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

💜 DSYTRF_RK — the manual page, but make it make sense

okay so bestie, this is a page from the Library Functions Manual and it's literally all about one Fortran subroutine: DSYTRF_RK. It lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dsytrf_rk.f — yes that's a whole entire path, she's thorough 💅. Buckle up, we're doing linear algebra.

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✨ SYNOPSIS ✨

Functions/Subroutines

• subroutine DSYTRF_RK (uplo, n, a, lda, e, ipiv, work, lwork, info) — DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

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🎀 Function/Subroutine Documentation 🎀

subroutine DSYTRF_RK (character uplo, integer n, double precision,
    dimension( lda, * ) a, integer lda, double precision, dimension( * ) e,
    integer, dimension( * ) ipiv, double precision, dimension( * ) work,
    integer lwork, integer info)

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

🌸 Purpose:

!> DSYTRF_RK computes the factorization of a real 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.
!> For more information see Further Details section.
!>

so yeah — she takes your matrix A and splits it into A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T), where U/L is unit upper/lower triangular, U**T/L**T is the transpose of U/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, calling Level 3 BLAS — icon behaviour ✨.

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🌈 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
!>

UPLO is a CHARACTER*1 — it tells us whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U' means upper triangular, = 'L' means lower triangular. pick your fighter 💅

N

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

N is an INTEGER — the order of the matrix A. Lowkey important: N >= 0.

A

!>          A is DOUBLE PRECISION 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:
!>            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.
!>

A is a DOUBLE PRECISION array, dimension (LDA,N). On entry it's the symmetric matrix A itself:

• 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, 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.

ngl she reuses the memory like a queen, nothing goes to waste 🫶

LDA

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

LDA is an INTEGER — the leading dimension of the array A. Constraint era: LDA >= max(1,N).

E

!>          E is DOUBLE PRECISION 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.
!>

E is a DOUBLE PRECISION array, dimension (N). On exit it holds 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 UPLO = 'U' or UPLO = 'L' cases. no crumbs left behind 💀

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 Further
!>          Details section.

IPIV is an INTEGER array, dimension (N). She's the one who explains the permutation matrix P in the factorization of matrix A, like this: the absolute value of IPIV(k) represents the index of the 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 — she's not gatekeeping, she's redirecting 🧚

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

!>          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.
!>

so, 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. periodt ✨

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

!>          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.
!>

same energy, other direction — 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. 💅

WORK

!>          WORK is DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ).
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

WORK is a DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. she's doing the most so you don't have to 🫶

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.
!>
!>          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.
!>

LWORK is an INTEGER — 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. sneaky little shortcut, love that for us 💖

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.
!>
!>               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.
!>

INFO is an INTEGER:

• = 0: successful exit. slay 🎀
• < 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. 😭 not the undefined behaviour

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 one and moves on, main character energy 👑

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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
!>

bestie the docs literally say TODO: put correct description and honestly? iconic. 😭

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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
!>
!>

• 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 dsytrf_rk.f.

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Author

Generated automatically by Doxygen for LAPACK from the source code. (yes, a robot wrote this, and she ate 🦋)

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okay that's the whole page bestie, you're officially a Bunch-Kaufman girlie now 💅

LAPACK — Version 3.12.0 · /home/abui...sytrf_rk.f(3) · mwah, go factor something symmetric 💖✨