SRC/slasyf_rk.f(3)

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💅 SRC/slasyf_rk.f 💅

Okay bestie, welcome to the Library Functions Manual page for SRC/slasyf_rk.f — this is the Fortran source-file page, and it's honestly lowkey iconic. Let's get into it 🫶

✨ NAME ✨

SRC/slasyf_rk.f — like, that's the file, that's the vibe, no notes 💅

🌸 SYNOPSIS 🌸

Functions/Subroutines

  • subroutine slasyf_rk (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info) — SLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method ✨

🦋 Function/Subroutine Documentation 🦋

subroutine slasyf_rk (character uplo, integer n, integer nb, integer kb, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv, real, dimension( ldw, * ) w, integer ldw, integer info)

SLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. It's giving auxiliary-routine main character energy 👑

💖 Purpose 💖

SLASYF_RK computes a partial factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method. The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. 🎀

SLASYF_RK is an auxiliary routine called by SSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). Blocked code, bestie — she's not doing it one element at a time, that would be embarrassing 💀

🎀 Parameters 🎀

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

N

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

NB

NB is INTEGER. The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks. Like, genuinely at least 2, she needs room for her 2-by-2 pivot blocks 🎀

KB

KB is INTEGER. The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.

A

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

On entry, the symmetric matrix A:

  • 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.
  • 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, contains:

  1. 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
  2. If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

Not me leaving the untouched triangle of A completely unbothered — she said "not referenced," periodt. 🙈

LDA

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

E

E is REAL array, dimension (N).

On exit, 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. Bestie, E(k) is getting zeroed no matter what your UPLO said, that's just the law 💀

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. 👑

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 submatrix A(1:N,N-KB+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,N-KB+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 submatrix A(1:N,N-KB+1:N). If -IPIV(k-1) = k-1, no interchange occurred.
  • c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
  • d) NOTE: Any entry IPIV(k) is always NONZERO on output. Every. Single. One. Nonzero. No cap. ✨

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 submatrix A(1:N,1:KB). 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 submatrix A(1:N,1:KB). 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 submatrix A(1:N,1:KB). If -IPIV(k+1) = k+1, no interchange occurred.
  • c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
  • d) NOTE: Any entry IPIV(k) is always NONZERO on output. ✨

W

W is REAL array, dimension (LDW,NB). 🦋

LDW

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

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.

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. DIVISION BY ZERO. Do not do that. That's the whole warning. 💀

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 remembers the first one and then moves on, respectfully 🙈

🌈 Author 🌈

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

💜 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

⭐ Definition at line 260 of file slasyf_rk.f. ⭐

👑 Author 👑

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Okay that's the whole page bestie, you ate that 💅✨ And for the sign-off: this little beauty is from LAPACK, Version 3.12.0, on the page SRC/slasyf_rk.f(3) — go forth and factorize responsibly 🎀🫶💖

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