SRC/zlatrs.f(3)

💅 SRC/zlatrs.f(3) — Library Functions Manual 🎀

🌸 NAME 🌸

SRC/zlatrs.f

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🦋 SYNOPSIS 🦋

Functions/Subroutines

• subroutine zlatrs — ZLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

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

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✨ Function/Subroutine Documentation ✨

💖 subroutine zlatrs 💖

subroutine zlatrs (character uplo, character trans, character diag,
    character normin, integer n, complex*16, dimension( lda, * ) a, integer
    lda, complex*16, dimension( * ) x, double precision scale, double
    precision, dimension( * ) cnorm, integer info)

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

🎀 Purpose 🎀

okay so bestie, here's the vibe: ZLATRS solves one of these 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. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTRSV 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.

no cap, that singular case matters: s goes to 0 and you get a non-trivial solution to A*x = 0. 💀

💅 Parameters 💅

• UPLO — CHARACTER*1. Tells you whether matrix A is upper or lower triangular.
  • = 'U': Upper triangular
  • = 'L': Lower triangular
• TRANS — CHARACTER*1. Says which operation is 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. Says whether or not matrix A is unit triangular.
  • = 'N': Non-unit triangular
  • = 'U': Unit triangular
• NORMIN — CHARACTER*1. Says whether CNORM has already 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 matrix A. N >= 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 (N). On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.
• SCALE — DOUBLE PRECISION. 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 — 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 — 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, ZTRSV 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 ZTRSV 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 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 ZTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

Definition at line 237 of file zlatrs.f.

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⭐ Author ⭐

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okay bestie, that's the whole page — the footer gives LAPACK Version 3.12.0 and SRC/zlatrs.f(3), and that's the cutest little source/version sign-off. mwah 💋✨