SRC/zhetrd_hb2st.F(3)

💖 SRC/zhetrd_hb2st.F 💖 (LAPACK manual page, bestie)

okay so. this is the manual page for SRC/zhetrd_hb2st.F, and it is lowkey iconic because it's the routine that takes your fancy complex Hermitian band matrix and serves it up as a real symmetric tridiagonal one. let's go 💅

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🌸 SYNOPSIS 🌸

✨ Functions/Subroutines ✨

subroutine zhetrd_hb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e,
    hous, lhous, work, lwork, info)

ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T

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👑 Function/Subroutine Documentation 👑

subroutine zhetrd_hb2st (character stage1, character vect, character uplo,
    integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer
    ldab, double precision, dimension( * ) d, double precision, dimension( *
    ) e, complex*16, dimension( * ) hous, integer lhous, complex*16,
    dimension( * ) work, integer lwork, integer info)

ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T

🦋 Purpose 🦋

ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation:

Q**H * A * Q = T.

🎀 Parameters 🎀

• STAGE1 — STAGE1 is CHARACTER*1
  • = 'N': 'No': to mention that the stage 1 of the reduction from dense to band using the zhetrd_he2hb routine was not called before this routine to reproduce AB. In other term this routine is called as standalone.
  • = 'Y': 'Yes': to mention that the stage 1 of the reduction from dense to band using the zhetrd_he2hb routine has been called to produce AB (e.g., AB is the output of zhetrd_he2hb.
• VECT — VECT is CHARACTER*1
  • = 'N': No need for the Housholder representation, and thus LHOUS is of size max(1, 4*N);
  • = 'V': the Householder representation is needed to either generate or to apply Q later on, then LHOUS is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE).
• UPLO — UPLO is CHARACTER*1
  • = '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.
• KD — 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 — 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, the diagonal elements of AB are overwritten by the diagonal elements of the tridiagonal matrix T; if KD > 0, the elements on the first superdiagonal (if UPLO = 'U') or the first subdiagonal (if UPLO = 'L') are overwritten by the off-diagonal elements of T; the rest of AB is overwritten by values generated during the reduction.

• LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD+1.
• D — D is DOUBLE PRECISION array, dimension (N). The diagonal elements of the tridiagonal matrix T.
• E — E is DOUBLE PRECISION array, dimension (N-1). The off-diagonal elements of the tridiagonal matrix T: E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
• HOUS — HOUS is COMPLEX*16 array, dimension LHOUS, that store the Householder representation.
• LHOUS — LHOUS is INTEGER. The dimension of the array HOUS. LHOUS = MAX(1, dimension). If LWORK = -1, or LHOUS=-1, then a query is assumed; the routine only calculates the optimal size of the HOUS array, returns this value as the first entry of the HOUS array, and no error message related to LHOUS is issued by XERBLA. LHOUS = MAX(1, dimension) where
  • dimension = 4*N if VECT='N'
  • not available now if VECT='H'
• WORK — WORK is COMPLEX*16 array, dimension LWORK.
• LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK = MAX(1, dimension). If LWORK = -1, or LHOUS=-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 = (2KD+1)*N + KD*NTHREADS
  • 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

💜 Author 💜

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

⭐ 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

🌈 Definition 🌈

Definition at line 228 of file zhetrd_hb2st.F.

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ngl bestie, the Householder stuff for VECT='V' is NOT AVAILABLE IN THIS RELEASE, so don't come for me when you query it 🙈

Author: Generated automatically by Doxygen for LAPACK from the source code.

Version 3.12.0 — Tue Nov 28 2023 12:08:42 — and that's the whole SRC/zhetrd_hb2st.F(3) page, periodt. go forth and tridiagonalize 💖✨🦋