STRRFS(3)

💅 STRRFS(3) — like, the LAPACK manual page, but make it make sense 🎀

Okay so bestie, real talk: this page lives in the Library Functions Manual, and it's the doc for /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/strrfs.f. Yes, that's the whole NAME section. It's giving "my address is my name" energy and honestly? Iconic. 💀

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✨ SYNOPSIS ✨

Functions/Subroutines — buckle up, there's exactly one function and she's a whole subroutine:

subroutine STRRFS (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx,
    ferr, berr, work, iwork, info)

That's her. That's STRRFS. 🫶

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🌸 Function/Subroutine Documentation 🌸

subroutine STRRFS (character uplo, character trans, character diag, integer
    n, integer nrhs, real, dimension( lda, * ) a, integer lda, real,
    dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer
    ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real,
    dimension( * ) work, integer, dimension( * ) iwork, integer info)

🦋 Purpose 🦋

!> STRRFS provides error bounds and backward error estimates for the !> solution to a system of linear equations with a triangular !> coefficient matrix. !> !> The solution matrix X must be computed by STRTRS or some other !> means before entering this routine. STRRFS does not do iterative !> refinement because doing so cannot improve the backward error.

So like, she doesn't solve anything for you, okay? She's the vibe-check-after-the-fact girl. You already have your solution matrix X — from STRTRS or literally any other means — and now STRRFS tells you how much to trust it. And before you ask: no iterative refinement, because apparently doing so cannot improve the backward error. Not me questioning the math, the math said what it said. 😭

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🧚 Parameters 🧚

• UPLO — CHARACTER*1. Which triangle are we even working with?

!> UPLO is CHARACTER*1 !> = 'U': A is upper triangular; !> = 'L': A is lower triangular.

• TRANS — CHARACTER*1. Specifies the form of the system of equations, i.e. what shape the problem is wearing today:

!> TRANS is CHARACTER*1 !> Specifies the form of the system of equations: !> = 'N': A * X = B (No transpose) !> = 'T': AT * X = B (Transpose) !> = 'C': AH * X = B (Conjugate transpose = Transpose)

• DIAG — CHARACTER*1. Is the diagonal doing its own thing or is it all ones?

!> DIAG is CHARACTER*1 !> = 'N': A is non-unit triangular; !> = 'U': A is unit triangular.

• N — INTEGER. The order of the matrix A. Constraint: N >= 0.

!> N is INTEGER !> The order of the matrix A. N >= 0.

• NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. Constraint: NRHS >= 0.

!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0.

• A — REAL array, dimension (LDA,N). The triangular matrix herself. Here's the tea 🍓:

!> A is REAL array, dimension (LDA,N) !> The triangular matrix A. If UPLO = 'U', the leading N-by-N !> upper triangular part of the array A contains the upper !> triangular matrix, and the strictly lower triangular part of !> A is not referenced. If UPLO = 'L', the leading N-by-N lower !> triangular part of the array A contains the lower triangular !> matrix, and the strictly upper triangular part of A is not !> referenced. If DIAG = 'U', the diagonal elements of A are !> also not referenced and are assumed to be 1.

(Translation: half of A is literally not invited to the party. And if DIAG = 'U', the diagonal is also not referenced and is assumed to be 1. Not referenced means NOT REFERENCED, bestie. Don't come for me. 🙈)

• LDA — INTEGER. The leading dimension of the array A. Constraint: LDA >= max(1,N).

!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N).

• B — REAL array, dimension (LDB,NRHS). The right hand side matrix B. That's it, that's the whole description. Short queen. 👑

!> B is REAL array, dimension (LDB,NRHS) !> The right hand side matrix B.

• LDB — INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,N).

!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N).

• X — REAL array, dimension (LDX,NRHS). The solution matrix X. Again: short and confident. 💅

!> X is REAL array, dimension (LDX,NRHS) !> The solution matrix X.

• LDX — INTEGER. The leading dimension of the array X. Constraint: LDX >= max(1,N).

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

• FERR — REAL array, dimension (NRHS). Okay this one's important, pay attention 🔥:

!> FERR is REAL array, dimension (NRHS) !> The estimated forward error bound for each solution vector !> X(j) (the j-th column of the solution matrix X). !> If XTRUE is the true solution corresponding to X(j), FERR(j) !> is an estimated upper bound for the magnitude of the largest !> element in (X(j) - XTRUE) divided by the magnitude of the !> largest element in X(j). The estimate is as reliable as !> the estimate for RCOND, and is almost always a slight !> overestimate of the true error.

So FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). Reliability? Same as the estimate for RCOND, and it's almost always a slight overestimate of the true error. She's dramatic, she rounds up, we love her anyway. ✨

• BERR — REAL array, dimension (NRHS). The glow-up metric:

!> BERR is REAL array, dimension (NRHS) !> The componentwise relative backward error of each solution !> vector X(j) (i.e., the smallest relative change in !> any element of A or B that makes X(j) an exact solution).

Like, the smallest relative change in any element of A or B that makes X(j) an exact solution. Basically: "how much would I have to fudge reality for this to be perfect?" 🦋

• WORK — REAL array, dimension (3*N). Scratch space. Three N's worth of it.

!> WORK is REAL array, dimension (3*N)

• IWORK — INTEGER array, dimension (N). Integer scratch space. One N's worth.

!> IWORK is INTEGER array, dimension (N)

• INFO — INTEGER. The status report, the final verdict:

!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value

So 0 means slay, and INFO = -i means the i-th argument had an illegal value. Named and shamed, one index at a time. 💀

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🌈 Author 🌈

These absolute legends:

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

Definition at line 180 of file strrfs.f.

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⭐ Author (again, yes really) ⭐

Generated automatically by Doxygen for LAPACK from the source code.

No human wrote this section and honestly you can tell. It's giving "the robot did its homework." 🤖

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Anyway bestie, that's the whole page 🫶 — auto-generated by Doxygen for LAPACK from the source code, straight outta LAPACK Version 3.12.0, and the file herself is /home/abui...RC/strrfs.f(3). Go get those error bounds, queen. 👑💖