CLATRS(3)

💅 CLATRS(3) — solving triangular systems with the scale factor set to prevent overflow 💅

okay so bestie, this is CLATRS(3) from the Library Functions Manual, and the path floating around near the top is /home/abui...RC/clatrs.f(3). Slay. ✨

🎀 NAME 🎀

The NAME section is literally just this source-file path, no cap:

/home/abuild/rpmbuild/BUILD/lapack-
static-3.12.0-build/lapack-3.12.0/SRC/clatrs.f

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

subroutine CLATRS (uplo, trans, diag, normin, n, a, lda, x, scale,
    cnorm, info)

CLATRS solves a triangular system of equations with the scale factor set to prevent overflow. Periodt. 💖

🌸 Function/Subroutine Documentation 🌸

Here's the full signature, bestie:

subroutine CLATRS (character uplo, character trans, character diag,
    character normin, integer n, complex, dimension( lda, * ) a, integer
    lda, complex, dimension( * ) x, real scale, real, dimension( * ) cnorm,
    integer info)

CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

💖 Purpose 💖

okay so CLATRS solves one of the triangular systems:

A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,

with scaling to prevent overflow. Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, A**H denotes the conjugate 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. Literally protecting your numbers from overflow drama. If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTRSV 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. Not me explaining singular matrices at 2am. 😭

🌈 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**H * x = s*b (Conjugate 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.
• A — 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 — INTEGER. The leading dimension of the array A. LDA >= max (1,N).
• X — COMPLEX 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 — REAL. The scaling factor s for the triangular system

A * x = s*b,  A**T * x = s*b,  or  A**H * 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 — 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.
• 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 🔥

A rough bound on x is computed; if that is less than overflow, CTRSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. Every single operation, bestie. 💅

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 CTRSV 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. Yes, that's the dramatic part. 💀

Similarly, a row-wise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for A upper triangular is:

for j = 1, ..., n
     x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
end

We simultaneously compute two bounds:

G(j) = bound on ( b(i) - A[1:i-1,i]' * 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 CTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

Definition at line 237 of file clatrs.f.

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the whole page, bestie 💖 This was LAPACK Version 3.12.0, from /home/abui...RC/clatrs.f(3) — the page doesn't list a date, so we're just gonna vibe with the version. Bye bestie 🫶✨