ZLATPS(3)

okay so bestie, this is the zlatps.f(3) page from the Library Functions Manual, and the page path is /home/abui...RC/zlatps.f(3). Buckle up, because ZLATPS is lowkey iconic: it solves triangular systems with the matrix held in packed storage, and it does the scaling thing so overflow stays uninvited. 💅

🌸 NAME 🌸

The source file for this page is literally:

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

That's the whole NAME section, no cap.

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

• subroutine ZLATPS (uplo, trans, diag, normin, n, ap, x, scale, cnorm, info) — ZLATPS solves a triangular system of equations with the matrix held in packed storage.

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💖 Function/Subroutine Documentation 💖

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

ZLATPS solves a triangular system of equations with the matrix held in packed storage.

🦋 Purpose 🦋

okay so the actual purpose is this: ZLATPS 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, where A is an upper or lower triangular matrix stored in packed form. Here 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 ZTPSV 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 — 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.
• AP — COMPLEX*16 array, dimension (N*(N+1)/2). The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=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 🔥

Here's the lore, bestie: a rough bound on x is computed; if that is less than overflow, ZTPSV 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 ZTPSV 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 ZTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

Definition at line 229 of file zlatps.f.

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💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code.

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Sign-off time, bestie: this is LAPACK Version 3.12.0, source zlatps.f(3), page path /home/abui...RC/zlatps.f(3), and the footer doesn't print a date, so that's the whole tea. Stay shiny! 💖✨🎀