CSYSV_RK(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 csysv_rk — from the LAPACK library manual, bestie

okay so this page is csysv_rk.f(3) and it lives in the Library Functions Manual section, and honestly? it's giving linear algebra main character energy. let's get into it ✨

🎀 NAME 🎀

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

yep, that's the whole name line, path and all — she's long, she's specific, she's that girl 🫶

✨ SYNOPSIS ✨

🔧 Functions/Subroutines

  • subroutine CSYSV_RK (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info) — the one and only. CSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices 💖

👑 Function/Subroutine Documentation 👑

subroutine CSYSV_RK (character uplo, integer n, integer nrhs, complex,
    dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer,
    dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb,
    complex, dimension( * ) work, integer lwork, integer info)

so like, CSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices. periodt. 💅

🌸 Purpose 🌸

okay so the actual purpose, straight up:

!> CSYSV_RK computes the solution to a complex system of linear
!> equations A * X = B, where A is an N-by-N symmetric matrix
!> and X and B are N-by-NRHS matrices.
!>
!> The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
!> to factor A as
!>    A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
!>    A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',
!> where U (or L) is unit upper (or lower) triangular matrix,
!> U**T (or L**T) is the transpose of U (or L), P is a permutation
!> matrix, P**T is the transpose of P, and D is symmetric and block
!> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
!>
!> CSYTRF_RK is called to compute the factorization of a complex
!> symmetric matrix.  The factored form of A is then used to solve
!> the system of equations A * X = B by calling BLAS3 routine CSYTRS_3.
!>

like, she factors, she solves, she calls CSYTRF_RK and then CSYTRS_3. the way that's a whole pipeline 😭🔥

💖 Parameters 💖

UPLO

!>          UPLO is CHARACTER*1
!>          Specifies whether the upper or lower triangular part of the
!>          symmetric matrix A is stored:
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored.
!>

pick your triangle bestie, upper or lower, that's the whole vibe check 🌈

N

!>          N is INTEGER
!>          The number of linear equations, i.e., the order of the
!>          matrix A.  N >= 0.
!>

number of linear equations = order of the matrix A. and N >= 0, no negativity allowed 🙈

NRHS

!>          NRHS is INTEGER
!>          The number of right hand sides, i.e., the number of columns
!>          of the matrix B.  NRHS >= 0.
!>

how many right hand sides you got? that's how many columns of B. also NRHS >= 0 💜

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, if INFO = 0, diagonal of the block diagonal
!>          matrix D and factors U or L  as computed by CSYTRF_RK:
!>            a) ONLY diagonal elements of the symmetric block diagonal
!>               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
!>               (superdiagonal (or subdiagonal) elements of D
!>                are stored on exit in array E), and
!>            b) If UPLO = 'U': factor U in the superdiagonal part of A.
!>               If UPLO = 'L': factor L in the subdiagonal part of A.
!>
!>          For more info see the description of CSYTRF_RK routine.
!>

the matrix herself 💅 on entry she's symmetric, on exit (if INFO = 0) she's carrying D's diagonal plus the factors U or L. the half you didn't pick? not referenced, she doesn't even look at it 👀

LDA

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

leading dimension of A, and LDA >= max(1,N). icon behavior ⭐

E

!>          E is COMPLEX array, dimension (N)
!>          On exit, contains the output computed by the factorization
!>          routine CSYTRF_RK, i.e. the superdiagonal (or subdiagonal)
!>          elements of the symmetric block diagonal matrix D
!>          with 1-by-1 or 2-by-2 diagonal blocks, where
!>          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
!>          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
!>
!>          NOTE: For 1-by-1 diagonal block D(k), where
!>          1 <= k <= N, the element E(k) is set to 0 in both
!>          UPLO = 'U' or UPLO = 'L' cases.
!>
!>          For more info see the description of CSYTRF_RK routine.
!>

she's the off-diagonal girlie of D 🦋 and super important: for a 1-by-1 diagonal block D(k) with 1 <= k <= N, E(k) gets set to 0 in both the UPLO = 'U' and UPLO = 'L' cases. no cap.

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D,
!>          as determined by CSYTRF_RK.
!>
!>          For more info see the description of CSYTRF_RK routine.
!>

all the receipts on the interchanges and D's block structure 🧚 as decided by CSYTRF_RK.

B

!>          B is COMPLEX array, dimension (LDB,NRHS)
!>          On entry, the N-by-NRHS right hand side matrix B.
!>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
!>

comes in as B, leaves as X (if INFO = 0). glow-up fr fr ✨

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B.  LDB >= max(1,N).
!>

leading dimension of B, LDB >= max(1,N). same energy as LDA, we love consistency 🫶

WORK

!>          WORK is COMPLEX array, dimension ( MAX(1,LWORK) ).
!>          Work array used in the factorization stage.
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

the work array for the factorization stage. on exit with INFO = 0, WORK(1) hands you the optimal LWORK — she's helpful like that 💖

LWORK

!>          LWORK is INTEGER
!>          The length of WORK.  LWORK >= 1. For best performance
!>          of factorization stage LWORK >= max(1,N*NB), where NB is
!>          the optimal blocksize for CSYTRF_RK.
!>
!>          If LWORK = -1, then a workspace query is assumed;
!>          the routine only calculates the optimal size of the WORK
!>          array for factorization stage, returns this value as
!>          the first entry of the WORK array, and no error message
!>          related to LWORK is issued by XERBLA.
!>

length of WORK, LWORK >= 1. want the factorization stage to slay? use LWORK >= max(1,N*NB) where NB is the optimal blocksize for CSYTRF_RK 🔥

and LWORK = -1?? that's the workspace query, bestie. the routine just computes the optimal size of WORK for the factorization stage, drops it in as the first entry of the WORK array, and XERBLA will not issue any error message related to LWORK. very demure, very mindful 🙈

INFO

!>          INFO is INTEGER
!>          = 0: successful exit
!>
!>          < 0: If INFO = -k, the k-th argument had an illegal value
!>
!>          > 0: If INFO = k, the matrix A is singular, because:
!>                 If UPLO = 'U': column k in the upper
!>                 triangular part of A contains all zeros.
!>                 If UPLO = 'L': column k in the lower
!>                 triangular part of A contains all zeros.
!>
!>               Therefore D(k,k) is exactly zero, and superdiagonal
!>               elements of column k of U (or subdiagonal elements of
!>               column k of L ) are all zeros. 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.
!>
!>               NOTE: INFO only stores the first occurrence of
!>               a singularity, any subsequent occurrence of singularity
!>               is not stored in INFO even though the factorization
!>               always completes.
!>

quick rundown:

  • = 0: successful exit, we ate 🎀
  • < 0: if INFO = -k, the k-th argument had an illegal value, oops 💀
  • > 0: if INFO = k, the matrix A is singular, because (UPLO = 'U') column k in the upper triangular part of A contains all zeros, or (UPLO = 'L') column k in the lower triangular part of A contains all zeros
⚠️ real talk: when that happens D(k,k) is exactly zero, and the superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros too. the factorization has been completed, but the block diagonal matrix D is exactly singular — and division by zero will occur if it's used to solve a system of equations. 😭 do not proceed, bestie.
📌 NOTE: INFO only stores the first occurrence of a singularity — any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes. the first one is the only one getting the spotlight 👑

🌸 Author 🌸

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

the whole squad, we stan 🫶

🦋 Contributors 🦋

!>
!>  December 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
!>
!>

December 2016 gang and September 2007 gang, both iconic, both credited, no erasure 💜

⭐ Definition ⭐

Definition at line 226 of file csysv_rk.f.

🧚 Author (the meta one) 🧚

Generated automatically by Doxygen for LAPACK from the source code. yes, a bot wrote the page about the subroutine. it's giving inception 💀✨

and that's the whole page, bestie!! 🎀🦋💖

LAPACK — Version 3.12.0, and the file is /home/abui.../csysv_rk.f(3). mwah, go solve something symmetric 💅👑

📦 where this page comes from