SRC/dlasyf_rk.f(3)

🌸 SRC/dlasyf_rk.f(3) — Library Functions Manual 🌸

okay so bestie, this is the <pink>SRC/dlasyf_rk.f(3)</pink> page from the <purple>Library Functions Manual</purple>, and it is literally all about DLASYF_RK doing a partial factorization of a real symmetric indefinite matrix with the bounded Bunch-Kaufman (rook) diagonal pivoting method. <rainbow>iconic linear algebra behavior` ✨

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✨ NAME ✨

SRC/dlasyf_rk.f 🎀

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🦋 SYNOPSIS 🦋

✨ Functions/Subroutines ✨

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

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💜 Function/Subroutine Documentation 💜

subroutine dlasyf_rk (character uplo, integer n, integer nb, integer kb,
    double precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( * ) e, integer, dimension( * ) ipiv, double precision,
    dimension( ldw, * ) w, integer ldw, integer info)

DLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method. <pink>slay.` 💅

🎀 Purpose 🎀

ok so here is the whole vibe: DLASYF_RK computes a partial factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method. The partial factorization has this form, bestie:

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. <cyan>periodt.` ✨

DLASYF_RK is an auxiliary routine called by DSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). <purple>main character energy fr fr.` 🫶

🍓 Parameters 🍓

• UPLO — UPLO is CHARACTER*1. It tells you whether the upper or lower triangular part of the symmetric matrix A is stored:
  • = 'U': Upper triangular 🌸
  • = 'L': Lower triangular 🦋
• N — N is INTEGER. It's the order of the matrix A. N >= 0. <mint>math bestie.` ✨
• NB — NB is INTEGER. This is 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. <yellow>important detail, no cap.` 💛
• KB — KB is INTEGER. This is the number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB. <blue>iykyk.` 🦋
• A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, it's 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, it 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. <green>she contains multitudes.` 🌈
• LDA — LDA is INTEGER. This is the leading dimension of the array A. LDA >= max(1,N). <pink>respect the bounds, bestie.` 💅
• E — E is DOUBLE PRECISION 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 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. <lavender>cute little zero.` 🎀
• 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. <hotpink>this is the whole tea right here.` 🍵
  • 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. <cyan>math check, slay.` ✨
    • d) NOTE: Any entry IPIV(k) is always NONZERO on output. <red>no sneaky zeros.` 💀
  • 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. <cyan>noted, bestie.` ✨
    • d) NOTE: Any entry IPIV(k) is always NONZERO on output. <red>again: no zeros.` 💀
• W — W is DOUBLE PRECISION array, dimension (LDW,NB). <purple>workspace, doing its thing.` 🫶
• LDW — LDW is INTEGER. This is the leading dimension of the array W. LDW >= max(1,N). <mint>bounds, bestie, bounds.` 💅
• INFO — INFO is INTEGER. Here's the status tea:
  • = 0: successful exit 🌈
  • < 0: If INFO = -k, the k-th argument had an illegal value <red>uh oh.` 😭
  • > 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. <red>this is the dangerous part, do not ignore it.` 💀
  • 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. <yellow>first one only, periodt.` ⭐

bestie, if INFO > 0, the factorization finished but D is exactly singular, and division by zero will occur if it's used to solve a system of equations. <red>that is not a vibe.` 😭

👑 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 dlasyf_rk.f.

⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code.

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<rainbow>LAPACK Version 3.12.0</rainbow>, source SRC/dlasyf_rk.f(3) — that's the footer, bestie, and we are done. <pink>mwah.` 💖🎀✨