SRC/clatps.f(3)

✨ Bestie, you're looking at SRC/clatps.f(3) from the Library Functions Manual — and this page is all about the CLATPS subroutine. Buckle up, it's giving linear algebra but make it cute. 💅

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🌸 NAME

SRC/clatps.f

That's the whole NAME section — iconic minimalism, no notes. 🎀

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✨ SYNOPSIS

🧚 Functions/Subroutines

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

CLATPS solves a triangular system of equations with the matrix held in packed storage. Slay. 💖

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💖 Function/Subroutine Documentation

💫 The subroutine

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

CLATPS solves a triangular system of equations with the matrix held in packed storage. Periodt. ✨

🌟 Purpose:

Okay so CLATPS 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, 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. No cap, that's the whole vibe.

If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTPSV 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

UPLO is CHARACTER*1.

Specifies whether the matrix A is upper or lower triangular.

• = 'U': Upper triangular
• = 'L': Lower triangular

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)

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.

N

N is INTEGER.

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

AP

AP is COMPLEX 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

X is 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

SCALE is 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. That's a big mood, so pay attention.

CNORM

CNORM is 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

INFO is 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.

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🔥 Further Details:

A rough bound on x is computed; if that is less than overflow, CTPSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. Safety first, 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 CTPSV 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. Not me reading that twice, but okay. 💀

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 CTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow). Math, but make it fashion. 🌈

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Definition at line 229 of file clatps.f.

📝 Author

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Cute sign-off, bestie: this page is from SRC/clatps.f, LAPACK Version 3.12.0, and the footer's SRC/clatps.f(3) is right there — the original footer didn't include a date, so I'm not inventing one. 🫶💖