SRC/ssytf2_rook.f(3)

💖 SRC/ssytf2_rook.f(3) — Library Functions Manual 💖

okay bestie, this page is from the Library Functions Manual, and it's literally all about SRC/ssytf2_rook.f(3). It's giving LAPACK unblocked factorization energy, and we are keeping every single detail. No cap. 🫶

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🌸 NAME 🌸

SRC/ssytf2_rook.f

That's the name, periodt. 💅

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✨ SYNOPSIS ✨

🧚 Functions/Subroutines

subroutine ssytf2_rook (uplo, n, a, lda, ipiv, info)
    SSYTF2_ROOK computes the factorization of a real symmetric
    indefinite matrix using the bounded Bunch-Kaufman
    ("rook") diagonal pivoting method (unblocked algorithm).

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👑 Function/Subroutine Documentation 👑

subroutine ssytf2_rook (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

SSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm). Slay. ✨

💖 Purpose 💖

SSYTF2_ROOK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:

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

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**T is the transpose of U, 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. Lowkey iconic. 🔥

🎀 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.
• 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, 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).
• IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D.
  • If UPLO = 'U':
    • 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 IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were interchanged, and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
  • If UPLO = 'L':
    • 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 IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were interchanged, and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
• INFO — INFO is INTEGER.
  • = 0: successful exit
  • < 0: if INFO = -k, the k-th argument had an illegal value
  • > 0: if INFO = k, D(k,k) 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.

⚠️ Bestie, if INFO = k > 0, then D(k,k) is exactly zero — the factorization finished, but D is exactly singular, and division by zero will occur if you use it to solve a system of equations. That's not a vibe. 💀

🌈 Author 🌈

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

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🦋 Further Details 🦋

💜 If UPLO = 'U'

Then A = U*D*U**T, 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).

🍓 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).

⭐ Contributors ⭐

• November 2013, 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, and A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

💅 Definition at line 193 of file ssytf2_rook.f. 💅

That's literally where the definition is, bestie. 🎀

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🌟 Author 🌟

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That's the whole page, bestie 💅 — footer vibes: LAPACK Version 3.12.0, source SRC/ssytf2_rook.f(3). Obsessed. 💖