SRC/dlatrs.f(3)

💅 SRC/dlatrs.f(3) 💅

okay so bestie, this page is from the LAPACK Library Functions Manual, and it's about SRC/dlatrs.f — the DLATRS routine. no cap, it's lowkey iconic: it solves a triangular system and scales things so overflow doesn't crash the party. ✨

🎀 NAME 🎀

SRC/dlatrs.f — yep, that's the file. DLATRS solves a triangular system of equations with the scale factor set to prevent overflow. periodt. 🔥

🦋 SYNOPSIS 🦋

Functions/Subroutines

• subroutine dlatrs (uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info) — ngl this is the whole vibe: DLATRS solves a triangular system of equations with the scale factor set to prevent overflow. 💖

🌈 Function/Subroutine Documentation 🌈

🫶 subroutine dlatrs 🫶

subroutine dlatrs (character uplo, character trans, character diag, character normin, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) x, double precision scale, double precision, dimension( * ) cnorm, integer info)

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

🎯 Purpose:

!>
!> DLATRS solves one of the triangular systems
!>
!>    A *x = s*b  or  A**T *x = s*b
!>
!> 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-element vectors, and s is a scaling factor, usually less than
!> or equal to 1, chosen so that the components of x will be less than
!> the overflow threshold.  If the unscaled problem will not cause
!> overflow, the Level 2 BLAS routine DTRSV is called.  If the matrix A
!> is singular (A(j,j) = 0 for some j), then s is set to 0 and a
!> non-trivial solution to A*x = 0 is returned.
!>

💖 Parameters 💖

• UPLO — UPLO is CHARACTER*1. Specifies whether the matrix A is upper or lower triangular. Values: = 'U': Upper triangular, = 'L': Lower triangular. ✨
• TRANS — TRANS is CHARACTER*1. Specifies the operation applied to A. Values: = '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. Values: = 'N': Non-unit triangular, = 'U': Unit triangular. 🎀
• NORMIN — NORMIN is CHARACTER*1. Specifies whether CNORM has been set or not. Values: = '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 matrix A. N >= 0. no cap.
• A — A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N). On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x. ✨
• SCALE — SCALE is DOUBLE PRECISION. The scaling factor s for the triangular system A * x = s*b or A**T* x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0. ⚠️
• CNORM — CNORM is 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. 🫶
• INFO — INFO is INTEGER. Values: = 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: 🔥

okay so the docs go full math mode here — this is the bound/scaling tea, bestie:

!>
!>  A rough bound on x is computed; if that is less than overflow, DTRSV
!>  is called, otherwise, specific code is used which checks for possible
!>  overflow or divide-by-zero at every operation.
!>
!>  A columnwise scheme is used for solving A*x = b.  The basic algorithm
!>  if A is lower triangular is
!>
!>       x[1:n] := b[1:n]
!>       for j = 1, ..., n
!>            x(j) := x(j) / A(j,j)
!>            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
!>       end
!>
!>  Define bounds on the components of x after j iterations of the loop:
!>     M(j) = bound on x[1:j]
!>     G(j) = bound on x[j+1:n]
!>  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
!>
!>  Then for iteration j+1 we have
!>     M(j+1) <= G(j) / | A(j+1,j+1) |
!>     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
!>            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
!>
!>  where CNORM(j+1) is greater than or equal to the infinity-norm of
!>  column j+1 of A, not counting the diagonal.  Hence
!>
!>     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
!>                  1<=i<=j
!>  and
!>
!>     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
!>                                   1<=i< j
!>
!>  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
!>  reciprocal of the largest M(j), j=1,..,n, is larger than
!>  max(underflow, 1/overflow).
!>
!>  The bound on x(j) is also used to determine when a step in the
!>  columnwise method can be performed without fear of overflow.  If
!>  the computed bound is greater than a large constant, x is scaled to
!>  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
!>  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
!>
!>  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
!>  algorithm for A upper triangular is
!>
!>       for j = 1, ..., n
!>            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
!>       end
!>
!>  We simultaneously compute two bounds
!>       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
!>       M(j) = bound on x(i), 1<=i<=j
!>
!>  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
!>  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
!>  Then the bound on x(j) is
!>
!>       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
!>
!>            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
!>                      1<=i<=j
!>
!>  and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater
!>  than max(underflow, 1/overflow).
!>

Definition at line 236 of file dlatrs.f.

🌸 Author 🌸

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