trsv(3)

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version 3.12.1 · 9 versions
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💅 trsv(3) — like, the whole trsv situation (part 1 of 2)

okay bestie gather round, this is a page from the Library Functions Manual and it's literally all about trsv, which is the triangular matrix-vector solve. it's a whole family of routines, one per flavour of numbers, and we're doing this properly so nothing gets skipped. let's go ✨

🎀 NAME

trsv — trsv: triangular matrix-vector solve

yep that's it, that's the whole vibe. it solves. triangularly. 💅

🌸 SYNOPSIS

✨ Functions

  • subroutine ctrsv (uplo, trans, diag, n, a, lda, x, incx) → CTRSV
  • subroutine dtrsv (uplo, trans, diag, n, a, lda, x, incx) → DTRSV
  • subroutine strsv (uplo, trans, diag, n, a, lda, x, incx) → STRSV
  • subroutine ztrsv (uplo, trans, diag, n, a, lda, x, incx) → ZTRSV

four routines, same argument list, different number types. it's giving matching friendship bracelets fr fr 🫶

🦋 Detailed Description

💜 Function Documentation

subroutine ctrsv</p> (character uplo, character trans, character diag, integer n, complex, dimension(lda,) a, integer lda, complex, dimension() x, integer incx)

👑 CTRSV

Purpose:

!>
!> CTRSV  solves one of the systems of equations
!>
!>    A*x = b,   or   A**T*x = b,   or   A**H*x = b,
!>
!> where b and x are n element vectors and A is an n by n unit, or
!> non-unit, upper or lower triangular matrix.
!>
!> No test for singularity or near-singularity is included in this
!> routine. Such tests must be performed before calling this routine.
!>

so like, CTRSV solves one of those three systems up there — A*x = b, or A**T*x = b, or A**H*x = b — where b and x are n element vectors and A is an n by n unit or non-unit, upper or lower triangular matrix.

⚠️ read this twice bestie: there is no test for singularity or near-singularity built into this routine. none. zero. you gotta do those tests yourself before you call it. don't come crying to CTRSV when your matrix is a mess 😭

Parameters:

  • UPLO — is CHARACTER*1. on entry, UPLO tells us whether the matrix is upper or lower triangular, like this:
    • UPLO = 'U' or 'u' → A is an upper triangular matrix.
    • UPLO = 'L' or 'l' → A is a lower triangular matrix.
  • TRANS — is CHARACTER*1. on entry, TRANS says which equation we're solving, as follows:
    • TRANS = 'N' or 'n' → A*x = b.
    • TRANS = 'T' or 't' → A**T*x = b.
    • TRANS = 'C' or 'c' → A**H*x = b.
  • DIAG — is CHARACTER*1. on entry, DIAG says whether or not A is unit triangular, like this:
    • DIAG = 'U' or 'u' → A is assumed to be unit triangular.
    • DIAG = 'N' or 'n' → A is not assumed to be unit triangular.
  • N — is INTEGER. on entry, N is the order of the matrix A. N must be at least zero.
  • A — is COMPLEX array, dimension ( LDA, N ). before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular matrix, and the strictly lower triangular part of A is not referenced. before entry with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular matrix, and the strictly upper triangular part of A is not referenced.
🫶 also note: when DIAG = 'U' or 'u', the diagonal elements of A are not referenced either — they're just assumed to be unity. iconic behaviour honestly.
  • LDA — is INTEGER. on entry, LDA is the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ).
  • X — is COMPLEX array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). before entry, the incremented array X must contain the n element right-hand side vector b. on exit, X is overwritten with the solution vector x. 🔥
  • INCX — is INTEGER. on entry, INCX is the increment for the elements of X. INCX must not be zero.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

!>
!>  Level 2 Blas routine.
!>
!>  -- Written on 22-October-1986.
!>     Jack Dongarra, Argonne National Lab.
!>     Jeremy Du Croz, Nag Central Office.
!>     Sven Hammarling, Nag Central Office.
!>     Richard Hanson, Sandia National Labs.
!>

that's a Level 2 BLAS routine, written way back on 22-October-1986 by Jack Dongarra (Argonne National Lab), Jeremy Du Croz (Nag Central Office), Sven Hammarling (Nag Central Office) and Richard Hanson (Sandia National Labs). literal legends, we stan 👑

Definition at line 148 of file ctrsv.f.

subroutine dtrsv</p> (character uplo, character trans, character diag, integer n, double precision, dimension(lda,) a, integer lda, double precision, dimension() x, integer incx)

💙 DTRSV

Purpose:

!>
!> DTRSV  solves one of the systems of equations
!>
!>    A*x = b,   or   A**T*x = b,
!>
!> where b and x are n element vectors and A is an n by n unit, or
!> non-unit, upper or lower triangular matrix.
!>
!> No test for singularity or near-singularity is included in this
!> routine. Such tests must be performed before calling this routine.
!>

DTRSV solves one of the two systems up there — note there's no A**H*x = b here, because these are DOUBLE PRECISION numbers and conjugate transpose is a whole complex-number thing, iykyk 💀 — where b and x are n element vectors and A is an n by n unit or non-unit, upper or lower triangular matrix.

⚠️ same deal as before, no cap: no test for singularity or near-singularity is included in this routine. such tests must be performed before calling this routine. do your homework first bestie.

Parameters:

  • UPLO — is CHARACTER*1. on entry, UPLO specifies whether the matrix is an upper or lower triangular matrix, as follows:
    • UPLO = 'U' or 'u' → A is an upper triangular matrix.
    • UPLO = 'L' or 'l' → A is a lower triangular matrix.
  • TRANS — is CHARACTER*1. on entry, TRANS specifies the equations to be solved, as follows:
    • TRANS = 'N' or 'n' → A*x = b.
    • TRANS = 'T' or 't' → A**T*x = b.
    • TRANS = 'C' or 'c' → A**T*x = b.
  • DIAG — is CHARACTER*1. on entry, DIAG specifies whether or not A is unit triangular, as follows:
    • DIAG = 'U' or 'u' → A is assumed to be unit triangular.
    • DIAG = 'N' or 'n' → A is not assumed to be unit triangular.
  • N — is INTEGER. on entry, N specifies the order of the matrix A. N must be at least zero.
  • A — is DOUBLE PRECISION array, dimension ( LDA, N ). before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular matrix and the strictly lower triangular part of A is not referenced. before entry with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced.
🦋 note that when DIAG = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity.
  • LDA — is INTEGER. on entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ).
  • X — is DOUBLE PRECISION array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). before entry, the incremented array X must contain the n element right-hand side vector b. on exit, X is overwritten with the solution vector x.
  • INCX — is INTEGER. on entry, INCX specifies the increment for the elements of X. INCX must not be zero.
🙈 fun formatting moment: in dtrsv this next bit is printed inside the INCX blurb instead of in its own section, but it's the same info, so here it is verbatim:
!>
!>  Level 2 Blas routine.
!>
!>  -- Written on 22-October-1986.
!>     Jack Dongarra, Argonne National Lab.
!>     Jeremy Du Croz, Nag Central Office.
!>     Sven Hammarling, Nag Central Office.
!>     Richard Hanson, Sandia National Labs.
!>

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 142 of file dtrsv.f.

subroutine strsv</p> (character uplo, character trans, character diag, integer n, real, dimension(lda,) a, integer lda, real, dimension() x, integer incx)

🍓 STRSV

Purpose:

!>
!> STRSV  solves one of the systems of equations
!>
!>    A*x = b,   or   A**T*x = b,
!>
!> where b and x are n element vectors and A is an n by n unit, or
!> non-unit, upper or lower triangular matrix.
!>
!> No test for singularity or near-singularity is included in this
!> routine. Such tests must be performed before calling this routine.
!>

STRSV is the REAL-numbers one 💖 — solves A*x = b or A**T*x = b, where b and x are n element vectors and A is an n by n unit or non-unit, upper or lower triangular matrix. again, no A**H situation, because real numbers don't do that.

⚠️ and again, for the people in the back: no test for singularity or near-singularity is included in this routine. such tests must be performed before calling this routine. periodt.

Parameters:

  • UPLO — is CHARACTER*1. on entry, UPLO specifies whether the matrix is an upper or lower triangular matrix, as follows:
    • UPLO = 'U' or 'u' → A is an upper triangular matrix.
    • UPLO = 'L' or 'l' → A is a lower triangular matrix.
  • TRANS — is CHARACTER*1. on entry, TRANS specifies the equations to be solved, as follows:
    • TRANS = 'N' or 'n' → A*x = b.
    • TRANS = 'T' or 't' → A**T*x = b.
    • TRANS = 'C' or 'c' → A**T*x = b.
  • DIAG — is CHARACTER*1. on entry, DIAG specifies whether or not A is unit triangular, as follows:
    • DIAG = 'U' or 'u' → A is assumed to be unit triangular.
    • DIAG = 'N' or 'n' → A is not assumed to be unit triangular.
  • N — is INTEGER. on entry, N specifies the order of the matrix A. N must be at least zero.

…and that's where this half of the page runs out — the A parameter of STRSV picks up in part 2, no spoilers 💅✨

  • A — a REAL array, dimension ( LDA, N ). Before you come in with UPLO = 'U' or 'u', the leading n by n upper triangular part of A has to actually contain the upper triangular matrix, and the strictly lower triangular part of A is not referenced (so like, ghost it, bestie, nobody's looking at it). Before entry with UPLO = 'L' or 'l', it flips: the leading n by n lower triangular part of A must contain the lower triangular matrix, and the strictly upper triangular part of A is not referenced. And note — when DIAG = 'U' or 'u', the diagonal elements of A are not referenced either, they're just assumed to be unity. Assumed!! The audacity, honestly. 💅
  • LDA — LDA is INTEGER. On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). Non-negotiable, fr fr.
  • X — X is REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element right-hand side vector b. On exit, X is overwritten with the solution vector x. Yes, overwritten — she takes the wheel and doesn't give it back. 🚗
  • INCX — INCX is INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Like, ever. Zero is not the vibe. 🙈

🫶 Author

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

🌸 Further Details:

Level 2 Blas routine.

And the origin story, because we love a receipt:

-- Written on 22-October-1986.
   Jack Dongarra, Argonne National Lab.
   Jeremy Du Croz, Nag Central Office.
   Sven Hammarling, Nag Central Office.
   Richard Hanson, Sandia National Labs.

Definition at line 148 of file strsv.f. Iconic. 👑

💜 ztrsv 💜

Okay bestie, now the complex one. Same energy, bigger numbers, more drama ✨

subroutine ztrsv (character uplo, character trans, character diag, integer
    n, complex*16, dimension(lda,*) a, integer lda, complex*16, dimension(*)
    x, integer incx)

🎯 Purpose:

ZTRSV solves one of the systems of equations A*x = b, or A**T*x = b, or A**H*x = b, where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular matrix.

And the warning that matters, no cap:

No test for singularity or near-singularity is included in this routine. Such tests must be performed before calling this routine.

So like… she will not be checking if your matrix is a mess. That's on you, babes. Do your own vibe check first. 😭

✨ Parameters

  • UPLO — UPLO is CHARACTER*1. On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix, as follows:
    • UPLO = 'U' or 'u' — A is an upper triangular matrix.
    • UPLO = 'L' or 'l' — A is a lower triangular matrix.
  • TRANS — TRANS is CHARACTER*1. On entry, TRANS specifies the equations to be solved, as follows:
    • TRANS = 'N' or 'n' — A*x = b.
    • TRANS = 'T' or 't' — A**T*x = b.
    • TRANS = 'C' or 'c' — A**H*x = b.
  • DIAG — DIAG is CHARACTER*1. On entry, DIAG specifies whether or not A is unit triangular, as follows:
    • DIAG = 'U' or 'u' — A is assumed to be unit triangular.
    • DIAG = 'N' or 'n' — A is not assumed to be unit triangular.
  • N — N is INTEGER. On entry, N specifies the order of the matrix A. N must be at least zero. Zero is allowed! Slay, an empty matrix, we respect her. 🎀
  • A — A is COMPLEX*16 array, dimension ( LDA, N ). Before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular matrix and the strictly lower triangular part of A is not referenced. Before entry with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced. Note that when DIAG = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity. Same deal as her real sister — assumed unity, don't @ her. 🦋
  • LDA — LDA is INTEGER. On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). Periodt. 🔥
  • X — X is COMPLEX*16 array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element right-hand side vector b. On exit, X is overwritten with the solution vector x. Again — overwritten. She ate and left no crumbs. 🍓
  • INCX — INCX is INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Hard no on zero, bestie. 💀

🫶 Author

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

🌈 Further Details:

Level 2 Blas routine.

Same legendary origin, same date, same four icons:

-- Written on 22-October-1986.
   Jack Dongarra, Argonne National Lab.
   Jeremy Du Croz, Nag Central Office.
   Sven Hammarling, Nag Central Office.
   Richard Hanson, Sandia National Labs.

Definition at line 148 of file ztrsv.f. 💖

🧚 Author

Generated automatically by Doxygen for LAPACK from the source code. Not a human, just a little robot doing her best 🫶

And that's the whole page, bestie — every last triangle, every last !> comment, all of it. This lil masterpiece is from LAPACK, Version 3.12.0, and yes the footer says trsv(3) because we're fancy and documented 💅✨ Go solve your triangular systems, you absolute main character 👑

📦 where this page comes from