SRC/clatrs3.f(3)

📦
version 3.12.1 · 3 versions
, in arch rolling

💅 NAME

Ok bestie, you're on the SRC/clatrs3.f(3) page in the Library Functions Manual 💅, and the main character is SRC/clatrs3.f — a routine that solves triangular systems while keeping the numbers from overflowing. It’s giving “math but make it safe” energy. ✨

✨ SYNOPSIS

🌟 Functions/Subroutines

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. No cap. 🫶

🧚 Function/Subroutine Documentation

🦋 subroutine clatrs3

Here’s the full signature, bestie, copied exactly so you don’t have to squint:

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. That’s the whole point, and honestly? Iconic. 💖

🌸 Purpose

Ok so CLATRS3 solves one of these 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. 🔥

🫶 Parameters

🌟 UPLO

UPLO is CHARACTER*1. It specifies whether the matrix A is upper or lower triangular.

  • = 'U': Upper triangular
  • = 'L': Lower triangular

🌟 TRANS

TRANS is CHARACTER*1. It 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. It specifies whether or not the matrix A is unit triangular.

  • = 'N': Non-unit triangular
  • = 'U': Unit triangular

🌟 NORMIN

NORMIN is CHARACTER*1. It 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. It’s the order of the matrix A. N >= 0. 🫡

🌟 NRHS

NRHS is INTEGER. It’s the number of columns of X. NRHS >= 0.

🌟 A

A is COMPLEX array, dimension (LDA,N). It’s 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. It’s the leading dimension of the array A. LDA >= max (1,N).

🌟 X

X is COMPLEX array, dimension (LDX,NRHS).

On entry, it’s the right hand side B of the triangular system. On exit, X is overwritten by the solution matrix X. Literally the glow-up happens in place. 💅

🌟 LDX

LDX is INTEGER. It’s 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. 🫠

🌟 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. It’s 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

Not much extra tea here, bestie — just the source pointer:

Definition at line 233 of file clatrs3.f.

🎀 Author

Generated automatically by Doxygen for LAPACK from the source code.

That’s the whole page, bestie 💖 — from LAPACK Version 3.12.0, SRC/clatrs3.f(3). Slay, stay scaled, and don’t overflow. ✨

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