SSYSV_RK(3)

💅 NAME

So the title up top says /home/abui.../ssysv_rk.f(3) and the manual it lives in is the Library Functions Manual — it's the LAPACK one, bestie, we're in the Fortran streets now. And the actual file it's documenting is:

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

That's the whole path, no cap, she lives deep in the build tree 🫶

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✨ SYNOPSIS ✨

Functions/Subroutines

subroutine SSYSV_RK (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work,
    lwork, info)

SSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices — that's the whole vibe, we solve systems here 🔥

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👑 Function/Subroutine Documentation 👑

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

SSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices ⭐

Purpose

okay so, here's the tea 💖

!> SSYSV_RK computes the solution to a real 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.
!>
!> SSYTRF_RK is called to compute the factorization of a real
!> symmetric matrix.  The factored form of A is then used to solve
!> the system of equations A * X = B by calling BLAS3 routine SSYTRS_3.
!>

like, the bounded Bunch-Kaufman (rook) diagonal pivoting method is doing the most and I'm obsessed 😭 she factors A into A = P*U*D*(U**T)*(P**T) when UPLO = 'U', or A = P*L*D*(L**T)*(P**T) when UPLO = 'L'. Then SSYTRF_RK handles the factorization and the BLAS3 routine SSYTRS_3 comes in to actually solve A * X = B. teamwork, periodt 💅

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.
!>

Your <italic>character</italic> with exactly one char of commitment 🎀 — 'U' means the upper triangle of A is stored, 'L' means the lower triangle is stored.

N

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

The number of linear equations aka the order of the matrix A. Rule: N >= 0 ✨

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, i.e. how many columns B has. Rule: NRHS >= 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, if INFO = 0, diagonal of the block diagonal
!>          matrix D and factors U or L  as computed by SSYTRF_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 DSYTRF_RK routine.
!>

the main character herself 👑 REAL array, dimension (LDA,N). On entry she's the symmetric matrix A:

• If UPLO = 'U': the leading N-by-N upper triangular part of A holds 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 holds the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced 🙈

On exit, if INFO = 0, you get the diagonal of the block diagonal matrix D and the factors U or L as computed by SSYTRF_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); (the 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 DSYTRF_RK routine. iykyk 🦋

LDA

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

Leading dimension of A. Rule: LDA >= max(1,N) ✅

E

!>          E is REAL array, dimension (N)
!>          On exit, contains the output computed by the factorization
!>          routine DSYTRF_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 DSYTRF_RK routine.
!>

REAL array, dimension (N). On exit she's holding the output computed by the factorization routine DSYTRF_RK — 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 a 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both the UPLO = 'U' and UPLO = 'L' cases. 🫶

For more info see the description of DSYTRF_RK routine.

IPIV

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

INTEGER array, dimension (N) — all the details of the interchanges and the block structure of D, as determined by SSYTRF_RK. For more info see the description of DSYTRF_RK routine 💜

B

!>          B is REAL 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.
!>

REAL array, dimension (LDB,NRHS). On entry she's the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, she's the N-by-NRHS solution matrix X. glow up of the century ✨

LDB

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

Leading dimension of B. Rule: LDB >= max(1,N) 🌸

WORK

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

REAL array, dimension ( MAX(1,LWORK) ) — the work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK 👑

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 DSYTRF_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.
!>

The length of WORK. Rule: LWORK >= 1. For best performance of the factorization stage, LWORK >= max(1,N*NB), where NB is the optimal blocksize for DSYTRF_RK 🔥

💡 If LWORK = -1, a workspace query is assumed — the routine only calculates the optimal size of the WORK array for the factorization stage, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. sneaky but valid 💅

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.
!>

The receipts line 🧾

• = 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. she logs the first red flag and moves on, honestly iconic 🚩

Author

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

the whole friend group showed up 🫶🎀

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 — 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 226 of file ssysv_rk.f. 📍

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🌈 Author

Generated automatically by Doxygen for LAPACK from the source code. (not a human girlie, a whole robot 💀)

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and that's the page bestie!! version 3.12.0 of LAPACK, straight out of /home/abui.../ssysv_rk.f(3) — mwah, go solve some symmetric systems, you're gonna slay 💖✨🎀🦋