SRC/zhetf2_rook.f(3)

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version 3.12.1 · 7 versions
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💅 SRC/zhetf2_rook.f — LAPACK's zhetf2_rook page, but make it make sense

okay so bestie, you opened the manual page for SRC/zhetf2_rook.f and it's giving "doxygen threw up on a Fortran comment block." don't panic 🫶 it's actually just one subroutine, and it's a genuinely iconic one. let's get into it ✨

✨ NAME ✨

SRC/zhetf2_rook.f — the home of zhetf2_rook, a real one 👑

🌸 SYNOPSIS 🌸

Functions/Subroutines

  • subroutine zhetf2_rook (uplo, n, a, lda, ipiv, info) — ZHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).

🦋 Function/Subroutine Documentation 🦋

subroutine zhetf2_rook (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).

💜 Purpose:

so what does it do, you ask. it computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:

A = U*D*U**H  or  A = L*D*L**H

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**H is the conjugate transpose of U, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. slay 💅

this is the unblocked version of the algorithm, and it calls Level 2 BLAS. no cap, that's the whole vibe.

🎀 Parameters 🎀

UPLO

UPLO is CHARACTER*1. it specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:

  • = 'U': Upper triangular
  • = 'L': Lower triangular

pick your fighter ⭐

N

N is INTEGER. the order of the matrix A. N >= 0. (yes bestie, zero is legal, it's giving "empty but valid" 🫶)

A

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

On entry, the Hermitian 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). it literally reinvents itself in place, main character energy 👑

LDA

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

⚠️ ok so that >= max(1,N) is not a suggestion, it's the law. disrespect it and the whole thing falls apart 😭

IPIV

IPIV is INTEGER array, dimension (N). the 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 inerchaged, 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, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
🙈 yes, "inerchaged" is spelled like that in the source docs. i'm not fixing it, i'm not that kind of girl. copy it as-is and move on. 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.
💀 the "divide by zero will occur" line is real and it's coming for you if you ignore INFO > 0. this is your roman empire moment. check your INFO.

👑 Author

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

🌈 Further Details: 🌈

if UPLO = 'U'

then A = U*D*U**H, 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**H, 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).

🧚 the way this is just "everything gets overwritten, but like, on purpose"? iconic. it's in-place and it's not sorry.

🍓 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, USA

⭐ Definition at line 193 of file zhetf2_rook.f.

🧚 Author (the doxygen one, not the humans)

Generated automatically by Doxygen for LAPACK from the source code.

🎀 ok that's the page! and for the sign-off: this whole thing is LAPACK, Version 3.12.0, straight out of SRC/zhetf2_rook.f — stay shiny, check your INFO, and don't forget the LDA >= max(1,N) bestie 💖✨
📦 where this page comes from