ZSYSVX(3)

💖 ZSYSVX(3) — bestie, it's the symmetric solve 💖

okay so real quick, the manual name here is zsysvx.f(3) and it lives at the very extra path /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zsysvx.f — like, that's the whole file, that's the NAME section, we're already done with it. slay. moving on ✨

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🌸 SYNOPSIS 🌸

Functions/Subroutines

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

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

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🦋 Function/Subroutine Documentation 🦋

subroutine ZSYSVX (character fact, character uplo, integer n, integer nrhs,
    complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
    ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex*16,
    dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x,
    integer ldx, double precision rcond, double precision, dimension( * )
    ferr, double precision, dimension( * ) berr, complex*16, dimension( * )
    work, integer lwork, double precision, dimension( * ) rwork, integer
    info)

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

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✨ Purpose ✨

no cap, here's the whole vibe, straight from the source:

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

so like — it solves your system and hands you the receipts on how much you should trust the answer. main character behaviour honestly 🫶

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🎀 Description 🎀

here's the step-by-step, she's a whole process:

!>
!> 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.
!>

like. factor, check, solve, refine. four steps, zero notes 💀 she's efficient.

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👑 Parameters 👑

🌈 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.  A, AF and IPIV will not be modified.
!>          = 'N':  The matrix A will be copied to AF and factored.
!>

basically: 'F' means "bestie I already factored it, it's in AF and IPIV, don't touch" and 'N' means "nope, copy A into AF and do the work". 🍓

💜 UPLO

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

pick your triangle, only one, we don't do both 💅

⭐ N

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

how many equations, aka how big is A. and yes N >= 0, we respect the constraint 🫶

🔥 NRHS

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

how many right hand sides, i.e. how many columns B and X got. NRHS >= 0, periodt.

🌸 A

!>          A is COMPLEX*16 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.
!>

the symmetric matrix herself 💖 note that the other triangle literally does not get read. 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. LDA >= max(1,N). no notes.

🎀 AF

!>          AF is COMPLEX*16 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 ZSYTRF.
!>
!>          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.
!>

so if FACT = 'F' it's input (already got the D and the multipliers from ZSYTRF, tyvm), and if FACT = 'N' it's output and comes back stuffed with exactly that. iconic either way ✨

💜 LDAF

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

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

🌈 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 ZSYTRF.
!>          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 ZSYTRF.
!>

okay this one's the swap-and-block gossip:

• IPIV(k) > 0 → rows and columns k and IPIV(k) got interchanged, and D(k,k) is a 1-by-1 diagonal block.
• UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0 → rows and columns k-1 and -IPIV(k) got interchanged, and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
• UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0 → rows and columns k+1 and -IPIV(k) got interchanged, and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

and if FACT = 'N', it's an output argument and on exit it holds all that same info, as determined by ZSYTRF. 🔍

🍓 B

!>          B is COMPLEX*16 array, dimension (LDB,NRHS)
!>          The N-by-NRHS right hand side matrix B.
!>

the N-by-NRHS right hand side matrix B. that's the whole thing. 🫶

⭐ LDB

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

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

🔥 X

!>          X is COMPLEX*16 array, dimension (LDX,NRHS)
!>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
!>

the payoff 💖 if INFO = 0 or INFO = N+1, this is your N-by-NRHS solution matrix X.

🌸 LDX

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

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

🦋 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.
!>

this is your trust-o-meter for A 🔍 if RCOND is less than the machine precision (especially if RCOND = 0), the matrix is singular to working precision, and you'll know because INFO > 0. keep an eye on it bestie.

🎀 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.
!>

one value per solution vector X(j) (the j-th column of X). if XTRUE is the true solution for 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). and it's as reliable as the RCOND estimate — almost always a slight overestimate of the true error. she's cautious, we love her 🌸

💜 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).
!>

the componentwise relative backward error of each X(j) — aka the smallest relative change in any element of A or B that would make X(j) an exact solution. tiny number, big meaning 🙈

🌈 WORK

!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

workspace bestie. on exit, if INFO = 0, WORK(1) hands you the optimal LWORK ✨

⭐ LWORK

!>          LWORK is INTEGER
!>          The length of WORK.  LWORK >= max(1,2*N), and for best
!>          performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where
!>          NB is the optimal blocksize for ZSYTRF.
!>
!>          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:

• LWORK >= max(1,2*N)
• for best performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where NB is the optimal blocksize for ZSYTRF

and if LWORK = -1, it's a workspace query! the routine only calculates the optimal size of the WORK array, returns that value as the first entry of WORK, and no error message related to LWORK is issued by XERBLA. so don't panic. it's just asking. 🧚

🌸 RWORK

!>          RWORK is DOUBLE PRECISION array, dimension (N)
!>

that's literally it. DOUBLE PRECISION, dimension (N). she's a simple icon 💅

🔥 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.
!>

the whole mood ring, decoded:

• = 0 → successful exit, we ate 👑
• < 0 → if INFO = -i, the i-th argument had an illegal value (read: you passed something cursed)
• > 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. ✨ she said "trust me".

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🌈 Author 🌈

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

the whole squad 🫶

Definition at line 282 of file zsysvx.f. 📍

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⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code. so like, don't come for the formatting, a robot did it 🤖✨

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signed, sealed, delivered by LAPACK, Version 3.12.0 — and that lil path at the bottom, /home/abui...RC/zsysvx.f(3), is the header/footer situation, consider it mentioned. mwah 💋 🎀