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

💜 CSYTRF_RK 💜

Okay bestie, welcome to the Library Functions Manual page for /home/abui...sytrf_rk.f(3) — yes that's the whole filename situation, we love an ellipsis era. Anyway. Sit down, hydrate, and let's talk about the complex symmetric indefinite factorization girlboss of your dreams. 🫶

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🌸 NAME 🌸

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

That's it. That's the name. She's a whole file path and she's thriving. 💅

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✨ SYNOPSIS ✨

Functions/Subroutines

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

CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method — and it's the BLAS3 blocked algorithm version, no cap. 🔥

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🦋 Function/Subroutine Documentation 🦋

The whole signature, in all her glory 🎀

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

CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). She's booked, busy, and blocked — literally.

👑 Purpose

Bestie. Listen. CSYTRF_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 a 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. Iconic block structure honestly.

This is the blocked version of the algorithm, calling Level 3 BLAS (so she's fast, she's efficient, she's that girl 💅). For more information see the Further Details section.

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🎀 Parameters 🎀

UPLO

UPLO is CHARACTER*1. It specifies whether the upper or lower triangular part of the symmetric matrix A is stored:

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

Pick one, queen. Don't be weird about it.

N

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

A

A is COMPLEX array, dimension (LDA,N).

On entry, the symmetric matrix A. Here's the tea 🍵:

• 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. (Left on read.)
• 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); (the 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.

LDA

LDA is INTEGER. The leading dimension of the array A. Constraint: LDA >= max(1,N). Respect the leading dimension, she has boundaries. 🚧

E

E is COMPLEX array, dimension (N).

On exit, it 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 a 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both the UPLO = 'U' and UPLO = 'L' cases. Zeroed. Periodt.

IPIV

IPIV is INTEGER array, dimension (N).

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 the 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.

💀 Yes, item 2) under UPLO = 'L' really does say k-1 and -IPIV(k-1) while item 1) counts by k+1. That's exactly what the source says, we are transcribing, not editing. Copying her exactly as she is. 🙈

WORK

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

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. She tells you what she needs. Communication! 🫶

LWORK

LWORK is INTEGER. The length of WORK.

• Constraint: LWORK >= 1.
• For best performance: LWORK >= N*NB, where NB is the block size returned by ILAENV. (Give her the space, she'll give you the speed. ✨)

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. Lowkey the politest option in the whole page.

INFO

INFO is INTEGER.

• = 0: successful exit 🎉
• < 0: If INFO = -k, the k-th argument had an illegal value. (Rude of you.)
• > 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. ⚠️ Like, genuinely. Danger.

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 logs the first red flag and then keeps going. Main character energy honestly.

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💖 Author 💖

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

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🧚 Further Details:

!> TODO: put correct description

Not me finding an actual TODO in the shipped documentation. She said "I'll finish it later" and later never came. 😭 We love an unfinished queen.

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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

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Definition at line 257 of file csytrf_rk.f. ✨

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⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code. No humans were harmed, the robots did it. 🤖

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Okay that's the whole page bestie, go forth and factorize responsibly! 🦋

🍓 LAPACK, Version 3.12.0 — page /home/abui...sytrf_rk.f(3). Bye, love u, see u at the next subroutine! 💜🎀