SRC/zhetrd_he2hb.f(3)

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💅 SRC/zhetrd_he2hb.f(3) — LAPACK Library Functions Manual 🦋

okay bestie, so this is one of those LAPACK pages — it's the zhetrd_he2hb.f man page from the LAPACK library, and honestly? she's kind of a big deal. Buckle up, we're going to talk about reducing complex Hermitian matrices to band-diagonal form and you're going to understand it. 🎀

✨ SYNOPSIS ✨

Functions/Subroutines 🌸

subroutine zhetrd_he2hb (uplo, n, kd, a, lda, ab, ldab, tau, work,
    lwork, info)

ZHETRD_HE2HB — that's her name, that's her whole brand. 👑

💖 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 🧚

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.

literally that's it, that's the whole vibe. she takes your Hermitian matrix and puts it in band-diagonal form via a unitary similarity transform. no cap. ✨

🎀 Parameters 🎀

  • UPLO — is CHARACTER*1. it tells you which triangle is doing the werk:
    • = 'U': the upper triangle of A is stored 💜
    • = 'L': the lower triangle of A is stored 💙
  • N — is INTEGER. the order of the matrix A. constraint era: N >= 0. bestie, zero is allowed, negative is NOT. 🚫
  • KD — is INTEGER. the number of superdiagonals of the reduced matrix if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. rule of law: KD >= 0. the reduced matrix lives in the array AB. 🌈
  • 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 (she's ignored, she's unbothered 💅). 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 below, iykyk. 👀
  • LDA — is INTEGER. the leading dimension of the array A. requirement: LDA >= max(1,N). ⭐
  • 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 — is INTEGER. the leading dimension of the array AB. requirement: LDAB >= KD+1. 🌸
  • TAU — is COMPLEX*16 array, dimension (N-KD). the scalar factors of the elementary reflectors (peep Further Details for the lore). 🦋
  • WORK — is COMPLEX*16 array, dimension (MAX(1,LWORK)).
    • on exit: if INFO = 0, or if LWORK = -1, WORK(1) returns the size of LWORK. 🔥
  • LWORK — is INTEGER. the dimension of the array WORK which should be calculated by a workspace query.
    • if N <= KD+1, LWORK >= 1, else 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 (she's off the hook, don't @ her 😌).
    • 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 — is INTEGER. the receipts:
    • = 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
if UPLO = 'U', the matrix Q is represented as a product of elementary reflectors text Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.
each H(i) has the form text 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 text Q = H(1) H(2) . . . H(k), where k = n-kd.
each H(i) has the form text 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': text ( 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).

Definition at line 243 of file zhetrd_he2hb.f. 📍

🌟 Author (the meta one) 🌟

Generated automatically by Doxygen for LAPACK from the source code. she was literally built by a machine, main character energy fr fr 🤖💖

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