DSGESV(3)

💖 DSGESV(3) — dsgesv.f(3) from the Library Functions Manual 💖

Okay bestie, welcome to your fave LAPACK mixed-precision moment. The manual name is literally dsgesv.f(3) and this whole thing is serving group-chat energy with every fact intact.

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💅 NAME 💅

This page is for the file:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dsgesv.f

That's it. That's the name. Slay.

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✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

       subroutine DSGESV (n, nrhs, a, lda, ipiv, b, ldb, x, ldx, work, swork,
           iter, info)

DSGESV computes the solution to system of linear equations A * X = B for GE matrices (mixed precision with iterative refinement). Yes ma'am, mixed precision and iterative refinement in one routine.

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🎀 Function/Subroutine Documentation 🎀

   subroutine DSGESV (integer n, integer nrhs, double precision, dimension(
       lda, * ) a, integer lda, integer, dimension( * ) ipiv, double precision,
       dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, *
       ) x, integer ldx, double precision, dimension( n, * ) work, real,
       dimension( * ) swork, integer iter, integer info)

DSGESV computes the solution to system of linear equations A * X = B for GE matrices (mixed precision with iterative refinement).

💜 Purpose 💜

Okay so here's the plot: DSGESV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices. Important distinction, bestie.

First it tries to factorize the matrix in SINGLE PRECISION and then use that factorization inside an iterative refinement procedure. The goal? A solution with DOUBLE PRECISION normwise backward error quality (see below). If that approach fails, it switches to a DOUBLE PRECISION factorization and solve. No drama, just fallback.

Real talk: iterative refinement is not going to be a winning strategy if the ratio

SINGLE PRECISION performance over DOUBLE PRECISION performance

is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ILAENV in the future. Up to now, we always try iterative refinement. Always. Periodt.

The iterative refinement process is stopped if

ITER > ITERMAX

or for all the RHS we have:

RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX

where:

• ITER is the number of the current iteration in the iterative refinement process
• RNRM is the infinity-norm of the residual
• XNRM is the infinity-norm of the solution
• ANRM is the infinity-operator-norm of the matrix A
• EPS is the machine epsilon returned by DLAMCH('Epsilon')

The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively. Cute constants, honestly.

🌸 Parameters 🌸

• N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. Zero is allowed, bestie.
• NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
• A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the N-by-N coefficient matrix A. On exit: if iterative refinement has been successfully used (INFO = 0 and ITER >= 0, see description below), then A is unchanged. If double precision factorization has been used (INFO = 0 and ITER < 0, see description below), then the array A contains the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored. So yes, A can get a whole new personality on exit.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• IPIV — IPIV is INTEGER array, dimension (N). The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i). Corresponds either to the single precision factorization (if INFO = 0 and ITER >= 0) or the double precision factorization (if INFO = 0 and ITER < 0). Pivot tea 🫖.
• B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
• LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• X — X is DOUBLE PRECISION array, dimension (LDX,NRHS). If INFO = 0, the N-by-NRHS solution matrix X. This is the moment.
• LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
• WORK — WORK is DOUBLE PRECISION array, dimension (N,NRHS). This array is used to hold the residual vectors. It's literally holding the receipts.
• SWORK — SWORK is REAL array, dimension (N*(N+NRHS)). This array is used to use the single precision matrix and the right-hand sides or solutions in single precision. Single-precision storage, doing its supporting-character job.
• ITER — ITER is INTEGER. The vibe check on iterative refinement:
  • < 0: iterative refinement has failed, double precision factorization has been performed.
    • -1: the routine fell back to full precision for implementation- or machine-specific reasons
    • -2: narrowing the precision induced an overflow, the routine fell back to full precision
    • -3: failure of SGETRF
    • -31: stop the iterative refinement after the 30th iterations
  • > 0: iterative refinement has been successfully used. Returns the number of iterations.
• INFO — INFO is INTEGER. Exit status tea:
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, U(i,i) computed in DOUBLE PRECISION is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. Not the slay we wanted 💀.

🦋 Author 🦋

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

🧚 Definition at line 🧚

Definition at line 193 of file dsgesv.f. That's the exact line, bestie, no guessing.

💫 Author (the auto-generated one) 💫

Generated automatically by Doxygen for LAPACK from the source code. So yes, the docs also have main-character energy from a machine.

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Sign-off: this page is from LAPACK Version 3.12.0, and the footer path is /home/abui...RC/dsgesv.f(3). The date? Not printed on the page, so I'm not inventing one — consider this your cute little footer moment instead: stay precise, check your LDA, and may your U never be exactly singular 💖✨🎀