latrs3(3)

📦
version 3.12.1 · 9 versions
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💅 latrs3(3) — triangular solve with robust scaling, level 3 ✨

okay so bestie, this is latrs3(3), and it is literally the level-3 glow-up of triangular solves with robust scaling. iconic behavior. 👑

🌸 NAME 🌸

latrs3 - latrs3: triangular solve with robust scaling, level 3 🦋

✨ SYNOPSIS ✨

🎀 Functions 🎀

subroutine clatrs3 (uplo, trans, diag, normin, n, nrhs, a, lda, x, ldx,
    scale, cnorm, work, lwork, info)
  • CLATRS3 — solves a triangular system of equations with the scale factors set to prevent overflow. 💖
subroutine dlatrs3 (uplo, trans, diag, normin, n, nrhs, a, lda, x, ldx,
    scale, cnorm, work, lwork, info)
  • DLATRS3 — solves a triangular system of equations with the scale factors set to prevent overflow. 💖
subroutine slatrs3 (uplo, trans, diag, normin, n, nrhs, a, lda, x, ldx,
    scale, cnorm, work, lwork, info)
  • SLATRS3 — solves a triangular system of equations with the scale factors set to prevent overflow. 💖
subroutine zlatrs3 (uplo, trans, diag, normin, n, nrhs, a, lda, x, ldx,
    scale, cnorm, work, lwork, info)
  • ZLATRS3 — solves a triangular system of equations with the scale factors set to prevent overflow. 💖

📚 Detailed Description 📚

This page is about to get into the function docs, bestie. buckle up. 🎢

🧮 Function Documentation 🧮

💖 subroutine clatrs3 (character uplo, character trans, character diag, character normin, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldx, * ) x, integer ldx, real, dimension( * ) scale, real, dimension( * ) cnorm, real, dimension( * ) work, integer lwork, integer info)

subroutine clatrs3 (character uplo, character trans, character diag,
    character normin, integer n, integer nrhs, complex, dimension( lda, * )
    a, integer lda, complex, dimension( ldx, * ) x, integer ldx, real,
    dimension( * ) scale, real, dimension( * ) cnorm, real, dimension( * )
    work, integer lwork, integer info)

CLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow. slay. 💅

🌟 Purpose 🌟

okay so CLATRS3 solves one of the triangular systems

A * X = B * diag(scale),  A**T * X = B * diag(scale), or
A**H * X = B * diag(scale)

with scaling to prevent overflow. here A is an upper or lower triangular matrix, A**T denotes the transpose of A, and A**H denotes the conjugate transpose of A. X and B are n-by-nrhs matrices, and scale is an nrhs-element vector of scaling factors. a scaling factor scale(j) is usually less than or equal to 1, chosen such that X(:,j) is less than the overflow threshold. if the matrix A is singular (A(j,j) = 0 for some j), then a non-trivial solution to A*X = 0 is returned. if the system is so badly scaled that the solution cannot be represented as (1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned. 😭

this is a BLAS-3 version of LATRS for solving several right hand sides simultaneously. periodt. 🔥

🧸 Parameters 🧸

  • UPLO — UPLO is CHARACTER*1. specifies whether the matrix A is upper or lower triangular.
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • TRANS — TRANS is CHARACTER*1. specifies the operation applied to A.
    • = 'N': Solve A * x = s*b (No transpose)
    • = 'T': Solve A**T* x = s*b (Transpose)
    • = 'C': Solve A**T* x = s*b (Conjugate transpose)
  • DIAG — DIAG is CHARACTER*1. specifies whether or not the matrix A is unit triangular.
    • = 'N': Non-unit triangular
    • = 'U': Unit triangular
  • NORMIN — NORMIN is CHARACTER*1. specifies whether CNORM has been set or not.
    • = 'Y': CNORM contains the column norms on entry
    • = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.
  • N — N is INTEGER. the order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. the number of columns of X. NRHS >= 0.
  • A — A is COMPLEX 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.
  • LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max (1,N).
  • X — X is COMPLEX array, dimension (LDX,NRHS). on entry, the right hand side B of the triangular system. on exit, X is overwritten by the solution matrix X.
  • LDX — LDX is INTEGER. the leading dimension of the array X. LDX >= max (1,N).
  • SCALE — SCALE is REAL array, dimension (NRHS). the scaling factor s(k) is for the triangular system A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k). if SCALE = 0, the matrix A is singular or badly scaled. if A(j,j) = 0 is encountered, a non-trivial vector x(:,k) that is an exact or approximate solution to A*x(:,k) = 0 is returned. if the system is so badly scaled that the solution cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0 is returned. no cap, this one matters. 🫶
  • CNORM — CNORM is REAL array, dimension (N). if NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. if TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. if NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.
  • WORK — WORK is REAL array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, WORK(1) returns the optimal size of WORK.
  • LWORK — LWORK is INTEGER. the dimension of the array WORK. if MIN(N,NRHS) = 0, LWORK >= 1, else LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N + NB - 1)/NB and NB is the optimal block size. if LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -k, the k-th argument had an illegal value

🌟 Author 🌟

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

💫 Further Details: 💫

⭐ Definition at line 233 of file clatrs3.f. ⭐

💖 subroutine dlatrs3 (character uplo, character trans, character diag, character normin, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) scale, double precision, dimension( * ) cnorm, double precision, dimension( * ) work, integer lwork, integer info)

subroutine dlatrs3 (character uplo, character trans, character diag,
    character normin, integer n, integer nrhs, double precision, dimension(
    lda, * ) a, integer lda, double precision, dimension( ldx, * ) x,
    integer ldx, double precision, dimension( * ) scale, double precision,
    dimension( * ) cnorm, double precision, dimension( * ) work, integer
    lwork, integer info)

DLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow. 💜

🌟 Purpose 🌟

okay so DLATRS3 solves one of the triangular systems

A * X = B * diag(scale)  or  A**T * X = B * diag(scale)

with scaling to prevent overflow. here A is an upper or lower triangular matrix, and A**T denotes the transpose of A. X and B are n by nrhs matrices, and scale is an nrhs element vector of scaling factors. a scaling factor scale(j) is usually less than or equal to 1, chosen such that X(:,j) is less than the overflow threshold. if the matrix A is singular (A(j,j) = 0 for some j), then a non-trivial solution to A*X = 0 is returned. if the system is so badly scaled that the solution cannot be represented as (1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned. 😭

this is a BLAS-3 version of LATRS for solving several right hand sides simultaneously. periodt. 🔥

🧸 Parameters 🧸

  • UPLO — UPLO is CHARACTER*1. specifies whether the matrix A is upper or lower triangular.
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • TRANS — TRANS is CHARACTER*1. specifies the operation applied to A.
    • = 'N': Solve A * x = s*b (No transpose)
    • = 'T': Solve A**T* x = s*b (Transpose)
    • = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)
  • DIAG — DIAG is CHARACTER*1. specifies whether or not the matrix A is unit triangular.
    • = 'N': Non-unit triangular
    • = 'U': Unit triangular
  • NORMIN — NORMIN is CHARACTER*1. specifies whether CNORM has been set or not.
    • = 'Y': CNORM contains the column norms on entry
    • = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.
  • N — N is INTEGER. the order of the matrix A. N >= 0.
  • NRHS — NRHS is INTEGER. the number of columns of X. NRHS >= 0.
  • A —
  • A — DOUBLE PRECISION array, dimension (LDA,N). The triangular matrix A itself, bestie. 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 A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. And if DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1. ✨
  • LDA — INTEGER. The leading dimension of the array A. Gotta be LDA >= max (1,N). No exceptions, periodt.
  • X — DOUBLE PRECISION array, dimension (LDX,NRHS). On entry, this is the right hand side B of the triangular system. On exit? X gets overwritten by the solution matrix X 💅 so like, copy it first if you needed the original, iykyk.
  • LDX — INTEGER. The leading dimension of the array X. LDX >= max (1,N). Same energy as LDA, different letter 🎀
  • SCALE — DOUBLE PRECISION array, dimension (NRHS). The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k)  or  A**T* x(:,k) = s(k)*b(:,k)

If SCALE = 0, the matrix A is singular or badly scaled 😭 If A(j,j) = 0 is encountered, a non-trivial vector x(:,k) that is an exact or approximate solution to A*x(:,k) = 0 is returned. And if the system is so badly scaled that the solution cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0 is returned. She's doing her best, okay 🫶

  • CNORM — DOUBLE PRECISION array, dimension (N).

If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm.

If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A. Cute little switch-up, love that for her 🌸

  • WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal size of WORK.
  • LWORK — INTEGER. The dimension of the array WORK.

If MIN(N,NRHS) = 0, LWORK >= 1, else

LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where
NBA = (N + NB - 1)/NB and NB is the optimal block size.

If LWORK = -1, then a workspace query is assumed 👀 the routine only calculates the optimal dimensions of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. So like, it won't yell at you. Respect.

  • INFO — INTEGER.
    • = 0: successful exit 🎉
    • < 0: if INFO = -k, the k-th argument had an illegal value 💀

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔍 Further Details:

Definition at line 232 of file dlatrs3.f.

🌈 slatrs3 — the REAL one 🌈

subroutine slatrs3 (character uplo, character trans, character diag,
    character normin, integer n, integer nrhs, real, dimension( lda, * ) a,
    integer lda, real, dimension( ldx, * ) x, integer ldx, real, dimension(
    * ) scale, real, dimension( * ) cnorm, real, dimension( * ) work,
    integer lwork, integer info)

SLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow. Single precision edition, she's just as iconic 💅

🎀 Purpose 🎀

SLATRS3 solves one of the triangular systems fortran A * X = B * diag(scale) or A**T * X = B * diag(scale) with scaling to prevent overflow. Here A is an upper or lower triangular matrix, A**T denotes the transpose of A. X and B are n by nrhs matrices and scale is an nrhs element vector of scaling factors. A scaling factor scale(j) is usually less than or equal to 1, chosen such that X(:,j) is less than the overflow threshold. If the matrix A is singular (A(j,j) = 0 for some j), then a non-trivial solution to A*X = 0 is returned. If the system is so badly scaled that the solution cannot be represented as (1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned. This is a BLAS-3 version of LATRS for solving several right hand sides simultaneously. 🚀

🍓 Parameters 🍓

  • UPLO — CHARACTER*1. Specifies whether the matrix A is upper or lower triangular.
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • TRANS — CHARACTER*1. Specifies the operation applied to A.
    • = 'N': Solve A * x = s*b (No transpose)
    • = 'T': Solve A**T* x = s*b (Transpose)
    • = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)
  • DIAG — CHARACTER*1. Specifies whether or not the matrix A is unit triangular.
    • = 'N': Non-unit triangular
    • = 'U': Unit triangular
  • NORMIN — CHARACTER*1. Specifies whether CNORM has been set or not.
    • = 'Y': CNORM contains the column norms on entry
    • = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.
  • N — INTEGER. The order of the matrix A. N >= 0.
  • NRHS — INTEGER. The number of columns of X. NRHS >= 0.
  • A — 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.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max (1,N).
  • X — REAL array, dimension (LDX,NRHS). On entry, the right hand side B of the triangular system. On exit, X is overwritten by the solution matrix X 💖
  • LDX — INTEGER. The leading dimension of the array X. LDX >= max (1,N).
  • SCALE — REAL array, dimension (NRHS). The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k)  or  A**T* x(:,k) = s(k)*b(:,k)

If SCALE = 0, the matrix A is singular or badly scaled. If A(j,j) = 0 is encountered, a non-trivial vector x(:,k) that is an exact or approximate solution to A*x(:,k) = 0 is returned. If the system is so badly scaled that the solution cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0 is returned.

  • CNORM — REAL array, dimension (N).

If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm.

If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.

  • WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. (Note the glow-up: this one says optimal LWORK, not optimal size of WORK — keep your eyes peeled, bestie 👀)
  • LWORK — INTEGER. The dimension of the array WORK.

If MIN(N,NRHS) = 0, LWORK >= 1, else

LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where
NBA = (N + NB - 1)/NB and NB is the optimal block size.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

  • INFO — INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -k, the k-th argument had an illegal value

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔍 Further Details:

Definition at line 231 of file slatrs3.f.

💜 zlatrs3 — complex*16, double precision energy 💜

subroutine zlatrs3 (character uplo, character trans, character diag,
    character normin, integer n, integer nrhs, complex*16, dimension( lda, *
    ) a, integer lda, complex*16, dimension( ldx, * ) x, integer ldx, double
    precision, dimension( * ) scale, double precision, dimension( * ) cnorm,
    double precision, dimension( * ) work, integer lwork, integer info)

ZLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow. Complex maths, still zero overflow drama 🦋

🫶 Purpose 🫶

and this is exactly where ZLATRS3 starts explaining herself—

💅 Parameters 💅

Okay bestie, this is the whole cast of arguments — every single one is doing something, so don’t skim. 💖

  • UPLO — CHARACTER*1. Tells us whether A is upper or lower triangular.
    • 'U': Upper triangular
    • 'L': Lower triangular
  • TRANS — CHARACTER*1. Says which operation is being applied to A.
    • 'N': Solve A * x = s*b (No transpose)
    • 'T': Solve A**T* x = s*b (Transpose)
    • 'C': Solve A**T* x = s*b (Conjugate transpose)
  • DIAG — CHARACTER*1. Is A unit triangular or not? That’s the whole question.
    • 'N': Non-unit triangular
    • 'U': Unit triangular
  • NORMIN — CHARACTER*1. Has CNORM already been set, or are we filling it in now?
    • 'Y': CNORM contains the column norms on entry
    • 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.
  • N — INTEGER. The order of the matrix A. N >= 0.
  • NRHS — INTEGER. The number of columns of X. NRHS >= 0.
  • A — COMPLEX*16 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.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max (1,N).
  • X — COMPLEX*16 array, dimension (LDX,NRHS). On entry, the right hand side B of the triangular system. On exit, X is overwritten by the solution matrix X.
  • LDX — INTEGER. The leading dimension of the array X. LDX >= max (1,N).
  • SCALE — DOUBLE PRECISION array, dimension (NRHS). The scaling factor s(k) is for the triangular system A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k). If SCALE = 0, the matrix A is singular or badly scaled. If A(j,j) = 0 is encountered, a non-trivial vector x(:,k) that is an exact or approximate solution to A*x(:,k) = 0 is returned. If the system is so badly scaled that solution cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0 is returned.
  • CNORM — DOUBLE PRECISION array, dimension (N).
    • If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm.
    • If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.
  • WORK — DOUBLE PRECISION array, dimension (LWORK). On exit, if INFO = 0, WORK(1) returns the optimal size of WORK.
  • LWORK — INTEGER. The dimension of the array WORK.
    • If MIN(N,NRHS) = 0, LWORK >= 1, else LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N + NB - 1)/NB and NB is the optimal block size.
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • INFO — INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -k, the k-th argument had an illegal value

👑 Author 👑

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

🔍 Further Details 🔍

Definition at line 233 of file zlatrs3.f.

🧚 Author (the generated-by one) 🧚

Generated automatically by Doxygen for LAPACK from the source code.

That’s the whole page, bestie 💅✨ The footer tags this as LAPACK Version 3.12.0 and latrs3(3) — no separate date printed on it, so we’re not inventing one 🫶

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