SLATPS(3)

📦 version 3.12.0, in opensuse-leap 16.0

🎀 SLATPS(3) 🎀

ok so first things first bestie — this page lives at /home/abui...RC/slatps.f(3) and it's part of the Library Functions Manual 💅 (yes the header is literally a path, don't @ me, LAPACK does what it wants).

✨ NAME ✨

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

like... that's the whole name section. it's giving "my full legal name is my file path" 😭 iconic honestly.

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine SLATPS (uplo, trans, diag, normin, n, ap, x, scale, cnorm, info)

SLATPS solves a triangular system of equations with the matrix held in packed storage. periodt. 💖

🦋 Function/Subroutine Documentation 🦋

subroutine SLATPS (character uplo, character trans, character diag,
    character normin, integer n, real, dimension( * ) ap, real, dimension( *
    ) x, real scale, real, dimension( * ) cnorm, integer info)

SLATPS solves a triangular system of equations with the matrix held in packed storage. (yes I said it twice, it's important, let it sink in 🫶)

💜 Purpose 💜

!> !> SLATPS solves one of the triangular systems !> !> A *x = s*b or AT*x = s*b !> !> with scaling to prevent overflow, where A is an upper or lower !> triangular matrix stored in packed form. Here AT 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 !> STPSV 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. !>

so like, the whole vibe is: solve A *x = s*b or A**T*x = s*b, but with a little scaling safety net so nothing explodes into overflow 🔥. A is upper or lower triangular and stored packed (no zeros stored, we're efficient 💅). If the unscaled version is chill, it just calls the Level 2 BLAS routine STPSV. and if A turns out singular — A(j,j) = 0 for some j — then s gets set to 0 and you get a non-trivial solution to A*x = 0 instead. plot twist but make it linear algebra 😭

⭐ Parameters ⭐

  • UPLO — CHARACTER*1. tells us if A is upper or lower triangular:
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • TRANS — CHARACTER*1. the operation applied to A:
    • = '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 — CHARACTER*1. is A unit triangular or not:
    • = 'N': Non-unit triangular
    • = 'U': Unit triangular
  • NORMIN — CHARACTER*1. has CNORM 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. (no negatives bestie, that's illegal, literally)
  • AP — REAL 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 lives in AP like this:
    • 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 — REAL 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, that's just how it is 🫶)
  • SCALE — REAL. 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 — 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.

(the whole squad, no cap 🫶)

🌈 Further Details 🌈

ok this is the part where it gets good, grab a snack 🍓

!> !> A rough bound on x is computed; if that is less than overflow, STPSV !> 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 STPSV 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. 😭 that's the emergency exit, respect it.

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 STPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow). slay 💅

🧚 Definition 🧚

Definition at line 227 of file slatps.f.

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

(so nobody "wrote" this page, a robot did, and honestly? she ate 🤖💖)

and that's the whole thing bestie ✨ — LAPACK, Version 3.12.0, live from /home/abui...RC/slatps.f(3) 🎀 go solve something triangular, ily, bye 💜

📦 where this page comes from