/home/abui...sytf2_rk.f(3)

💅 NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ssytf2_rk.f — bestie, this is a Library Functions Manual page for one very specific Fortran file, and yes the whole path is the name, slay 💖

---

✨ SYNOPSIS ✨

Functions/Subroutines

• subroutine SSYTF2_RK (uplo, n, a, lda, e, ipiv, info) — SSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

---

🎀 Function/Subroutine Documentation 🎀

subroutine SSYTF2_RK (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv, integer info)

SSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

Purpose

ok so the whole point of this icon 👑:

!> SSYTF2_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 unblocked version of the algorithm, calling Level 2 BLAS.
!> For more information see Further Details section.
!>

literally that's the whole flex: you hand it a symmetric matrix, it hands you back P, U/L, and D. the unblocked version — Level 2 BLAS only, no cap 🔥

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
!>

one character, two moods — 'U' for upper triangular, 'L' for lower triangular 🦋

N

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

the order of matrix A, and it has to satisfy N >= 0 or we're not doing this 💅

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:
!>            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 REAL array of dimension (LDA,N) and it goes on a whole journey 😭 on entry it's the symmetric matrix A: if UPLO = 'U' the leading N-by-N upper triangular part holds the upper triangular part of A, and the strictly lower triangular part is not referenced; if UPLO = 'L' the leading N-by-N lower triangular part holds the lower triangular part of A, and the strictly upper triangular part is not referenced.

on exit it's a whole different girl:

• 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.  LDA >= max(1,N).
!>

the leading dimension of A. requirement: LDA >= max(1,N) — no exceptions bestie 🫶

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.
!>

REAL 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 the UPLO = 'U' and UPLO = 'L' cases. 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 Further
!>          Details section.
!>

INTEGER array, dimension (N). IPIV describes the permutation matrix P in the factorization of matrix A like this: the absolute value of IPIV(k) is the index of the row and column that got interchanged with the k-th row and column. the value of UPLO describes the order in which the interchanges were applied. and 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.

(and yes, case b) item 2 for 'L' really does say k-1 and -IPIV(k-1) in the source — copying it exactly, iykyk 🧚)

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.
!>

INTEGER, and it's the vibe check at the end:

• = 0: successful exit, we ate 💅
• < 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.

Author

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

Further Details:

!> TODO: put further details
!>

(the source literally just says TODO: put further details — iconic honestly, leaving it exactly as-is 🫶)

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
!>
!>  01-01-96 - Based on modifications by
!>    J. Lewis, Boeing Computer Services Company
!>    A. Petitet, Computer Science Dept.,
!>                Univ. of Tenn., Knoxville abd , USA
!>

shoutout to the 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; and 01-01-96 — based on modifications by J. Lewis, Boeing Computer Services Company, and A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA. (typos preserved, that's love 💜)

Definition at line 240 of file ssytf2_rk.f.

---

Author

Generated automatically by Doxygen for LAPACK from the source code. 🤖✨

---

🌸 and that's the whole page bestie — LAPACK, Version 3.12.0, LAPACK Version 3.12.0 /home/abui...sytf2_rk.f(3). go factor yourself a symmetric indefinite matrix, you've got this 🎀💖🦋