SRC/zhbevx_2stage.f(3)

🦋 SRC/zhbevx_2stage.f(3) 🦋

okay so bestie, this page is from the Library Functions Manual, and it’s fully about SRC/zhbevx_2stage.f. it’s giving Fortran subroutine energy, and we’re here for it. 💅

---

💖 NAME 💖

SRC/zhbevx_2stage.f — that’s the name, that’s the file, no extra lore needed. ✨

---

✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

bestie, the function list has exactly one main character: zhbevx_2stage, a subroutine with a truly iconic amount of arguments. 🎀

       subroutine zhbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq,
           vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork,
           ifail, info)

ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices — yes, OTHER is uppercase in the source, and we respect the source. 💜

🛠️ Function/Subroutine Documentation 🛠️

   subroutine zhbevx_2stage (character jobz, character range, character uplo,
       integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer
       ldab, complex*16, dimension( ldq, * ) q, integer ldq, double precision
       vl, double precision vu, integer il, integer iu, double precision
       abstol, integer m, double precision, dimension( * ) w, complex*16,
       dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work,
       integer lwork, double precision, dimension( * ) rwork, integer,
       dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

Bestie, meet the full signature. ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. Yes, “OTHER matrices” is the exact phrase, and yes, it’s a little mysterious. 💅

🌸 Purpose 🌸

Okay so the glow-up of this routine is: it computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. You can pick which eigenvalues/eigenvectors you want by specifying either a range of values or a range of indices for the desired eigenvalues. It’s giving selection-menu energy, honestly. periodt. 🔥

💅 Parameters 💅

• JOBZ is CHARACTER*1.
  • = 'N': Compute eigenvalues only;
  • = 'V': Compute eigenvalues and eigenvectors. Not available in this release.

🫶 Heads up, bestie: JOBZ = 'V' literally says “Not available in this release.” Do not come for me, I’m just the messenger. 💀

• 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.
• UPLO is CHARACTER*1.
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N is INTEGER.
  • The order of the matrix A. N >= 0.
• KD is INTEGER.
  • The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
• AB is COMPLEX*16 array, dimension (LDAB, N).
  • On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
    • if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    • if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
  • On exit, AB is overwritten by values generated during the reduction to tridiagonal form.
• LDAB is INTEGER.
  • The leading dimension of the array AB. LDAB >= KD + 1.
• Q is COMPLEX*16 array, dimension (LDQ, N).
  • If JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form.
  • If JOBZ = 'N', the array Q is not referenced.
• LDQ is INTEGER.
  • The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
• VL is DOUBLE PRECISION.
  • If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
• VU is DOUBLE PRECISION.
  • If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
• 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 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 is DOUBLE PRECISION.
  • 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 AB to tridiagonal form.
  • Lowkey this tolerance stuff is highkey important: eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
  • See by Demmel and Kahan, LAPACK Working Note #3. (That’s the exact reference line, double space and all — I’m not fixing it, that’s not my job. 💅)
• 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 is DOUBLE PRECISION array, dimension (N).
  • The first M elements contain the selected eigenvalues in ascending order.
• Z is COMPLEX*16 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. 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.
• LDZ is INTEGER.
  • The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
• WORK is COMPLEX*16 array, dimension (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 = (2KD+1)*N + KD*NTHREADS where KD is the size of the band. 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 sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. iykyk 🙈
• RWORK is DOUBLE PRECISION array, dimension (7*N).
• IWORK is INTEGER array, dimension (5*N).
• IFAIL is INTEGER array, dimension (N).
  • If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.
• INFO is INTEGER.
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.

vibe check: INFO = 0 is the slay. Anything else? Read the bullet, bestie. 😭

👑 Author 👑

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

no cap, that’s the whole author squad. 🫶

📚 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

Obsessed with a good citation list, honestly. ⭐

Definition at line 323 of file zhbevx_2stage.f.

🧾 Author 🧾

Generated automatically by Doxygen for LAPACK from the source code.

---

That’s the whole page, bestie! The footer’s giving LAPACK Version 3.12.0, file SRC/zhbevx_2stage.f(3) — mwah, go slay your eigenvalues. 💖✨🎀🍓🧚💜