWORK >= 1 when N <= 1; otherwise...
if JOBZ = 'N' and N > 1, LWORK must be queried, and 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 with KD = blocking size of the reduction, FACTOPTNB = the blocking used by the QR or LQ algorithm (usually FACTOPTNB=128 is a good choice), and NTHREADS = number of threads used when openMP compilation is enabled, otherwise =1.
if JOBZ = 'V' and N > 1, LWORK must be queried — not yet available 😭
and if LWORK = -1, it's a workspace query: the routine only computes the optimal size of the WORK array, drops it as the first entry of WORK, and XERBLA stays quiet about LWORK. no error message, no drama.

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.
!>

so INFO = 0 slay, INFO < 0 means argument -i was illegal 🙈, and INFO > 0 means i eigenvectors didn't converge and their indices are chilling in IFAIL.

🌈 Author 🌈

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

the whole squad, honestly. 🫶

⭐ 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 papers, three links, all the 2stage lore. go read them, i believe in you. 📚✨

Definition

Definition at line 303 of file cheevx_2stage.f.

🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole page, bestie!! 💖 this beauty is from LAPACK, Version 3.12.0, and it lives at SRC/cheevx_2stage.f(3). go forth and compute those Hermitian eigenvalues, you've got this. 🦋🌸👑

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms