SRC/ssytrd_sb2st.F(3)

📦 version 3.12.0, in centos-stream 10, fedora 43, fedora 44

✨ SRC/ssytrd_sb2st.F(3) ✨

okay bestie, welcome to the LAPACK manual page for SRC/ssytrd_sb2st.F — this is the one that takes your chunky real symmetric band matrix and shrinks it down to tridiagonal form, and honestly? she's iconic for it. 💅

🌸 NAME 🌸

SRC/ssytrd_sb2st.F — no cap, that's the file, that's the name, we move.

🦋 SYNOPSIS 🦋

Functions/Subroutines

  • subroutine ssytrd_sb2st (stage1, vect, uplo, n, kd, ab, ldab, d, e, hous, lhous, work, lwork, info)
    • SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T

💖 Function/Subroutine Documentation 💖

🎀 subroutine ssytrd_sb2st

subroutine ssytrd_sb2st (character stage1, character vect, character uplo,
    integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab,
    real, dimension( * ) d, real, dimension( * ) e, real, dimension( * )
    hous, integer lhous, real, dimension( * ) work, integer lwork, integer
    info)

SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T

⭐ Purpose:

SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T by a orthogonal similarity transformation: Q**T * 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 ssytrd_sy2sb 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 ssytrd_sy2sb routine has been called to produce AB (e.g., AB is the output of ssytrd_sy2sb.
  • 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 REAL array, dimension (LDAB,N)
    • On entry, the upper or lower triangle of the symmetric 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 REAL array, dimension (N)
    • The diagonal elements of the tridiagonal matrix T.
  • E — E is REAL 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 REAL 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 REAL 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 at line 228 of file ssytrd_sb2st.F.

🌟 Author

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