/home/abui...tf2_rook.f(3)

💅 NAME

okay so this is the man page for /home/abui...tf2_rook.f(3) — and the actual source file it's describing is the full path bestie, hold on to it:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zsytf2_rook.f

it lives in the Library Functions Manual section, and it's all about ZSYTF2_ROOK 💖

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✨ SYNOPSIS

🦋 Functions/Subroutines

• subroutine ZSYTF2_ROOK (uplo, n, a, lda, ipiv, info) — this one computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (the unblocked algorithm). yes that's a mouthful, we love her anyway 💅

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🌸 Function/Subroutine Documentation

subroutine ZSYTF2_ROOK (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm). periodt ✨

🎯 Purpose

ok so here's the whole vibe — ZSYTF2_ROOK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman () 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. 🧚

and like, this is the unblocked version of the algorithm, calling Level 2 BLAS. no cap. 🔥

🧸 Parameters

UPLO

UPLO is CHARACTER*1. it 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. it's the order of the matrix A. N >= 0. literally that's it. 💀

A

A is COMPLEX*16 array, dimension (LDA,N).

On entry, it's the symmetric matrix A, okay so listen:

• 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, you get the block diagonal matrix D and the multipliers used to obtain the factor U or L (scroll down for further details bestie, it's all there 💖).

LDA

LDA is INTEGER. it's the leading dimension of the array A. LDA >= max(1,N). slay. ✨

IPIV

IPIV is INTEGER array, dimension (N). it holds the details of the interchanges and the block structure of D. and it gets specific, so pay attention:

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 inerchaged, 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 inerchaged, and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

notice how it's k-1 for upper and k+1 for lower? that's not a typo, that's the whole point, iykyk 🌈

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 do NOT ignore a positive INFO and then go solve a system with it. that's division by zero. that's the bad place. fr fr.

👑 Author

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

🔥 Further Details

ok this is the iconic part, the actual structure breakdown 🌸

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

look at that lineage. generations of mathematicians. we stan legends 🫶👑

📌 Definition

Definition at line 193 of file zsytf2_rook.f. ⭐

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🌈 Author

Generated automatically by Doxygen for LAPACK from the source code. (so if the formatting's a little quirky, blame the robot, not me 💅)

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and that's the whole page bestie!! sign-off time: this cutie is from LAPACK, Version 3.12.0, and the page itself is /home/abui...tf2_rook.f(3) 🦋💖 go factor some symmetric indefinite matrices, you absolute icon 🎀✨