ZSPTRF(3)

💅 NAME

ok so bestie, this whole page is literally about /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zsptrf.f — that's the file, that's the whole vibe, it's a LAPACK source file and she's the main character here ✨

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🦋 SYNOPSIS

Functions/Subroutines

subroutine ZSPTRF (uplo, n, ap, ipiv, info)

ZSPTRF — like, that's her name, say it with your chest 💖

🌸 Function/Subroutine Documentation

subroutine ZSPTRF (character uplo, integer n, complex*16, dimension( * ) ap,
    integer, dimension( * ) ipiv, integer info)

ZSPTRF, again, iconic behavior, we love a signature reveal 👑

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✨ Purpose: (the why, the lore)

ok so picture this 🎀

ZSPTRF computes the factorization of a complex symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method:

   A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

like, she's out here splitting A into pieces. it's giving factorization slay 🔥

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🌈 Parameters (the cast of characters)

• UPLO

UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

so you gotta tell her which triangle you're vibing with, upper or lower, no in-between 💅

• N

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

the order of the matrix, and she does NOT accept negative energy, N >= 0 periodt.

• AP

AP is COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric 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. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L, stored as a packed triangular matrix overwriting A (see below for further details).

ok so AP comes in as your triangle, all packed columnwise into one linear array (dimension (N*(N+1)/2) btw), and then on exit she's been completely rewritten in place with D and the multipliers — highkey a glow-up AND a takeover at the same time 😭

• IPIV

IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

this one is the receipts 📋 — every swap, every block. positive means a 1-by-1 block and rows/cols k and IPIV(k) swapped. negative means 2-by-2 block energy, and the index you're looking at depends on whether UPLO = 'U' (peep IPIV(k-1)) or UPLO = 'L' (peep IPIV(k+1)). iykyk 💀

• INFO

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

the vibe check at the end 🫶 0 = we ate. negative = bestie, argument i was illegal, fix it. positive = the factorization finished BUT D(i,i) is exactly zero, meaning D is singular and you will literally divide by zero if you try to solve with it. that's a real warning, do not play with it 🚨

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👑 Author

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

the whole squad, credited, as they should be 💜

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🧚 Further Details: (the deep lore, buckle up)

5-96 - Based on modifications by J. Lewis, Boeing Computer Services Company

ok so if UPLO = 'U', then A = U*D*U**T, where

   U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

            (   I    v    0   )   k-s
    U(k) =  (   0    I    0   )   s
            (   0    0    I   )   n-k
               k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

so k counts down from n to 1, steps of 1 or 2, and the overwrite map is very specific — memorize it or perish 💀

now flip it: if UPLO = 'L', then A = L*D*L**T, where

   L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

            (   I    0     0   )  k-1
    L(k) =  (   0    I     0   )  s
            (   0    v     I   )  n-k-s+1
               k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

same energy but this time k goes up from 1 to n, steps of 1 or 2. yes the indices are different, yes you have to read them carefully, no you cannot skip this 🍓

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📍 Definition at line 157 of file zsptrf.f.

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💖 Author

Generated automatically by Doxygen for LAPACK from the source code.

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aaaand that's the page bestie 🫶 this whole thing is LAPACK, Version 3.12.0, and it lives at /home/abui...RC/zsptrf.f(3) — mwah, go factor something symmetric, you've got this 💅✨🎀