gesv_mixed(3)

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version 3.12.1 · 9 versions
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🎀 gesv_mixed 🎀

okay bestie, welcome to gesv_mixed(3) — this is a real one, so grab a snack 🍓. This is the page for the mixed precision "factor and solve" girls, and honestly? she ate. ✨

💖 NAME 💖

gesv_mixed — gesv: factor and solve, mixed precision. that's the whole name, no notes, periodt 💅

✨ SYNOPSIS ✨

Functions

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

and here are the actual subroutine signatures, copied with love and zero typos 🫶

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

🌸 DETAILED DESCRIPTION 🌸

so like, the "Detailed Description" chapter is here, lowkey just vibing with no extra prose of its own. everything spicy lives down in the function docs ⬇️ — that's where the actual content is, iykyk 💜

🦋 FUNCTION DOCUMENTATION 🦋

👑 dsgesv

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) — say it back, bestie.

🌈 Purpose

okay so here's the tea ☕ — 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.

DSGESV first attempts to factorize the matrix in SINGLE PRECISION and use this factorization within an iterative refinement procedure to produce a solution with DOUBLE PRECISION normwise backward error quality (see below 👀). If the approach fails the method switches to a DOUBLE PRECISION factorization and solve. no cap.

the 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. (she's trying her best, okay 😭)

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. locked in 🔒

🧚 Parameters

  • N

N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.

  • 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. 💅

  • 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).

  • 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. the moment of truth ✨

  • 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.

  • 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. (yes, that's the sentence, she's doing her best — single precision storage doing double duty 🫶)

  • ITER

ITER is INTEGER.

  • < 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

so like, read the sign, it's giving status report 💀

  • INFO

INFO is INTEGER.

  • = 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. 💀

🏆 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

📍 Definition at line 193 of file dsgesv.f.

🌟 zcgesv

subroutine zcgesv (integer n, integer nrhs, complex*16, dimension( lda, * )
    a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension(
    ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx,
    complex*16, dimension( n, * ) work, complex, dimension( * ) swork,
    double precision, dimension( * ) rwork, integer iter, integer info)

ZCGESV computes the solution to system of linear equations A * X = B for GE matrices (mixed precision with iterative refinement) — the complex sibling, equally iconic 🦋

🌈 Purpose

ZCGESV computes the solution to a complex system of linear equations

A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

ZCGESV first attempts to factorize the matrix in COMPLEX and use this factorization within an iterative refinement procedure to produce a solution with COMPLEX*16 normwise backward error quality (see below 👀). If the approach fails the method switches to a COMPLEX*16 factorization and solve.

the iterative refinement is not going to be a winning strategy if the ratio COMPLEX performance over COMPLEX*16 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.

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. same numbers, still locked in 🔒

🧚 Parameters

  • N

N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.

  • 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 COMPLEX*16 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. 💅

  • 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).

  • B

B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0, the N-by-NRHS solution matrix X. the payoff ✨

  • LDX

LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).

  • WORK

WORK 💖

WORK is a COMPLEX*16 array, dimension (N,NRHS). It's literally the holding spot for residual vectors. 🫶

SWORK ✨

SWORK is a COMPLEX array, dimension (N*(N+NRHS)). This one is used to use the single precision matrix and the right-hand sides or solutions in single precision. Slay. 🎀

RWORK 🌸

RWORK is a DOUBLE PRECISION array, dimension (N). Short and sweet. 💅

ITER 🌀

ITER is an INTEGER. Here's the whole vibe:

  • < 0: iterative refinement has failed, COMPLEX*16 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 CGETRF.
    • -31: stop the iterative refinement after the 30th iterations.
  • > 0: iterative refinement has been successfully used. Returns the number of iterations. Periodt. ✅

INFO 🚨

INFO is an INTEGER. The ultimate vibe check:

  • = 0: successful exit. 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value. 🙈
  • > 0: if INFO = i, U(i,i) computed in COMPLEX*16 is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. 💀

👑 Author

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

Definition at line 199 of file zcgesv.f.

🌈 Author (Doxygen's version)

Generated automatically by Doxygen for LAPACK from the source code.

Okay bestie, that's the whole page! LAPACK Version 3.12.0, gesv_mixed(3) — footer energy, but make it cute. Catch you never! 💖🦋🎀
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