SRC/zlatps.f(3)

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version 3.12.1 · 4 versions
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💅 SRC/zlatps.f(3) — Library Functions Manual 🎀

okay so bestie, buckle up — this is the manual page for SRC/zlatps.f, straight out of the Library Functions Manual, and it's genuinely iconic. It's giving "linear algebra but make it safe from overflow" and honestly? Obsessed. ✨

🌸 NAME 🌸

SRC/zlatps.f — the file, the legend, the whole vibe 💖

🦋 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. no cap, that's literally what it does 🫶

👑 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. periodt. 💅

✨ Purpose ✨

ZLATPS solves one of the triangular systems fortran 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. 😭 (she's doing her best and honestly, so are we)

🌈 Parameters 🌈

UPLO

UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. - = 'U': Upper triangular - = 'L': Lower triangular

tl;dr: upper or lower, pick your fighter 💀

TRANS

TRANS is 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)

three whole modes, main character energy fr fr 🔥

DIAG

DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. - = 'N': Non-unit triangular - = 'U': Unit triangular

NORMIN

NORMIN is 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.

it's giving "did you already do the homework or should I" 🙈

N

N is INTEGER The order of the matrix A. N >= 0.

zero is legal, negative is not, that's the vibe check ⭐

AP

AP is 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.

packed storage is lowkey the closet organizer of linear algebra — everything folded, nothing wasted 💜

X

X is 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.

she gets replaced, iconic behaviour, no notes 🦋

SCALE

SCALE is DOUBLE PRECISION The scaling factor s for the triangular system fortran 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.
⚠️ warning bestie: if SCALE comes back as 0, that matrix is singular or badly scaled. not a drill 😭

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.
⚠️ yeah so those "greater than or equal to" bits are on you — feed her a norm that's too small and things go sideways 🙈

INFO

INFO is INTEGER - = 0: successful exit - < 0: if INFO = -k, the k-th argument had an illegal value

so if it's negative it's snitching on which argument was messy 💀

🫶 Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

(the whole squad, credited, as they should be 💅)

🌸 Further Details: 🌸

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 fortran 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 fortran 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 fortran G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and fortran |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 fortran 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 fortran 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).

like… the math is mathing, the bounds are bounding, we love a safety-first queen 🧚‍♀️⭐

🎀 Definition 🎀

Definition at line 229 of file zlatps.f.

💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code. (yes, a robot wrote the page, but the vibes are all ours 🫶)

and that's the whole page, bestie — go forth and solve those triangular systems 💅✨

sign-off: this cutie came from LAPACK, Version 3.12.0, page SRC/zlatps.f(3) 🎀💖 no cap, that's the whole footer and she ate 🌸

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