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

📦 version 3.12.0, in opensuse-leap 16.0

💅 /home/abui...trf_rook.f(3) — Library Functions Manual 🎀

okay so bestie, welcome to the glow-up zone ✨ this is a page from the Library Functions Manual, and it's literally all about csytrf_rook.f — yes the whole filename is the NAME section, we don't make the rules, we just slay within them 🫶

The full source path it came from, fr fr no edits:

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

✨ SYNOPSIS ✨

Functions/Subroutines 🦋

  • subroutine CSYTRF_ROOK (uplo, n, a, lda, ipiv, work, lwork, info) — that's the baddie, that's the main character, CSYTRF_ROOK.

🌸 Function/Subroutine Documentation 🌸

💖 subroutine CSYTRF_ROOK (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer lwork, integer info) 💖

CSYTRF_ROOK — say it with your chest 👑

🧚 Purpose:

!> !> CSYTRF_ROOK computes the factorization of a complex symmetric matrix A !> using the bounded Bunch-Kaufman () diagonal pivoting method. !> The form of the factorization is !> !> 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, and D is symmetric and block diagonal with !> 1-by-1 and 2-by-2 diagonal blocks. !> !> This is the blocked version of the algorithm, calling Level 3 BLAS. !>

so it factors a complex symmetric matrix A using the bounded Bunch-Kaufman (that empty parens is in the original, don't come for me 🙈) diagonal pivoting method — giving you either 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, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. also!! it's the blocked version of the algorithm, calling Level 3 BLAS ⚡ that's the performance era, we love her.

🍓 Parameters (the whole roster, bestie, nobody gets left out)

  • UPLO — UPLO is CHARACTER*1
!> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !>

so 'U' means upper triangle of A is stored, 'L' means lower triangle of A is stored. pick your fighter 💅

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

it's the order of the matrix A, and it has to be N >= 0. negative N? not today bestie 💀

  • A — A is COMPLEX 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). !>

on entry it's your 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 (left on read 🙈). 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L are living here — see the Further Details below for the whole saga 🦋

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

the leading dimension of the array A, and it needs LDA >= max(1,N). periodt.

  • 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 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. !>

this is the receipts on 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 (yes, "inerchaged" — that typo is canon, it's in the source, we respect the lore 😭), 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 (same typo, she's consistent 💀), and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK))
!> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>

on exit, if INFO = 0, then WORK(1) returns the optimal LWORK. she's giving helpful 🫶

  • 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. !>

the length of WORK, and LWORK >= 1. for best performance, LWORK >= N*NB, where NB is the block size returned by ILAENV 🔥

⚠️ important one bestie: 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. so -1 is the "just tell me the size, don't yell at me" mode 🙈
  • 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. !>
  • = 0: successful exit, we ate 💅
  • < 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, clock that.

👑 Author

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

the whole squad, iconic 💜

🌈 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). !>

okay so let me break this down for you like we're on facetime 🎀

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). okay the way that's literally just bookkeeping bestie, don't overthink it 💅

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). look at her go 😭✨

🧾 Contributors:

!> !> June 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 !> !>

so credit where it's due 🫶:

  • June 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

📍 Definition at line 207 of file csytrf_rook.f.

🌟 Author

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and that's the whole page, bestie!! 💅✨ this beauty is from LAPACK, Version 3.12.0, and the file we've been screaming about is /home/abui...trf_rook.f(3) 🎀 mwah, go factor some complex symmetric matrices, you've got this 👑🦋🌸

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