heev_2stage(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
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3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

💖 NAME 💖

okay bestie, this is heev_2stage(3) from the Library Functions Manual — the whole {he,sy}ev_2stage eig / QR iteration situation. It’s giving “two-stage eigensolver glow-up” for HE and SY matrices, and honestly? iconic. ✨

🌸 SYNOPSIS 🌸

✨ Functions ✨

subroutine cheev_2stage (jobz, uplo, n, a, lda, w, work, lwork, rwork,
    info)
     CHEEV_2STAGE computes the eigenvalues and, optionally, the left
    and/or right eigenvectors for HE matrices

subroutine dsyev_2stage (jobz, uplo, n, a, lda, w, work, lwork, info)
     DSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or
    right eigenvectors for SY matrices

subroutine ssyev_2stage (jobz, uplo, n, a, lda, w, work, lwork, info)
     SSYEV_2STAGE computes the eigenvalues and, optionally, the left
    and/or right eigenvectors for SY matrices

subroutine zheev_2stage (jobz, uplo, n, a, lda, w, work, lwork, rwork,
    info)
     ZHEEV_2STAGE computes the eigenvalues and, optionally, the left
    and/or right eigenvectors for HE matrices

🦋 Detailed Description 🦋

okay so this is the detailed-description zone, and the function docs start right here. scroll vibes only, bestie 💅

👑 Function Documentation 👑

💜 `cheev_2stage` 💜

subroutine cheev_2stage (character jobz, character uplo, integer n, complex,
    dimension( lda, * ) a, integer lda, real, dimension( * ) w, complex,
    dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer
    info)

CHEEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

✨ Purpose: ✨

CHEEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A using the 2stage technique for the reduction to tridiagonal.

💅 Parameters 💅

JOBZ — JOBZ is CHARACTER*1
- = 'N': Compute eigenvalues only;
- = 'V': Compute eigenvalues and eigenvectors. Not available in this release.
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 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, if JOBZ = 'V', then if INFO = 0, A contains the orthonormal eigenvectors of the matrix A. If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 💀
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
W — W is REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
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, 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.

RWORK — RWORK is REAL array, dimension (max(1, 3*N-2)).
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.

🌟 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 187 of file cheev_2stage.f.

💙 `dsyev_2stage` 💙

subroutine dsyev_2stage (character jobz, character uplo, integer n, double
    precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( * ) w, double precision, dimension( * ) work, integer lwork,
    integer info)

DSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

✨ Purpose: ✨

DSYEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A using the 2stage technique for the reduction to tridiagonal.

💅 Parameters 💅

JOBZ — JOBZ is CHARACTER*1
- = 'N': Compute eigenvalues only;
- = 'V': Compute eigenvalues and eigenvectors. Not available in this release.
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, if JOBZ = 'V', then if INFO = 0, A contains the orthonormal eigenvectors of the matrix A. If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 💀
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
W — W is DOUBLE PRECISION array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
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, dimension) where
      dimension = max(stage1,stage2) + (KD+1)*N + 2*N
                = N*KD + N*max(KD+1,FACTOPTNB)
                  + max(2*KD*KD, KD*NTHREADS)
                  + (KD+1)*N + 2*N

💅 `INFO` 💅

It's INTEGER, and it's basically the final vibe check on the whole operation 💅

= 0 — successful exit, we ate and left no crumbs ✨
< 0 — if INFO = -i, the i-th argument had an illegal value 🙈 (bestie you passed something cursed)
> 0 — if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero 😭 she tried, she really did, but it didn't converge

🫶 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 these iconic papers (yes, we're citing, we're academic, we're that girl 🎀):

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 181 of file dsyev_2stage.f.

🦋 subroutine `ssyev_2stage`

subroutine ssyev_2stage (character jobz, character uplo, integer n, real,
    dimension( lda, * ) a, integer lda, real, dimension( * ) w, real,
    dimension( * ) work, integer lwork, integer info)

SSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices ✨

Single precision besties, this one's for you — the S is SY and the SY is symmetric 💜

🎀 Purpose:

SSYEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A using the 2stage technique for the reduction to tridiagonal. That's the whole flex, no cap 🔥

🌸 Parameters

JOBZ

JOBZ is CHARACTER*1

= 'N' — Compute eigenvalues only;
= 'V' — Compute eigenvalues and eigenvectors. Not available in this release. 😭 (so close, and yet)

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 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, if JOBZ = 'V', then if INFO = 0, A contains the orthonormal eigenvectors of the matrix A.

If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 💀 she's gone, don't come looking

LDA

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

W

W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order. Look at her, all sorted and organized 👑

WORK

WORK is REAL array, dimension 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, dimension) where
        dimension = max(stage1,stage2) + (KD+1)*N + 2*N
                  = N*KD + N*max(KD+1,FACTOPTNB)
                    + max(2*KD*KD, KD*NTHREADS)
                    + (KD+1)*N + 2*N

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. Ngl that's a nice little loophole ✨

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 😭

🫶 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 (same iconic bibliography, she's consistent):

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 181 of file ssyev_2stage.f.

💜 subroutine `zheev_2stage`

subroutine zheev_2stage (character jobz, character uplo, integer n,
    complex*16, dimension( lda, * ) a, integer lda, double precision,
    dimension( * ) w, complex*16, dimension( * ) work, integer lwork, double
    precision, dimension( * ) rwork, integer info)

ZHEEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices 🌈

Complex bestie hours 💅 Z means double complex, HE means Hermitian — not symmetric, Hermitian, the conjugate-transpose queen 👑

🎀 Purpose:

ZHEEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A using the 2stage technique for the reduction to tridiagonal.

🌸 Parameters

JOBZ

JOBZ is CHARACTER*1

= 'N' — Compute eigenvalues only;
= 'V' — Compute eigenvalues and eigenvectors. Not available in this release. 😭

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 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, if JOBZ = 'V', then if INFO = 0, A contains the orthonormal eigenvectors of the matrix A.

If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. RIP to that triangle, she served her purpose 💀

LDA

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

W

W is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order. Note she's DOUBLE PRECISION even though the matrix is COMPLEX*16 — eigenvalues of a Hermitian matrix are real, that's just how she is 🧚

WORK

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

💖 `LWORK`

Okay so LWORK is INTEGER and it’s literally the length of the array WORK, no cap. LWORK >= 1, when N <= 1; otherwise:

If JOBZ = 'N' and N > 1, LWORK must be queried. The formula is:

LWORK = MAX(1, 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

And the little glossary for that formula, bestie:

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 🫠

Query alert 🚨 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.

🦋 `RWORK`

RWORK is DOUBLE PRECISION array, dimension (max(1, 3*N-2)) — that’s the whole mood, periodt.

⭐ `INFO`

INFO is INTEGER and it’s the final vibe check:

= 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 💀

👑 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

Definition at line 187 of file zheev_2stage.f.

🤖 Author

Generated automatically by Doxygen for LAPACK from the source code.

That’s the heev_2stage(3) page, bestie — source/version: LAPACK Version 3.12.0; date: not shown in this footer, so I’m not inventing one 💅✨

📦 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

heev_2stage(3)

💖 NAME 💖

🌸 SYNOPSIS 🌸

✨ Functions ✨

🦋 Detailed Description 🦋

👑 Function Documentation 👑

💜 cheev_2stage 💜

✨ Purpose: ✨

💅 Parameters 💅

🌟 Author 🌟

🔥 Further Details: 🔥

💙 dsyev_2stage 💙

✨ Purpose: ✨

💅 Parameters 💅

💅 INFO 💅

🫶 Author

📚 Further Details:

🦋 subroutine ssyev_2stage

🎀 Purpose:

🌸 Parameters

🫶 Author

📚 Further Details:

💜 subroutine zheev_2stage

🎀 Purpose:

🌸 Parameters

💖 LWORK

🦋 RWORK

⭐ INFO

👑 Author

📚 Further Details:

📌 Definition

🤖 Author

💜 `cheev_2stage` 💜

💙 `dsyev_2stage` 💙

💅 `INFO` 💅

🦋 subroutine `ssyev_2stage`

💜 subroutine `zheev_2stage`

💖 `LWORK`

🦋 `RWORK`

⭐ `INFO`