SRC/dgsvj1.f(3)

okay so this is SRC/dgsvj1.f(3) from the Library Functions Manual — no cap, we're diving into the DGSVJ1 pre-processor for dgesvj, and it's giving "Jacobi rotations but make it targeted" 💅✨

💖 NAME 💖

SRC/dgsvj1.f

dgsvj1 — DGSVJ1 pre-processor for the routine dgesvj, applies Jacobi rotations targeting only particular pivots. That's the whole vibe of this page, bestie 🫶

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine dgsvj1 (jobv, m, n, n1, a, lda, d, sva, mv, v, ldv, eps,
    sfmin, tol, nsweep, work, lwork, info)

DGSVJ1 pre-processor for the routine dgesvj, applies Jacobi rotations targeting only particular pivots. Yup, that's the one-line summary, and it ate 💅

🛠️ Function/Subroutine Documentation 🛠️

subroutine dgsvj1 (character*1 jobv, integer m, integer n, integer n1,
       double precision, dimension( lda, * ) a, integer lda, double precision,
       dimension( n ) d, double precision, dimension( n ) sva, integer mv,
       double precision, dimension( ldv, * ) v, integer ldv, double precision
       eps, double precision sfmin, double precision tol, integer nsweep,
       double precision, dimension( lwork ) work, integer lwork, integer info)

DGSVJ1 pre-processor for the routine dgesvj, applies Jacobi rotations targeting only particular pivots. This is the full signature, periodt 🌸

🎯 Purpose

DGSVJ1 is called from DGESVJ as a pre-processor and that is its main purpose. It applies Jacobi rotations in the same way as DGESVJ does, but it targets only particular pivots and it does not check convergence (stopping criterion). Few tuning parameters (marked by [TP]) are available for the implementer 🫶

🌈 Further Details 🌈

DGSVJ1 applies few sweeps of Jacobi rotations in the column space of the input M-by-N matrix A. The pivot pairs are taken from the (1,2) off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The block-entries (tiles) of the (1,2) off-diagonal block are marked by the [x]'s in the following scheme:

   | *  *  * [x] [x] [x]|
   | *  *  * [x] [x] [x]|    Row-cycling in the nblr-by-nblc [x] blocks.
   | *  *  * [x] [x] [x]|    Row-cyclic pivoting inside each [x] block.
   |[x] [x] [x] *  *  * |
   |[x] [x] [x] *  *  * |
   |[x] [x] [x] *  *  * |

In terms of the columns of A, the first N1 columns are rotated 'against' the remaining N-N1 columns, trying to increase the angle between the corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter. The number of sweeps is given in NSWEEP and the orthogonality threshold is given in TOL 🦋

🧚 Parameters 🧚

• JOBV — JOBV is CHARACTER*1. Specifies whether the output from this procedure is used to compute the matrix V:
  • = 'V': the product of the Jacobi rotations is accumulated by postmultiplying the N-by-N array V. (See the description of V.)
  • = 'A': the product of the Jacobi rotations is accumulated by postmultiplying the MV-by-N array V. (See the descriptions of MV and V.)
  • = 'N': the Jacobi rotations are not accumulated.
• M — M is INTEGER. The number of rows of the input matrix A. M >= 0.
• N — N is INTEGER. The number of columns of the input matrix A. M >= N >= 0.
• N1 — N1 is INTEGER. N1 specifies the 2 x 2 block partition, the first N1 columns are rotated 'against' the remaining N-N1 columns of A.
• A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, M-by-N matrix A, such that A*diag(D) represents the input matrix. On exit, A_onexit * D_onexit represents the input matrix A*diag(D) post-multiplied by a sequence of Jacobi rotations, where the rotation threshold and the total number of sweeps are given in TOL and NSWEEP, respectively. (See the descriptions of N1, D, TOL and NSWEEP.)
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• D — D is DOUBLE PRECISION array, dimension (N). The array D accumulates the scaling factors from the fast scaled Jacobi rotations. On entry, A*diag(D) represents the input matrix. On exit, A_onexit*diag(D_onexit) represents the input matrix post-multiplied by a sequence of Jacobi rotations, where the rotation threshold and the total number of sweeps are given in TOL and NSWEEP, respectively. (See the descriptions of N1, A, TOL and NSWEEP.)
• SVA — SVA is DOUBLE PRECISION array, dimension (N). On entry, SVA contains the Euclidean norms of the columns of the matrix A*diag(D). On exit, SVA contains the Euclidean norms of the columns of the matrix onexit*diag(D_onexit).
• MV — MV is INTEGER. If JOBV = 'A', then MV rows of V are post-multiplied by a sequence of Jacobi rotations. If JOBV = 'N', then MV is not referenced.
• V — V is DOUBLE PRECISION array, dimension (LDV,N). If JOBV = 'V', then N rows of V are post-multiplied by a sequence of Jacobi rotations. If JOBV = 'A', then MV rows of V are post-multiplied by a sequence of Jacobi rotations. If JOBV = 'N', then V is not referenced.
• LDV — LDV is INTEGER. The leading dimension of the array V, LDV >= 1. If JOBV = 'V', LDV >= N. If JOBV = 'A', LDV >= MV.
• EPS — EPS is DOUBLE PRECISION. EPS = DLAMCH('Epsilon').
• SFMIN — SFMIN is DOUBLE PRECISION. SFMIN = DLAMCH('Safe Minimum').
• TOL — TOL is DOUBLE PRECISION. TOL is the threshold for Jacobi rotations. For a pair A(:,p), A(:,q) of pivot columns, the Jacobi rotation is applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.
• NSWEEP — NSWEEP is INTEGER. NSWEEP is the number of sweeps of Jacobi rotations to be performed.
• WORK — WORK is DOUBLE PRECISION array, dimension (LWORK).
• LWORK — LWORK is INTEGER. LWORK is the dimension of WORK. LWORK >= M.
• INFO — INFO is INTEGER.
  • = 0: successful exit.
  • < 0: if INFO = -i, then the i-th argument had an illegal value.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌸 Contributors

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

🧚 Definition

Definition at line 234 of file dgsvj1.f.

💖 Author (meta edition) 💖

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole page, bestie! Cute little footer sign-off: LAPACK, Version 3.12.0, and the source file SRC/dgsvj1.f(3). (No date in the footer, so no invented date — just vibes.) Go forth and rotate those pivots 💅✨