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

📦 version 3.12.0, in opensuse-leap 16.0

✨ CSYRFSX ✨

okay so this is the man page for /home/abui.../csyrfsx.f(3) — the header line literally says Library Functions Manual, and the source file lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/csyrfsx.f, which is a lot of path, but slay 🎀

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/csyrfsx.f — yes bestie, the NAME section is literally just the whole file path. no notes. iconic behavior 💅

🦋 SYNOPSIS 🦋

Functions/Subroutines

here's the routine, bestie, one bullet, one queen 👑

subroutine CSYRFSX (uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv, s, b,
    ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp,
    nparams, params, work, rwork, info)

CSYRFSX — that's her. that's the whole name. we love a single-name moment 🫶

⭐ Function/Subroutine Documentation ⭐

subroutine CSYRFSX (character uplo, character equed, integer n, integer
    nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension(
    ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, real,
    dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex,
    dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * )
    berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm,
    real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real,
    dimension( * ) params, complex, dimension( * ) work, real, dimension( *
    ) rwork, integer info)

CSYRFSX (again, because she's the main character 💖).

💜 Purpose 💜

okay so CSYRFSX improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, AND it hands you error bounds and backward error estimates for that solution. it's giving overachiever in the best way.

in addition to the normwise error bound, the code also gives you the maximum componentwise error bound if it can. for the deets on those error bounds, go read the comments on ERR_BNDS_NORM and ERR_BNDS_COMP — they spill the whole tea ☕

also important, bestie: the original system of linear equations may have been equilibrated before you called this routine, and that's exactly what the arguments EQUED and S are about (more on them below). if that happened, the solution and error bounds you get back are for the original unequilibrated system. so like, don't panic when the numbers look different, it's intentional ✨

some optional parameters are bundled up in the PARAMS array. those settings decide how refinement gets performed, but ngl the defaults are usually fine. if the defaults are fine for you, you can pass NPARAMS = 0, and that stops the source code from even touching the PARAMS argument. lowkey elegant 🎀

🌈 Parameters 🌈

okay this is the big one — every single argument, no skips, periodt 💅

  • UPLO — UPLO is CHARACTER*1. it tells you which triangle of A is actually stored:
    • = 'U': the upper triangle of A is stored;
    • = 'L': the lower triangle of A is stored.
  • EQUED — EQUED is CHARACTER*1. it specifies the form of equilibration that was done to A before calling this routine. this one matters, because it's needed to compute the solution and the error bounds correctly:
    • = 'N': no equilibration
    • = 'Y': both row and column equilibration — i.e., A has been replaced by diag(S) * A * diag(S). the right hand side B has been changed accordingly.
  • N — N is INTEGER. the order of the matrix A. constraint: N >= 0.
  • NRHS — NRHS is INTEGER. the number of right hand sides, i.e., the number of columns of the matrices B and X. constraint: NRHS >= 0.
  • A — A is COMPLEX 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. so like… only half of her is doing any work, and honestly? relatable 😭
  • LDA — LDA is INTEGER. the leading dimension of the array A. constraint: LDA >= max(1,N).
  • AF — AF is COMPLEX array, dimension (LDAF,N). the factored form of the matrix A. AF 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 CSYTRF. so CSYTRF did the work earlier and this is the receipts 🧾
  • LDAF — LDAF is INTEGER. the leading dimension of the array AF. constraint: LDAF >= max(1,N).
  • IPIV — IPIV is INTEGER array, dimension (N). all the details of the interchanges and the block structure of D, as determined by CSYTRF. iykyk 🫶
  • S — S is REAL array, dimension (N). the scale factors for A. if EQUED = 'Y', A is multiplied on the left and right by diag(S).
    • S is an input argument if FACT = 'F'; otherwise, S is an output argument.
    • if FACT = 'F' and EQUED = 'Y', each element of S must be positive.
    • if S is output, each element of S is a power of the radix.
    • if S is input, each element of S should be a power of the radix, to ensure a reliable solution and reliable error estimates.
    • scaling by powers of the radix does not cause rounding errors — unless the result underflows or overflows.
    • > ⚠️ okay so listen: rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, and that produces error estimates that may not be reliable. so like, keep your scaling clean, bestie 💀
  • B — B is COMPLEX array, dimension (LDB,NRHS). the right hand side matrix B. that's her.
  • LDB — LDB is INTEGER. the leading dimension of the array B. constraint: LDB >= max(1,N).
  • X — X is COMPLEX array, dimension (LDX,NRHS). on entry, the solution matrix X, as computed by CGETRS. on exit, the improved solution matrix X. glow-up arc ✨
  • LDX — LDX is INTEGER. the leading dimension of the array X. constraint: LDX >= max(1,N).
  • RCOND — RCOND is REAL. reciprocal scaled condition number. it's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if equilibration was done).
    • > 🚨 if this is less than the machine precision — in particular, if it is zero — then the matrix is singular to working precision.
    • but heads up: the error may still be small even if this number is very small and the matrix appears ill-conditioned. so don't spiral just because the number looks scary 😌
  • BERR — BERR is REAL array, dimension (NRHS). componentwise relative backward error. more precisely, it's 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. tiny change, big meaning 🎀
  • N_ERR_BNDS — N_ERR_BNDS is INTEGER. the number of error bounds to return for each right hand side and each type (normwise or componentwise). see ERR_BNDS_NORM and ERR_BNDS_COMP below. 👇
  • ERR_BNDS_NORM — ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS). for each right-hand side, this array contains info about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

normwise relative error in the ith solution vector:

        max_j (abs(XTRUE(j,i) - X(j,i)))
       ------------------------------
             max_j abs(X(j,i))

the array is indexed by the type of error information, as described below. there currently are up to three pieces of information returned.

the first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

the second index in ERR_BNDS_NORM(:,err) contains the following three fields:

  • err = 1 — boolean. trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — error bound: the estimated forward error, almost certainly within a factor of 10 of the true error, so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). this error bound should only be trusted if the previous boolean is true.
  • err = 3 — reciprocal condition number: estimated normwise reciprocal condition number. compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is . (yeah — the source text literally just stops there mid-sentence, bestie, I'm not making that up 😭💀) these reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.

see Lapack Working Note 165 for further details and extra cautions. 📓

  • ERR_BNDS_COMP — and this one's coming at you in the next part, bestie, don't go anywhere 💖🦋

ERR_BNDS_COMP is a REAL array, dimension (NRHS, N_ERR_BNDS). Okay bestie, for each right-hand side this array is holding the tea on various error bounds and condition numbers for the componentwise relative error, and that error is defined like this:

                   abs(XTRUE(j,i) - X(j,i))
            max_j ----------------------
                         abs(X(j,i))

So the array is indexed by the right-hand side i (because the componentwise relative error depends on it) and by the type of error info described below. There are currently up to three pieces of info returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains these three fields:

  • err = 1 — boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — Reciprocal condition number: estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is . These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions.

NPARAMS is INTEGER. It specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. Slay, defaults only.

PARAMS is a REAL array, dimension NPARAMS. It specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters.

  • PARAMS(LA_LINRX_ITREF_I = 1) — Whether to perform iterative refinement or not. Default: 1.0
    • = 0.0: No refinement is performed, and no error bounds are computed.
    • = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
    • (other values are reserved for future use)
  • PARAMS(LA_LINRX_ITHRESH_I = 2) — Maximum number of residual computations allowed for refinement. Default: 10
    • Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy.
  • PARAMS(LA_LINRX_CWISE_I = 3) — Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence)

WORK is a COMPLEX array, dimension (2*N). 💅

RWORK is a REAL array, dimension (2*N). ✨

INFO is INTEGER.

  • = 0: Successful exit. The solution to every right-hand side is guaranteed.
  • < 0: If INFO = -i, the i-th argument had an illegal value.
  • > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
  • = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.

👑 AUTHOR 👑

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

Definition at line 398 of file csyrfsx.f.

Author: Generated automatically by Doxygen for LAPACK from the source code.

Okay that’s the whole tea, bestie 🫶 — footer vibe is LAPACK Version 3.12.0, page /home/abui.../csyrfsx.f(3), and that’s the source/version-and-date-ish sign-off energy. Bye 💖✨

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