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

✨ /home/abui.../dsyrfsx.f(3) ✨

okay so this is a page from the Library Functions Manual, and the file it's documenting lives at that whole path up there. bestie, it's a Fortran thing, it's from LAPACK, and it's about to make your linear algebra so much better. buckle up 💅

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🎀 NAME 🎀

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

that's the whole name section, no cap — it's literally just the file path. iconic minimalism honestly 🦋

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🌸 SYNOPSIS 🌸

Functions/Subroutines

• subroutine DSYRFSX — the one and only star of this page ✨

subroutine DSYRFSX (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, iwork, info)

DSYRFSX — that's her name, that's her whole vibe 💖

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💜 Function/Subroutine Documentation 💜

here's the full signature with all the types spelled out, because the compiler is a lil picky about that:

subroutine DSYRFSX (character uplo, character equed, integer n, integer
    nrhs, double precision, dimension( lda, * ) a, integer lda, double
    precision, dimension( ldaf, * ) af, integer ldaf, integer, dimension( *
    ) ipiv, double precision, dimension( * ) s, double precision, dimension(
    ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x,
    integer ldx, double precision rcond, double precision, dimension( * )
    berr, integer n_err_bnds, double precision, dimension( nrhs, * )
    err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp,
    integer nparams, double precision, dimension( * ) params, double
    precision, dimension( * ) work, integer, dimension( * ) iwork, integer
    info)

DSYRFSX 🫶

🔥 Purpose 🔥

okay so what does she do? DSYRFSX improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, and she also provides error bounds and backward error estimates for the solution. so you get your answer and receipts on how wrong it might be. icon behaviour 🔥

on top of the normwise error bound, the code provides a maximum componentwise error bound if possible. the comments for ERR_BNDS_NORM and ERR_BNDS_COMP have the deets on those error bounds — keep scrolling, bestie.

the original system of linear equations may have been equilibrated before calling this routine, and EQUED and S (below) are how she finds out about that. in that case, the solution and error bounds returned are for the original unequilibrated system. like, she undoes the glow-up so you see the real thing ✨

💫 some optional parameters are bundled in the PARAMS array. those settings determine how refinement is performed, but often the defaults are perfectly acceptable. if the defaults are acceptable, you can pass NPARAMS = 0, which stops the source code from ever touching the PARAMS argument. respectfully, she doesn't even look at it.

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🍓 Parameters 🍓

📌 UPLO

UPLO is CHARACTER*1

• = 'U' — upper triangle of A is stored 🬀
• = 'L' — lower triangle of A is stored 🬂

pick a lane, bestie, pick a triangle.

📌 EQUED

EQUED is CHARACTER*1

specifies the form of equilibration that was done to A before calling this routine. this is needed to compute the solution and error bounds correctly — so don't lie to her about it 🙈

• = '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. N >= 0. the main character's size 💅

📌 NRHS

NRHS is INTEGER

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

📌 A

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.

(the parts she don't reference are simply not her business 🫶)

📌 LDA

LDA is INTEGER

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

📌 AF

AF is DOUBLE PRECISION 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 DSYTRF.

📌 LDAF

LDAF is INTEGER

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

📌 IPIV

IPIV is INTEGER array, dimension (N)

details of the interchanges and the block structure of D as determined by DSYTRF. the gossip, basically 💀

📌 S

S is DOUBLE PRECISION 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 error estimates — scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows.

⚠️ rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, which produces error estimates that may not be reliable. and nobody wants untrustworthy error estimates, ngl 😭

📌 B

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

the right hand side matrix B. what you're actually solving for, queen 👑

📌 LDB

LDB is INTEGER

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

📌 X

X is DOUBLE PRECISION array, dimension (LDX,NRHS)

on entry, the solution matrix X, as computed by DGETRS. on exit, the improved solution matrix X. she takes your answer and makes it better 💅✨

📌 LDX

LDX is INTEGER

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

📌 RCOND

RCOND is DOUBLE PRECISION

reciprocal scaled condition number. this is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). if this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision.

🦋 note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. like, don't panic-ghost her just because RCOND is tiny, iykyk.

📌 BERR

BERR is DOUBLE PRECISION array, dimension (NRHS)

componentwise relative backward error. this is 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). so, how much would you have to fudge reality to be right 🫣

📌 N_ERR_BNDS

N_ERR_BNDS is INTEGER

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 DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

for each right-hand side, this array contains information 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) * dlamch('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) * dlamch('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) * dlamch('Epsilon') to determine if the error estimate is … (the source text just cuts off there, bestie, that's not me gatekeeping 💀). 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 is exactly where part 1 runs out of road — the ERR_BNDS_COMP deets are rolling in with part 2, so stay tuned 🧚‍♀️

ERR_BNDS_COMP — okay so this one is a DOUBLE PRECISION array with dimension (NRHS, N_ERR_BNDS). For each right-hand side, it’s packing information about various error bounds and condition numbers for the componentwise relative error, which is defined like this:

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

The array is indexed by the right-hand side i (because the componentwise relative error depends on it) and by the type of error information described below. There are currently up to three pieces of info returned for each right-hand side. If componentwise accuracy isn’t requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed at all. 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) * dlamch('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) * dlamch('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) * dlamch('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 — this is an INTEGER. It specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used.

PARAMS — this is a DOUBLE PRECISION array with 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: 1.0D+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 — this is a DOUBLE PRECISION array with dimension (4*N).

IWORK — this is an INTEGER array with dimension (N).

INFO — this is an INTEGER.

• = 0: successful exit. The solution to every right-hand side is guaranteed.
• < 0: if INFO = -i, the ith 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 dsyrfsx.f.

✨ Author (the generated one) ✨

Generated automatically by Doxygen for LAPACK from the source code.

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That’s the whole vibe check, bestie! This page is LAPACK Version 3.12.0, generated from the source at /home/abui.../dsyrfsx.f(3). Stay iconic 💅✨