ZHEGV_2STAGE(3)

🌸 `zhegv_2stage.f(3)` — Library Functions Manual energy 🌸

okay so bestie, this is the zhegv_2stage.f(3) page from the Library Functions Manual, and it's LAPACK 3.12.0 business. It's giving serious numerical linear algebra, but we're gonna make it make sense. 💖✨

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zhegv_2stage.f — that's the file this whole page is about, no cap. 💅

✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

       subroutine ZHEGV_2STAGE (itype, jobz, uplo, n, a, lda, b, ldb, w, work,
           lwork, rwork, info)
           ZHEGV_2STAGE

💅 Function/Subroutine Documentation 💅

Here's the full signature, character-for-character, because we do not play with exact types:

   subroutine ZHEGV_2STAGE (integer itype, character jobz, character uplo,
       integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16,
       dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) w,
       complex*16, dimension( * ) work, integer lwork, double precision,
       dimension( * ) rwork, integer info)
       ZHEGV_2STAGE

🎀 Purpose: 🎀

ZHEGV_2STAGE computes all the eigenvalues, and optionally the eigenvectors, of a complex generalized Hermitian-definite eigenproblem. The forms are literally A*x=(lambda)*B*x, A*Bx=(lambda)*x, and B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, and B is also positive definite. It uses the 2stage technique for the reduction to tridiagonal — and per the page, that showed higher performance on recent architecture and for large sizes N>2000. Slay. 🔥

🌸 Parameters 🌸

💖 ITYPE 💖

ITYPE is INTEGER and it specifies the problem type to be solved:

= 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.

🎀 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 — the order of the matrices A and B. N >= 0.

🍓 A 🍓

A is COMPLEX*16 array, dimension (LDA, N).

On entry, it's 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. 💀

🧚 LDA 🧚

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

💜 B 💜

B is COMPLEX*16 array, dimension (LDB, N).

On entry, it's 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 DOUBLE PRECISION array, dimension (N). If INFO = 0, the eigenvalues in ascending order. 👑

✨ 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 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 DOUBLE PRECISION 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: ZPOTRF or ZHEEV returned an error code:
- <= N: if INFO = i, ZHEEV 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.

💖 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 230 of file zhegv_2stage.f.

👑 Author 👑

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