DSYSVX(3)

💖 DSYSVX(3) — the `dsysvx.f` era 💖

okay so first things first, bestie: this page lives in the Library Functions Manual under the name dsysvx.f(3), and the source file it's documenting is literally at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dsysvx.f. that's the whole address. she's got a long name and she's worth it 🎀

✨ NAME ✨

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

like, that's her. that's the file. no cap, the NAME section is just the path and honestly? iconic 💅

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine DSYSVX (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb,
    x, ldx, rcond, ferr, berr, work, lwork, iwork, info)

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

👑 Function/Subroutine Documentation 👑

subroutine DSYSVX (character fact, character uplo, integer n, integer nrhs,
    double precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv,
    double precision, dimension( ldb, * ) b, integer ldb, double precision,
    dimension( ldx, * ) x, integer ldx, double precision rcond, double
    precision, dimension( * ) ferr, double precision, dimension( * ) berr,
    double precision, dimension( * ) work, integer lwork, integer,
    dimension( * ) iwork, integer info)

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

🧚 Purpose 🧚

ok so, straight from the source:

!>
!> DSYSVX uses the diagonal pivoting factorization to compute 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.
!>
!> Error bounds on the solution and a condition estimate are also
!> provided.
!>

translation: she solves your system, and she also hands you the receipts — error bounds and a condition estimate. main character behaviour, fr fr ✨

💅 Description 💅

the steps she runs through, in order (this is the whole choreography, no skipping):

!>
!> The following steps are performed:
!>
!> 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
!>    The form of the factorization is
!>       A = U * D * U**T,  if UPLO = 'U', or
!>       A = L * D * L**T,  if UPLO = 'L',
!>    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.
!>
!> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
!>    returns with INFO = i. Otherwise, the factored form of A is used
!>    to estimate the condition number of the matrix A.  If the
!>    reciprocal of the condition number is less than machine precision,
!>    INFO = N+1 is returned as a warning, but the routine still goes on
!>    to solve for X and compute error bounds as described below.
!>
!> 3. The system of equations is solved for X using the factored form
!>    of A.
!>
!> 4. Iterative refinement is applied to improve the computed solution
!>    matrix and calculate error bounds and backward error estimates
!>    for it.
!>

so: factor (if you didn't already hand her a factorization) → vibe check on the condition number → solve → refine and bound. she does the whole glow-up in four steps 🌈

🍓 Parameters 🍓

buckle up bestie, there's a lot of them, and every single one matters:

FACT

!>          FACT is CHARACTER*1
!>          Specifies whether or not the factored form of A has been
!>          supplied on entry.
!>          = 'F':  On entry, AF and IPIV contain the factored form of
!>                  A.  AF and IPIV will not be modified.
!>          = 'N':  The matrix A will be copied to AF and factored.
!>

'F' — you're handing her the factorization already done: AF and IPIV hold it on entry, and she will not modify them 🫶
'N' — she copies A into AF and factors it herself 💪

UPLO

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

'U' — upper triangle of A is the one that's stored
'L' — lower triangle of A is the one that's stored

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

how many equations you got — aka the order of matrix A. must be N >= 0.

NRHS

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

the number of right hand sides = the number of columns in B and X. must be NRHS >= 0.

!>          A is DOUBLE PRECISION array, dimension (LDA,N)
!>          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.
!>

DOUBLE PRECISION array, dimension (LDA,N) — the symmetric matrix herself. if UPLO = 'U', the leading N-by-N upper triangular part carries the upper triangle and the strictly lower triangular part is not referenced 🙈. if UPLO = 'L', it's the other way round: leading N-by-N lower triangular part holds the lower triangle, and the strictly upper triangular part is not referenced.

LDA

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

leading dimension of A. LDA >= max(1,N) — no exceptions, bestie.

!>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
!>          If FACT = 'F', then AF is an input argument and on entry
!>          contains the block diagonal matrix D and the multipliers used
!>          to obtain the factor U or L from the factorization
!>          A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.
!>
!>          If FACT = 'N', then AF is an output argument and on exit
!>          returns the block diagonal matrix D and the multipliers used
!>          to obtain the factor U or L from the factorization
!>          A = U*D*U**T or A = L*D*L**T.
!>

DOUBLE PRECISION array, dimension (LDAF,N).

if FACT = 'F' → input. on entry she expects the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T, as computed by DSYTRF
if FACT = 'N' → output. on exit she returns that same block diagonal matrix D and the multipliers for U or L from A = U*D*U**T or A = L*D*L**T

LDAF

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

leading dimension of AF. LDAF >= max(1,N).

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          If FACT = 'F', then IPIV is an input argument and on entry
!>          contains details of the interchanges and the block structure
!>          of D, as determined by DSYTRF.
!>          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 UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
!>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
!>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
!>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
!>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
!>
!>          If FACT = 'N', then IPIV is an output argument and on exit
!>          contains details of the interchanges and the block structure
!>          of D, as determined by DSYTRF.
!>

INTEGER array, dimension (N). the tea ☕:

if FACT = 'F' → input, and on entry it holds the details of the interchanges plus the block structure of D, as determined by DSYTRF
if IPIV(k) > 0 → rows and columns k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1 diagonal block
if UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0 → rows and columns k-1 and -IPIV(k) were interchanged, and D(k-1:k,k-1:k) is a 2-by-2 diagonal block
if UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0 → rows and columns k+1 and -IPIV(k) were interchanged, and D(k:k+1,k:k+1) is a 2-by-2 diagonal block
if FACT = 'N' → output, and on exit it holds those same interchange details and block structure of D, as determined by DSYTRF

!>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
!>          The N-by-NRHS right hand side matrix B.
!>

DOUBLE PRECISION array, dimension (LDB,NRHS) — the N-by-NRHS right hand side matrix B. your actual problem, basically 💖

LDB

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

leading dimension of B. LDB >= max(1,N).

!>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
!>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
!>

DOUBLE PRECISION array, dimension (LDX,NRHS) — if INFO = 0 or INFO = N+1, this is the N-by-NRHS solution matrix X. the payoff ✨

LDX

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

leading dimension of X. LDX >= max(1,N).

RCOND

!>          RCOND is DOUBLE PRECISION
!>          The estimate of the reciprocal condition number of the matrix
!>          A.  If RCOND is less than the machine precision (in
!>          particular, if RCOND = 0), the matrix is singular to working
!>          precision.  This condition is indicated by a return code of
!>          INFO > 0.
!>

DOUBLE PRECISION — the estimate of the reciprocal condition number of matrix A. if RCOND is less than machine precision (in particular, if RCOND = 0), the matrix is singular to working precision, and that whole situation is flagged by a return code of INFO > 0 ⚠️

FERR

!>          FERR is DOUBLE PRECISION array, dimension (NRHS)
!>          The estimated forward error bound for each solution vector
!>          X(j) (the j-th column of the solution matrix X).
!>          If XTRUE is the true solution corresponding to X(j), FERR(j)
!>          is an estimated upper bound for the magnitude of the largest
!>          element in (X(j) - XTRUE) divided by the magnitude of the
!>          largest element in X(j).  The estimate is as reliable as
!>          the estimate for RCOND, and is almost always a slight
!>          overestimate of the true error.
!>

DOUBLE PRECISION array, dimension (NRHS) — the estimated forward error bound for each solution vector X(j) (that's the j-th column of the solution matrix X). if XTRUE is the true solution corresponding to X(j), then FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). the estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error 🫶

BERR

!>          BERR is DOUBLE PRECISION array, dimension (NRHS)
!>          The componentwise relative backward error of each solution
!>          vector X(j) (i.e., the smallest relative change in
!>          any element of A or B that makes X(j) an exact solution).
!>

DOUBLE PRECISION array, dimension (NRHS) — the componentwise relative backward error of each solution vector X(j), i.e. the smallest relative change in any element of A or B that makes X(j) an exact solution. she's telling you how much she'd have to lie to be right, respectfully 💀

WORK

!>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

DOUBLE PRECISION array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, then WORK(1) returns the optimal LWORK 🎀

LWORK

!>          LWORK is INTEGER
!>          The length of WORK.  LWORK >= max(1,3*N), and for best
!>          performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where
!>          NB is the optimal blocksize for DSYTRF.
!>
!>          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.
!>

INTEGER — the length of WORK. LWORK >= max(1,3*N), and for best performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where NB is the optimal blocksize for DSYTRF 🔥

and if LWORK = -1, 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. she's just window shopping, no commitment 💅

IWORK

!>          IWORK is INTEGER array, dimension (N)
!>

INTEGER array, dimension (N). that's it. that's the whole description. short queen 🌸

INFO

!>          INFO is INTEGER
!>          = 0: successful exit
!>          < 0: if INFO = -i, the i-th argument had an illegal value
!>          > 0: if INFO = i, and i is
!>                <= N:  D(i,i) is exactly zero.  The factorization
!>                       has been completed but the factor D is exactly
!>                       singular, so the solution and error bounds could
!>                       not be computed. RCOND = 0 is returned.
!>                = N+1: D is nonsingular, but RCOND is less than machine
!>                       precision, meaning that the matrix is singular
!>                       to working precision.  Nevertheless, the
!>                       solution and error bounds are computed because
!>                       there are a number of situations where the
!>                       computed solution can be more accurate than the
!>                       value of RCOND would suggest.
!>

INTEGER. the vibe check of return codes:

= 0 — successful exit, we love to see it 👑
< 0 — if INFO = -i, the i-th argument had an illegal value 🙈
> 0 — if INFO = i, and i is:
- <= N — D(i,i) is exactly zero. the factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned 💀
- = N+1 — D is nonsingular, but RCOND is less than machine precision, meaning the matrix is singular to working precision. nevertheless, the solution and error bounds are computed, because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest 🧚

🌟 Author 🌟

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

Definition at line 281 of file dsysvx.f. 📍

🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code. she was literally born from the source, no ghostwriters 🦋

💖 and that's the whole page bestie — LAPACK, Version 3.12.0, from /home/abui...RC/dsysvx.f(3). go forth and factor your symmetric matrices, you absolute icon ✨👑

📦 where this page comes from

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