ZHETRD_HE2HB(3)

💖 NAME 💖

okay so bestie, this is the Library Functions Manual page for ZHETRD_HE2HB(3) — the source file is literally /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zhetrd_he2hb.f and the routine is ZHETRD_HE2HB. That's the whole name section, no cap. 💅

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

       subroutine ZHETRD_HE2HB (uplo, n, kd, a, lda, ab, ldab, tau, work,
           lwork, info)
           ZHETRD_HE2HB

🫶 Function/Subroutine Documentation 🫶

   subroutine ZHETRD_HE2HB (character uplo, integer n, integer kd, complex*16,
       dimension( lda, * ) a, integer lda, complex*16, dimension( ldab, * ) ab,
       integer ldab, complex*16, dimension( * ) tau, complex*16, dimension( * )
       work, integer lwork, integer info)
       ZHETRD_HE2HB

🎯 Purpose 🎯

Okay so listen, bestie: ZHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian band-diagonal form AB by a unitary similarity transformation:

Q**H * A * Q = AB.

That's the whole glow-up. ✨

🧸 Parameters 🧸

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 reduced matrix if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
• The reduced matrix is stored in the array AB.

A

• A is COMPLEX*16 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, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
• On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details.

LDA

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

AB

• AB is COMPLEX*16 array, dimension (LDAB,N)
• On exit, 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).

LDAB

• LDAB is INTEGER
• The leading dimension of the array AB. LDAB >= KD+1.

TAU

• TAU is COMPLEX*16 array, dimension (N-KD)
• The scalar factors of the elementary reflectors (see Further Details).

WORK

• WORK is COMPLEX*16 array, dimension (LWORK)
• On exit, if INFO = 0, or if LWORK=-1, WORK(1) returns the size of LWORK.

LWORK

• LWORK is INTEGER
• The dimension of the array WORK which should be calculated by a workspace query. LWORK = MAX(1, LWORK_QUERY)
• 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.
• LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD where FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice otherwise putting LWORK=-1 will provide the size of WORK.

INFO

• INFO is INTEGER
• = 0: successful exit
• < 0: if INFO = -i, the i-th argument had an illegal value

bestie, that < 0 case is your "something was lowkey illegal in the arguments" alarm, okay? Don't ignore it. 💀

🎓 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

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If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors

Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in A(i,i+kd+1:n), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in A(i+kd+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
  (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
  (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
  (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
  (                            ab    )              (  v1     v2     v3     ab/v4 ab )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).

The raw source block ends with:

.fi

Definition at line 241 of file zhetrd_he2hb.f.

✍️ Author ✍️

Generated automatically by Doxygen for LAPACK from the source code.

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Cute sign-off, bestie: this whole page is from LAPACK Version 3.12.0, footer source /home/abui...trd_he2hb.f(3). Byeee, go reduce something gorgeously 💖🎀🦋