/home/abui...d_2stage.f(3)

💅 NAME 💅

ok so this whole page is literally about one file, and its name is a whole journey:

/home/abuild/rpmbuild/BUILD/lapack-
static-3.12.0-build/lapack-3.12.0/SRC/dsytrd_2stage.f

like yes bestie that is the entire name section, we don't make the rules 🎀 it's from the LAPACK library functions manual (section 3), and the star of the show lives in that file.

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✨ SYNOPSIS ✨

Functions/Subroutines

• subroutine DSYTRD_2STAGE (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info) — that's our girl 💖

DSYTRD_2STAGE

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🦋 Function/Subroutine Documentation 🦋

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

🎀 Purpose 🎀

!> !> DSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric !> tridiagonal form T by a orthogonal similarity transformation: !> Q1T Q2T* A * Q2 * Q1 = T. !>

like, it takes your real symmetric matrix A and squishes it down into real symmetric tridiagonal form T using an orthogonal similarity transformation — the whole glow-up is literally 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). !>

so VECT is a CHARACTER*1, and it's giving "what do you actually want from me":

• 'N' — no need for the Housholder representation, especially not for the second stage (that's the Band to tridiagonal part), so LHOUS2 is of size max(1, 4*N).
• 'V' — the Householder representation IS needed, either to generate Q1 Q2 or to apply Q1 Q2, and then LHOUS2 is to be queried and computed. BUT — and this is the important part — (NOT AVAILABLE IN THIS RELEASE) 😭

UPLO

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

UPLO is CHARACTER*1 and it's just picking a side:

• '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. !>

N is INTEGER, it's the order of the matrix A, and the vibe check is 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. !>

ok this one has a whole storyline, bestie. A is a DOUBLE PRECISION array, dimension (LDA,N).

On entry, it's your 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, the plot twists:

• 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 below, iykyk 👑

LDA

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

LDA is INTEGER, the leading dimension of the array A, and it must satisfy LDA >= max(1,N).

D

!> D is DOUBLE PRECISION array, dimension (N) !> The diagonal elements of the tridiagonal matrix T. !>

D is a DOUBLE PRECISION array, dimension (N) — the diagonal elements of the tridiagonal matrix T. main character energy tbh ⭐

E

!> E is DOUBLE PRECISION array, dimension (N-1) !> The off-diagonal elements of the tridiagonal matrix T. !>

E is a DOUBLE PRECISION array, dimension (N-1) — the off-diagonal elements of the tridiagonal matrix T.

TAU

!> TAU is DOUBLE PRECISION array, dimension (N-KD) !> The scalar factors of the elementary reflectors of !> the first stage (see Further Details). !>

TAU is a DOUBLE PRECISION array, dimension (N-KD) — the scalar factors of the elementary reflectors of the first stage (see Further Details).

HOUS2

!> HOUS2 is DOUBLE PRECISION array, dimension (LHOUS2) !> Stores the Householder representation of the stage2 !> band to tridiagonal. !>

HOUS2 is a DOUBLE PRECISION array, dimension (LHOUS2) — it stores the Householder representation of the stage2 band to tridiagonal. literally its whole job 💅

LHOUS2

!> LHOUS2 is INTEGER !> The dimension of the array HOUS2. !> 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. !>

LHOUS2 is INTEGER, the dimension of the array HOUS2. Important deets:

• 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 😭 (again! not yet!)

WORK

!> WORK is DOUBLE PRECISION array, dimension (LWORK) !>

WORK is a DOUBLE PRECISION array, dimension (LWORK). that's it, that's the whole vibe 🧚

LWORK

!> LWORK is INTEGER !> The dimension of the array WORK. 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. !>

LWORK is INTEGER, the dimension of the array WORK, and 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.

INFO

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

INFO is INTEGER, the ultimate vibe report:

• = 0 — successful exit, slay 🎉
• < 0 — if INFO = -i, the i-th argument had an illegal value 💀

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👑 Author 👑

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

the whole squad collab 💜

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🔥 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 !> !>

so real ones who built this: implemented by Azzam Haidar 🫶 and all details are available on technical report, SC11, SC13 papers. the receipts:

• 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

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Definition at line 222 of file dsytrd_2stage.f. 📍

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🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code. (yes the docs wrote themselves, we love an autonomous queen 🤖✨)

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and that's the whole page bestie!! 🎀 this was LAPACK, Version 3.12.0, and the footer path was /home/abui...d_2stage.f(3) — mwah, go reduce something symmetric 💖🦋🌸