hetrd_2stage(3)

📦

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🦋 `hetrd_2stage(3)` 🦋

okay bestie, so this is hetrd_2stage from the Library Functions Manual, and it's literally the glow-up page for the {he,sy}trd_2stage crew: reduction to tridiagonal, 2-stage. it's a long one so we're doing it in parts — buckle up 💅

✨ NAME ✨

hetrd_2stage — {he,sy}trd_2stage: reduction to tridiagonal, 2-stage

🌸 SYNOPSIS 🌸

🎀 Functions 🎀

   subroutine chetrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2,
       lhous2, work, lwork, info)
       CHETRD_2STAGE
   subroutine dsytrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2,
       lhous2, work, lwork, info)
       DSYTRD_2STAGE
   subroutine ssytrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2,
       lhous2, work, lwork, info)
       SSYTRD_2STAGE
   subroutine zhetrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2,
       lhous2, work, lwork, info)
       ZHETRD_2STAGE

four icons, one vibe check 🫶

💖 Detailed Description 💖

like, that's the whole overview — every deets lives down in the Function Documentation. keep scrolling, we got you.

👑 Function Documentation 👑

🌟 `subroutine chetrd_2stage` 🌟

subroutine chetrd_2stage (character vect, character uplo, integer n,
    complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d,
    real, dimension( * ) e, complex, dimension( * ) tau, complex, dimension(
    * ) hous2, integer lhous2, complex, dimension( * ) work, integer lwork,
    integer info)

CHETRD_2STAGE — main character energy, no cap 💜

🧚 Purpose:

!>
!> CHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric
!> tridiagonal form T by a unitary similarity transformation:
!> Q1**H Q2**H* A * Q2 * Q1 = T.
!>

so it takes your complex Hermitian A and turns it into a real symmetric tridiagonal T — iconic behavior, honestly. the transformation is a unitary similarity one, and the whole thing goes Q1**H Q2**H* A * Q2 * Q1 = T 🌀

🍓 Parameters

VECT — VECT is CHARACTER*1
- = 'N': No need for the Housholder representation, in particular for the second stage (Band to tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
- = 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then LHOUS2 is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).

ok so the 'V' option? it's literally not available in this release, bestie. don't come for me, I'm just the messenger 😭

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, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
- On exit, if UPLO = 'U', the band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array TAU, represent the unitary matrix Q1 as a product of elementary reflectors; if UPLO = 'L', the diagonal and band subdiagonal of A are over- written by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array TAU, represent the unitary matrix Q1 as a product of elementary reflectors. See Further Details.

heads up: on exit your A gets rewritten, like fully repurposed. AB and TAU and Q1 all move in. so don't act surprised when your original matrix is gone 💀 See Further Details.

LDA — LDA is INTEGER — The leading dimension of the array A. LDA >= max(1,N).
D — D is REAL array, dimension (N) — The diagonal elements of the tridiagonal matrix T. the main diagonal bestie, the moment ✨
E — E is REAL array, dimension (N-1) — The off-diagonal elements of the tridiagonal matrix T. yes it's N-1, that's not a typo, that's just how off-diagonals roll 🦋
TAU — TAU is COMPLEX array, dimension (N-KD) — The scalar factors of the elementary reflectors of the first stage (see Further Details). — note the N-KD size, periodt.
HOUS2 — HOUS2 is COMPLEX array, dimension (MAX(1,LHOUS2)) — Stores the Householder representation of the stage2 band to tridiagonal. stage 2's whole personality lives in here 🎀
LHOUS2 — LHOUS2 is INTEGER — The dimension of the array HOUS2. LHOUS2 >= 1.
- If LWORK = -1, or LHOUS2=-1, then a query is assumed; the routine only calculates the optimal size of the HOUS2 array, returns this value as the first entry of the HOUS2 array, and no error message related to LHOUS2 is issued by XERBLA.
- If VECT='N', LHOUS2 = max(1, 4*n);
- if VECT='V', option not yet available.
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)) — On exit, if INFO = 0, WORK(1) returns the optimal LWORK. so peep WORK(1) on the way out, it's telling you the tea 💅
LWORK — LWORK is INTEGER — The dimension of the array WORK.
- If N = 0, LWORK >= 1, else LWORK = MAX(1, dimension).
- If LWORK = -1, or LHOUS2 = -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.
- LWORK = MAX(1, dimension) where

dimension   = max(stage1,stage2) + (KD+1)*N
            = N*KD + N*max(KD+1,FACTOPTNB)
              + max(2*KD*KD, KD*NTHREADS)
              + (KD+1)*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.

ok that formula is giving very much "trust me bestie" but it's all there fr — KD, FACTOPTNB, NTHREADS, the whole squad 🌈

INFO — INFO is INTEGER
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value

so if INFO comes back negative, the i-th argument was messy and got called out. i-th argument, not vibe, keep up 💀

🌸 Author

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

🦋 Further Details:

!>
!>  Implemented by Azzam Haidar.
!>
!>  All details are available on technical report, SC11, SC13 papers.
!>
!>  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 227 of file chetrd_2stage.f. 📍

🌟 `subroutine dsytrd_2stage` 🌟

subroutine dsytrd_2stage (character vect, character uplo, integer n, double
    precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( * ) d, double precision, dimension( * ) e, double precision,
    dimension( * ) tau, double precision, dimension( * ) hous2, integer
    lhous2, double precision, dimension( * ) work, integer lwork, integer
    info)

DSYTRD_2STAGE — the double precision sibling, equally iconic 🔥

🧚 Purpose:

!>
!> DSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric
!> tridiagonal form T by a orthogonal similarity transformation:
!> Q1**T Q2**T* A * Q2 * Q1 = T.
!>

real symmetric A in, real symmetric tridiagonal T out, orthogonal similarity all the way: Q1**T Q2**T* A * Q2 * Q1 = T 💫

🍓 Parameters

VECT — VECT is CHARACTER*1
- = 'N': No need for the Housholder representation, in particular for the second stage (Band to tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
- = 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then LHOUS2 is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).

again — 'V' is (NOT AVAILABLE IN THIS RELEASE). 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.
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, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
- On exit, if UPLO = 'U', the band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array TAU, represent the orthogonal matrix Q1 as a product of elementary reflectors; if UPLO = 'L', the diagonal and band subdiagonal of A are over- written by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array TAU, represent the orthogonal matrix Q1 as a product of elementary reflectors. See Further Details.

same deal as before: on exit, A is not your A anymore. AB and TAU and Q1 took over. See Further Details. 🫶

LDA — LDA is INTEGER — The leading dimension of the array A. LDA >= max(1,N).
D — (the next part picks up right here ✨)
D — DOUBLE PRECISION array, dimension (N). The diagonal elements of the tridiagonal matrix T. That's the main-character diagonal right there. ✨
E — DOUBLE PRECISION array, dimension (N-1). The off-diagonal elements of the tridiagonal matrix T. The sidekicks, the bits just off the main axis, we love them. 💫
TAU — DOUBLE PRECISION array, dimension (N-KD). The scalar factors of the elementary reflectors of the first stage — peep the Further Details for the full lore. 🎀
HOUS2 — DOUBLE PRECISION array, dimension (MAX(1,LHOUS2)). Stores the Householder representation of the stage2 band-to-tridiagonal. It's the storage closet for Q2's reflectors, basically. 🧚
LHOUS2 — INTEGER. The dimension of the array HOUS2. LHOUS2 >= 1. Ngl, careful with this one:

If LWORK = -1, or LHOUS2 = -1, then a query is assumed — the routine only calculates the optimal size of the HOUS2 array, returns this value as the first entry of the HOUS2 array, and no error message related to LHOUS2 is issued by XERBLA. So like, it won't yell at you, but that's because you asked for a query, not because you're safe. 🙈

If VECT='N', LHOUS2 = max(1, 4*n); if VECT='V', option not yet available. 💀

WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. Free intel. 🫶
LWORK — INTEGER. The dimension of the array WORK. If N = 0, LWORK >= 1, else LWORK = MAX(1, dimension).

If LWORK = -1, or LHOUS2 = -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. Same deal as above — query mode, no scolding. 🦋

LWORK = MAX(1, dimension) where

dimension   = max(stage1,stage2) + (KD+1)*N
            = N*KD + N*max(KD+1,FACTOPTNB)
              + max(2*KD*KD, KD*NTHREADS)
              + (KD+1)*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 — and NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. So yes, the formula is giving final boss of buffer sizing energy, but every term is right there. 🔥

INFO — INTEGER.
- = 0: successful exit. Slay. 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value. Not slay. 💀

👑 Author (yes, all of them, they all showed up)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🎓 Further Details

Implemented by Azzam Haidar. Legend behavior, honestly.

All details are available on technical report, SC11, SC13 papers.

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 227 of file dsytrd_2stage.f.

🌸 subroutine ssytrd_2stage 🌸

subroutine ssytrd_2stage (character vect, character uplo, integer n, real,
    dimension( lda, * ) a, integer lda, real, dimension( * ) d, real,
    dimension( * ) e, real, dimension( * ) tau, real, dimension( * ) hous2,
    integer lhous2, real, dimension( * ) work, integer lwork, integer info)

SSYTRD_2STAGE — ok bestie, this is the same routine but in single precision REAL instead of DOUBLE PRECISION. Everything you just read? Same soul, different word size. 💅

✨ Purpose ✨

SSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric tridiagonal form T by a orthogonal similarity transformation:

Q1**T Q2**T* A * Q2 * Q1 = T.

Two stages, two Q's, one very iconic tridiagonal T at the end. 🦋

🎀 Parameters 🎀

VECT — CHARACTER*1.
- = 'N': No need for the Housholder representation, in particular for the second stage (Band to tridiagonal) and thus LHOUS2 is of size max(1, 4*N). Keep it light, keep it simple. ⭐
- = 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then LHOUS2 is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE). Yeah. Not yet. Don't come for it. 😭
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. Zero is valid, zero is fine, zero is a whole mood. 🍓
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, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. So like, only one half is canon, the other half is ignored. 🙈

On exit, if UPLO = 'U', the band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array TAU, represent the orthogonal matrix Q1 as a product of elementary reflectors; if UPLO = 'L', the diagonal and band subdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array TAU, represent the orthogonal matrix Q1 as a product of elementary reflectors. See Further Details.

Heads up: your input matrix does not survive this. It gets overwritten with AB bits and Q1's reflectors. That's the whole point, but don't say nobody warned you. 💀

LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). Non-negotiable. 💅
D — REAL array, dimension (N). The diagonal elements of the tridiagonal matrix T. 🌟
E — REAL array, dimension (N-1). The off-diagonal elements of the tridiagonal matrix T. ✨
TAU — REAL array, dimension (N-KD). The scalar factors of the elementary reflectors of the first stage (see Further Details). 🎀
HOUS2 — REAL array, dimension (MAX(1,LHOUS2)). Stores the Householder representation of the stage2 band to tridiagonal. 🧚
LHOUS2 — INTEGER. The dimension of the array HOUS2. LHOUS2 >= 1.

If LWORK = -1, or LHOUS2 = -1, then a query is assumed; the routine only calculates the optimal size of the HOUS2 array, returns this value as the first entry of the HOUS2 array, and no error message related to LHOUS2 is issued by XERBLA. Query mode = XERBLA stays quiet. Do not mistake that for approval. 🫶

If VECT='N', LHOUS2 = max(1, 4*n); if VECT='V', option not yet available. 😭

WORK — REAL array, dimension (LWORK). The scratch space. Give it what it asks for. 🍓
LWORK — INTEGER. The dimension of the array WORK. If N = 0, LWORK >= 1, else LWORK = MAX(1, dimension).

LWORK = MAX(1, dimension) where

dimension   = max(stage1,stage2) + (KD+1)*N
            = N*KD + N*max(KD+1,FACTOPTNB)
              + max(2*KD*KD, KD*NTHREADS)
              + (KD+1)*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 — and NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. Yes it's the same formula. Yes it still slaps. 🔥

INFO — INTEGER.
- = 0: successful exit. We love a clean exit. 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value. Go count your arguments, bestie. 💀

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🎓 Further Details

(hang tight — this is where the page hands the mic to the next chunk, the whole scholarly bibliography flex continues over there 💜)

Implemented by Azzam Haidar — periodt. All the deets are available on the technical report, SC11, SC13 papers, and the receipts are literally right here:

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 226 of file ssytrd_2stage.f.

Okay bestie, next up is the zhetrd_2stage subroutine — say hi to the icon 💅

subroutine zhetrd_2stage (character vect, character uplo, integer n,
    complex*16, dimension( lda, * ) a, integer lda, double precision,
    dimension( * ) d, double precision, dimension( * ) e, complex*16,
    dimension( * ) tau, complex*16, dimension( * ) hous2, integer lhous2,
    complex*16, dimension( * ) work, integer lwork, integer info)

This one is called ZHETRD_2STAGE, and she is serving ✨

💖 Purpose 💖

ZHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q1**H Q2**H* A * Q2 * Q1 = T.

🎀 Parameters 🎀

VECT — VECT is CHARACTER*1
- = 'N': No need for the Housholder representation, in particular for the second stage (Band to tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
- = 'V': the Householder representation is needed to either generate Q1 Q2 or to apply Q1 Q2, then LHOUS2 is to be queried and computed. (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, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if UPLO = 'U', the band superdiagonal of A are overwritten by the corresponding elements of the internal band-diagonal matrix AB, and the elements above the KD superdiagonal, with the array TAU, represent the unitary matrix Q1 as a product of elementary reflectors; if UPLO = 'L', the diagonal and band subdiagonal of A are over-written by the corresponding elements of the internal band-diagonal matrix AB, and the elements below the KD subdiagonal, with the array TAU, represent the unitary matrix Q1 as a product of elementary reflectors. See Further Details.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
D — D is DOUBLE PRECISION array, dimension (N). The diagonal elements of the tridiagonal matrix T.
E — E is DOUBLE PRECISION array, dimension (N-1). The off-diagonal elements of the tridiagonal matrix T.
TAU — TAU is COMPLEX*16 array, dimension (N-KD). The scalar factors of the elementary reflectors of the first stage (see Further Details).
HOUS2 — HOUS2 is COMPLEX*16 array, dimension (MAX(1,LHOUS2)). Stores the Householder representation of the stage2 band to tridiagonal.
LHOUS2 — LHOUS2 is INTEGER. The dimension of the array HOUS2. LHOUS2 >= 1.

If LWORK = -1, or LHOUS2 = -1, then a query is assumed; the routine only calculates the optimal size of the HOUS2 array, returns this value as the first entry of the HOUS2 array, and no error message related to LHOUS2 is issued by XERBLA. If VECT='N', LHOUS2 = max(1, 4*n); if VECT='V', option not yet available.

WORK — WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER. The dimension of the array WORK. If N = 0, LWORK >= 1, else LWORK = MAX(1, dimension).

          dimension   = max(stage1,stage2) + (KD+1)*N
                      = N*KD + N*max(KD+1,FACTOPTNB)
                        + max(2*KD*KD, KD*NTHREADS)
                        + (KD+1)*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, and NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.

INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

👑 Author 👑

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

✨ Further Details ✨

Implemented by Azzam Haidar. All details are available on technical report, SC11, SC13 papers.

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 227 of file zhetrd_2stage.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

Okay that's the whole page, bestie — this cutie is LAPACK Version 3.12.0 and the hetrd_2stage(3) tag is holding it down. No cap, periodt. 💅✨

📦 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

hetrd_2stage(3)

🦋 hetrd_2stage(3) 🦋

✨ NAME ✨

🌸 SYNOPSIS 🌸

🎀 Functions 🎀

💖 Detailed Description 💖

👑 Function Documentation 👑

🌟 subroutine chetrd_2stage 🌟

🧚 Purpose:

🍓 Parameters

🌸 Author

🦋 Further Details:

🌟 subroutine dsytrd_2stage 🌟

🧚 Purpose:

🍓 Parameters

👑 Author (yes, all of them, they all showed up)

🎓 Further Details

🌸 subroutine ssytrd_2stage 🌸

✨ Purpose ✨

🎀 Parameters 🎀

👑 Author

🎓 Further Details

💖 Purpose 💖

🎀 Parameters 🎀

👑 Author 👑

✨ Further Details ✨

👑 Author 👑

🦋 `hetrd_2stage(3)` 🦋

🌟 `subroutine chetrd_2stage` 🌟

🌟 `subroutine dsytrd_2stage` 🌟