/home/abui...r_2stage.f(3)

💖 /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ssyevr_2stage.f 💖

Okay so this is from the LAPACK Library Functions Manual, and the star of the page is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ssyevr_2stage.f. Yes, that whole path is the name. We love a main-character filename. 🎀

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✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ssyevr_2stage.f

That's it. That's the name. The whole absolute path. No cap, bestie. 🫶

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🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine SSYEVR_2STAGE (jobz, range, uplo, n, a, lda, vl, vu, il, iu,
           abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
            SSYEVR_2STAGE computes the eigenvalues and, optionally, the left
           and/or right eigenvectors for SY matrices

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🦋 Function/Subroutine Documentation 🦋

✨ subroutine SSYEVR_2STAGE (...) ✨

subroutine SSYEVR_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, integer, dimension( * )
    isuppz, real, dimension( * ) work, integer lwork, integer, dimension( *
    ) iwork, integer liwork, integer info)

Bestie, this routine computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices. It's giving “I contain multitudes, and also a bunch of eigenvalues.” 💅

🎀 Purpose 🎀

Okay so SSYEVR_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. Obsessed with the flexibility, honestly. ✨

SSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call to SSYTRD. Then, whenever possible, SSYEVR_2STAGE calls SSTEMR to compute the eigenspectrum using Relatively Robust Representations. SSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. 🧚

For each unreduced block (submatrix) of T:

• (a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general.
• (b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d).
• (c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy.
• (d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter ABSTOL. 💖

For more details, see SSTEMR's documentation and:

• Inderjit S. Dhillon and Beresford N. Parlett: Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
• Inderjit Dhillon and Beresford Parlett: SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
• Inderjit Dhillon: , Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1 : SSYEVR_2STAGE calls SSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. SSYEVR_2STAGE calls SSTEBZ and SSTEIN on non-ieee machines and when partial spectrum requests are made.

Normal execution of SSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner. 😭

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🍓 Parameters 🍓

JOBZ

JOBZ is CHARACTER*1

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

tiny warning, bestie: 'V' is not available in this release. The audacity, honestly. 💀

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.

For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and SSTEIN are called. 🌀

UPLO

UPLO is CHARACTER*1

• = 'U': Upper triangle of A is stored;
• = 'L': Lower triangle of A is stored.

Pick a side, any side. 💅

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, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. RIP to that triangle, 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.

See by Demmel and Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, , LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy. 🌈

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)

The first M elements contain the selected eigenvalues in ascending order. Slay, sorted. ⭐

Z

Z is 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 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. Supplying N columns is always safe. 🫶

LDZ

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

ISUPPZ

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )

The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of SSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the orthogonal transformations applied by SORMTR. Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1. 🦋

WORK

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 👑

LWORK

LWORK

Okay bestie, this is the workspace-size boss fight. LWORK is INTEGER — the dimension of the array WORK.

If JOBZ = 'N' and N > 1, LWORK must be queried. Then:

LWORK = MAX(1, 26*N, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 5*N
          = N*KD + N*max(KD+1,FACTOPTNB)
            + max(2*KD*KD, KD*NTHREADS)
            + (KD+1)*N + 5*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 — yes, that’s the exact tea. 🫖

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

IWORK is INTEGER array, dimension (MAX(1,LIWORK)).

On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. (Yep, it says LWORK — copy that exactly, bestie. 💅)

LIWORK

LIWORK is INTEGER — the dimension of the array IWORK. LIWORK >= max(1,10*N).

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.

INFO

INFO is INTEGER.

• = 0: successful exit
• < 0: if INFO = -i, the i-th argument had an illegal value
• > 0: Internal error

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👑 Author 👑

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

🎀 Contributors: 🎀

• Inderjit Dhillon, IBM Almaden, USA
• Osni Marques, LBNL/NERSC, USA
• Ken Stanley, Computer Science Division, University of California at Berkeley, USA
• Jason Riedy, Computer Science Division, University of California at Berkeley, USA

🧚 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 378 of file ssyevr_2stage.f.

🤖 Author 🤖

Generated automatically by Doxygen for LAPACK from the source code.

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💖 That’s the whole vibe, bestie: LAPACK Version 3.12.0, source /home/abui...r_2stage.f(3) — date? not printed on this footer, just version and source, and honestly that’s iconic. Stay slay and size your workspaces correctly! 💅✨