SSYTRF(3)

💜 NAME 💜

ok so bestie, this whole page is literally about one file, and its name is a whole entire path, no cap:

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

that's it. that's the name. she's long, she's dramatic, she's giving "full government name" energy 💅 and yes, this is the ssytrf.f manpage from the LAPACK library, version 3.12.0 — we'll get to that at the end 🎀

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✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

• subroutine SSYTRF (uplo, n, a, lda, ipiv, work, lwork, info) — SSYTRF ✨ (she's the moment, she's the icon, she's the whole reason we're here)

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🌸 Function/Subroutine Documentation 🌸

ok so here's the actual, full, unredacted signature — read it slowly, it's a lot but we got this 🫶

subroutine SSYTRF (character uplo, integer n, real, dimension( lda, * ) a,
    integer lda, integer, dimension( * ) ipiv, real, dimension( * ) work,
    integer lwork, integer info)

👑 SSYTRF 👑

💖 Purpose 💖

bestie. listen. SSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. the form of the factorization is:

A = U**T*D*U  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. literally so iconic 🌈

this is the blocked version of the algorithm, calling Level 3 BLAS. highkey the upgraded, faster, main-character version of the vibe 🔥

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🎀 Parameters 🎀

• UPLO — UPLO is CHARACTER*1
  • = 'U': the upper triangle of A is stored 🌸
  • = 'L': the lower triangle of A is stored 🦋
• N — N is INTEGER
  • the order of the matrix A. N >= 0. periodt.
• 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. (iykyk — only half the matrix matters, the other half is just vibes 💀)
  • On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details 👀).
• LDA — LDA is INTEGER
  • the leading dimension of the array A. LDA >= max(1,N). no pressure, but also do not mess this one up 🙈
• IPIV — IPIV is INTEGER array, dimension (N)
  • details of the interchanges and the block structure of D. so:
  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
  • If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
• WORK — WORK is REAL array, dimension (MAX(1,LWORK))
  • On exit, if INFO = 0, WORK(1) returns the optimal LWORK. she's telling you what she wants, listen to her 💅
• 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. (a little peek behind the curtain, no commit required ✨)
• INFO — INFO is INTEGER
  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value 😭
  • > 0: if INFO = i, D(i,i) is exactly zero. 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. 🚨 so like, she finished the job but the result is lowkey cursed — do NOT go solving systems with it or you'll get the math equivalent of a 404.

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🧚 Author 🧚

these absolute legends:

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

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⭐ Further Details ⭐

ok this is the part where she gets into the lore. stay with me 🫶

If UPLO = 'U', then A = U**T*D*U, where

U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    v    0   )   k-s
  U(k) =  (   0    I    0   )   s
          (   0    0    I   )   n-k
             k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). 💀 she reuses the memory, she's efficient, she's frugal, she's THAT girl.

If UPLO = 'L', then A = L*D*L**T, where

L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    0     0   )  k-1
  L(k) =  (   0    I     0   )  s
          (   0    v     I   )  n-k-s+1
             k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). 🌈 same energy, opposite triangle, she contains multitudes.

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🌟 Definition at line 181 of file ssytrf.f. 🌟

(yes, line 181, go look, she's right there 👑)

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🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code. (not a human, not a vibe curator, just Doxygen doing her little thing 🤖✨)

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and that's the whole page bestie!! 💖 this was LAPACK, Version 3.12.0, and the header up top pointed at that ssytrf.f path — so like, source + version + date all accounted for, we did it, we ate, no crumbs left 🍓🫶 mwah 💋