CHEGV_2STAGE(3)

💅 NAME 💅

So bestie, this page is literally about the file /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/chegv_2stage.f — and it's straight out of the Library Functions Manual, page /home/abui...gv_2stage.f(3). Slay, we're in the LAPACK source tree now.

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✨ SYNOPSIS ✨

Functions/Subroutines 🦋

subroutine CHEGV_2STAGE (itype, jobz, uplo, n, a, lda, b, ldb, w, work,
    lwork, rwork, info)

CHEGV_2STAGE — yeah that's the whole vibe check of this page, we stan one (1) subroutine.

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🎀 Function/Subroutine Documentation 🎀

subroutine CHEGV_2STAGE (integer itype, character jobz, character uplo,
    integer n, complex, dimension( lda, * ) a, integer lda, complex,
    dimension( ldb, * ) b, integer ldb, real, dimension( * ) w, complex,
    dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer
    info)

CHEGV_2STAGE ✨

Purpose 🌸

Okay so picture this 👑 — CHEGV_2STAGE computes all the eigenvalues, and optionally the eigenvectors, of a complex generalized Hermitian-definite eigenproblem. The possible forms are:

A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.

Here A and B are assumed to be Hermitian, and B is also positive definite. No cap. 💜

This routine uses the 2stage technique for the reduction to tridiagonal, which showed higher performance on recent architecture and for large sizes N>2000. It's giving main character energy on big matrices fr fr 🔥

Parameters 🌈

ITYPE

ITYPE is INTEGER. It specifies the problem type to be solved, and yes you have to pick one:

= 1:  A*x = (lambda)*B*x
= 2:  A*B*x = (lambda)*x
= 3:  B*A*x = (lambda)*x

JOBZ

JOBZ is CHARACTER*1:

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

Heads up bestie — 'V' is Not available in this release. Do not come for me, I'm just the messenger 😭

UPLO

UPLO is CHARACTER*1:

= 'U':  Upper triangles of A and B are stored;
= 'L':  Lower triangles of A and B are stored.

N

N is INTEGER. It's the order of the matrices A and B. 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 matrix Z of eigenvectors. The eigenvectors are normalized as follows:

if ITYPE = 1 or 2, Z**H*B*Z = I;
if ITYPE = 3, Z**H*inv(B)*Z = I.

If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of A, including the diagonal, is destroyed. 💀 Like, gone gone. So don't be out here needing that data afterwards, ok?

LDA

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

B

B is COMPLEX array, dimension (LDB, N).

On entry, the Hermitian positive definite matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization:

B = U**H*U or B = L*L**H.

LDB

LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N). ⭐

W

W is REAL array, dimension (N).

If INFO = 0, the eigenvalues in ascending order. Iconic behaviour honestly, sorted and everything 💅

WORK

WORK is COMPLEX array, dimension (MAX(1,LWORK)).

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

LWORK

LWORK is INTEGER. It's 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), and 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. Lowkey the neatest trick in the book ✨

RWORK

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

INFO

INFO is INTEGER. The receipts:

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  CPOTRF or CHEEV returned an error code:
   <= N:  if INFO = i, CHEEV failed to converge;
          i off-diagonal elements of an intermediate
          tridiagonal form did not converge to zero;
   > N:   if INFO = N + i, for 1 <= i <= N, then the leading
          principal minor of order i of B is not positive.
          The factorization of B could not be completed and
          no eigenvalues or eigenvectors were computed.

Okay so if INFO comes back > N, B was not positive definite in some leading principal minor of order i, the factorization could not be completed, and no eigenvalues or eigenvectors were computed. That's a hard no, bestie 😭

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 absolute bangers (yes I'm citing sources, academic queen behaviour 👑):

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 230 of file chegv_2stage.f. 📍

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🌟 AUTHOR 🌟

Generated automatically by Doxygen for LAPACK from the source code. Not a human, just a very efficient little robot 🦋

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And that's the whole page, bestie! This was LAPACK, Version 3.12.0, filed under /home/abui...gv_2stage.f(3). Go forth and query your LWORK, ily 💖✨