/home/abui...sytrf_rk.f(3)

🦋 ZSYTRF_RK — the manual page, but make it make sense 🦋

okay so real talk, this is a page from the Library Functions Manual (yes, that's the manual name, bestie, and it's giving very "I live in /home/abui...sytrf_rk.f(3)" energy). It's a LAPACK routine, and honestly? She's a lot. But we're gonna get through it together. Grab a strawberry 🍓 and let's go.

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✨ NAME ✨

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

like... that's literally the whole name. that's the file. that's her address. iconic of her to be a whole path ngl 💅

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🌈 SYNOPSIS 🌈

Functions/Subroutines

subroutine ZSYTRF_RK (uplo, n, a, lda, e, ipiv, work, lwork, info)

ZSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). no cap, that's the whole vibe. 🔥

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👑 Function/Subroutine Documentation 👑

subroutine ZSYTRF_RK (character uplo, integer n, complex*16, dimension( lda,
    * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( *
    ) ipiv, complex*16, dimension( * ) work, integer lwork, integer info)

ZSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

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🎀 Purpose 🎀

okay so here's the actual assignment. ZSYTRF_RK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where:

• U (or L) is unit upper (or lower) triangular matrix,
• U**T (or L**T) is the transpose of U (or L),
• P is a permutation matrix,
• P**T is the transpose of P,
• and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS. bestie she's BLOCKED and she's Level 3, the way that she batches her work 😭🔥

For more information see the Further Details section. (spoiler: the Further Details section is a jumpscare, we'll get there 🙈)

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💖 Parameters 💖

every single one, miss girl, don't skip. periodt.

UPLO

UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U':  Upper triangular
= 'L':  Lower triangular

so like, which half of A do you actually care about? that's the whole question. 'U' = upper triangular, 'L' = lower triangular. pick one, be decisive. 💅

N

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

the order of the matrix A. and N >= 0 — zero is legal, don't come for her.

A

A is COMPLEX*16 array, dimension (LDA,N)

on entry, the symmetric matrix A. here's the tea on how she's stored:

• If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. (not referenced!! she doesn't even look at it, rude but efficient)
• If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, A contains:

• a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
• b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

she reuses her own storage. recycling queen 🧚

LDA

LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).

the leading dimension of the array A. LDA >= max(1,N). it's giving strict but fair.

E

E is COMPLEX*16 array, dimension (N)

on exit, E contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where:

• If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
• If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.

no thoughts, just zeros in the right spots, we love a clean edge case. ✨

IPIV

IPIV is INTEGER array, dimension (N)

okay this one is the whole lore drop, buckle up 🫶

IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see Further Details section.

If UPLO = 'U' (in factorization order, k decreases from N to 1):

• a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred.
• b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
  1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
  2. If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
• c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
• d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = 'L' (in factorization order, k increases from 1 to N):

• a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred.
• b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
  1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
  2. If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
• c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
• d) NOTE: Any entry IPIV(k) is always NONZERO on output.

yeah. she's detailed. she's thorough. she has notes. iykyk 😭💀

WORK

WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ).

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. she tells you her ideal size. queen of communicating her needs. 👑

LWORK

LWORK is INTEGER
The length of WORK.  LWORK >=1.  For best performance
LWORK >= N*NB, where NB is the block size returned
by ILAENV.

so LWORK is the length of WORK. LWORK >= 1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. not XERBLA minding her own business for once 🫶

INFO

INFO is INTEGER
= 0: successful exit

< 0: If INFO = -k, the k-th argument had an illegal value

> 0: If INFO = k, the matrix A is singular, because:
       If UPLO = 'U': column k in the upper
       triangular part of A contains all zeros.
       If UPLO = 'L': column k in the lower
       triangular part of A contains all zeros.

so if INFO = 0 that's a successful exit — we slayed, we ate, no notes 🎀

if INFO < 0: if INFO = -k, the k-th argument had an illegal value. someone handed her a bad argument and she's calling it out by number.

if INFO > 0: if INFO = k, the matrix A is singular, because:

• If UPLO = 'U': column k in the upper triangular part of A contains all zeros.
• If UPLO = 'L': column k in the lower triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. 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. 💀 not the division by zero bestie, that's not the vibe!!

NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.

she reports the first problem and then keeps going. honestly? respect. the grind doesn't stop. 🔥

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🌸 Author 🌸

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

four whole institutions collaborated on this and that's honestly the most group-project-to-ever-group-project. 🫶

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⭐ Further Details: ⭐

TODO: put correct description

okay so this is not a bit, that's literally what the manpage says. the Further Details section is a TODO. she's a work in progress and she's honest about it. the way that I respect the transparency 🙈💖

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🧚 Contributors: 🧚

• December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
• September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

shoutout to the whole squad, dates and departments included, because we don't erase people's credit in this house 💜

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🎀 Definition at line 257 of file zsytrf_rk.f. 🎀

like, if you really want to go read her in the raw source, line 257, be my guest, she's right there. 👑

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🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code. so nobody wrote this prose, a robot did, and honestly the robot ate. 🤖✨

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LAPACK — Version 3.12.0 🌸

and that's the page!! footer says the source/version is Version 3.12.0 and the page is /home/abui...sytrf_rk.f(3) — so like, mwah, that's her signature, we're done here. go forth and factorize your complex symmetric indefinite matrices, bestie, you've got this. 🦋💅🔥