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

📦 version 3.12.0, in opensuse-leap 16.0

🌸 /home/abui...extended.f(3) 🌸

okay bestie, buckle up — this is a page from the Library Functions Manual and the file it's about is literally this whole path, no cap:

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

yes that's the actual NAME section. it's giving "my name is my entire address" energy 💀 but that's literally how this page labels itself, so there we go.

✨ SYNOPSIS ✨

so under Functions/Subroutines we've got exactly one icon:

subroutine ZLA_HERFSX_EXTENDED (prec_type, uplo, n, nrhs, a, lda, af,
    ldaf, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms,
    err_bnds_norm, err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh,
    rthresh, dz_ub, ignore_cwise, info)

and the one-liner tea on it: ZLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. 💅 like yes she does all of that.

🦋 Function/Subroutine Documentation 🦋

okay, the full queen in all her glory — the declaration is long, so here it is verbatim, absolutely not about to paraphrase a type signature 😤:

subroutine ZLA_HERFSX_EXTENDED (integer prec_type, character uplo, integer
    n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda,
    complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( *
    ) ipiv, logical colequ, double precision, dimension( * ) c, complex*16,
    dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldy, * ) y,
    integer ldy, double precision, dimension( * ) berr_out, integer n_norms,
    double precision, dimension( nrhs, * ) err_bnds_norm, double precision,
    dimension( nrhs, * ) err_bnds_comp, complex*16, dimension( * ) res,
    double precision, dimension( * ) ayb, complex*16, dimension( * ) dy,
    complex*16, dimension( * ) y_tail, double precision rcond, integer
    ithresh, double precision rthresh, double precision dz_ub, logical
    ignore_cwise, integer info)

ZLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

🎀 Purpose: 🎀

!> ZLA_HERFSX_EXTENDED improves the computed solution to a system of !> linear equations by performing extra-precise iterative refinement !> and provides error bounds and backward error estimates for the solution. !> This subroutine is called by ZHERFSX to perform iterative refinement. !> In addition to normwise error bound, the code provides maximum !> componentwise error bound if possible. See comments for ERR_BNDS_NORM !> and ERR_BNDS_COMP for details of the error bounds. Note that this !> subroutine is only responsible for setting the second fields of !> ERR_BNDS_NORM and ERR_BNDS_COMP.

so real talk: she's the refinement girlie called in by ZHERFSX. she gives you a normwise error bound, plus a maximum componentwise error bound when that's possible. and lowkey important — she's only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP. not the whole array. just that one field. periodt. 💖

✨ Parameters ✨

okay this is the part where every single arg gets its moment, and we are not skipping a single one 🫶

🌈 PREC_TYPE

!> PREC_TYPE is INTEGER !> Specifies the intermediate precision to be used in refinement. !> The value is defined by ILAPREC(P) where P is a CHARACTER and P !> = 'S': Single !> = 'D': Double !> = 'I': Indigenous !> = 'X' or 'E': Extra

tl;dr: it's an INTEGER that picks the intermediate precision used in refinement, and the value comes from ILAPREC(P) where P is a CHARACTER. the mapping is: 'S' → Single, 'D' → Double, 'I' → Indigenous, 'X' or 'E' → Extra. four-vibe menu, iconic. ⭐

🌈 UPLO

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

a CHARACTER*1 — 'U' means the upper triangle of A is stored, 'L' means the lower triangle is stored. pick a side bestie, there's only two 💅

🌈 N

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

INTEGER, number of linear equations = the order of matrix A, and the rule is N >= 0. zero is allowed, negative is not, we don't do that here 🚫

🌈 NRHS

!> NRHS is INTEGER !> The number of right-hand-sides, i.e., the number of columns of the !> matrix B.

INTEGER, the number of right-hand-sides — which is just the number of columns of matrix B. same thing, different font 💖

🌈 A

!> A is COMPLEX*16 array, dimension (LDA,N) !> On entry, the N-by-N matrix A.

COMPLEX*16 array, dimension (LDA,N). on entry, it's the N-by-N matrix A. the main character of the whole situation 👑

🌈 LDA

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

INTEGER, leading dimension of the array A, and the rule is LDA >= max(1,N). no shortcuts, we respect the leading dimension in this house 🧚

🌈 AF

!> AF is COMPLEX*16 array, dimension (LDAF,N) !> The block diagonal matrix D and the multipliers used to !> obtain the factor U or L as computed by ZHETRF.

COMPLEX*16 array, dimension (LDAF,N). this one's holding the block diagonal matrix D plus the multipliers used to obtain the factor U or L, as computed by ZHETRF. she's the factorization bestie 🎀

🌈 LDAF

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

INTEGER, leading dimension of the array AF, with the rule LDAF >= max(1,N). yup, same energy, different letter 🦋

🌈 IPIV

!> IPIV is INTEGER array, dimension (N) !> Details of the interchanges and the block structure of D !> as determined by ZHETRF.

INTEGER array, dimension (N). it carries the details of the interchanges and the block structure of D, as determined by ZHETRF. the receipts, basically 🧾

🌈 COLEQU

!> COLEQU is LOGICAL !> If .TRUE. then column equilibration was done to A before calling !> this routine. This is needed to compute the solution and error !> bounds correctly.

LOGICAL. if it's .TRUE., then column equilibration was done to A before calling this routine — and this is needed to compute the solution and error bounds correctly. not optional juice, it genuinely matters 😭

🌈 C

!> C is DOUBLE PRECISION array, dimension (N) !> The column scale factors for A. If COLEQU = .FALSE., C !> is not accessed. If C is input, each element of C 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, producing error estimates that may not be !> reliable.

DOUBLE PRECISION array, dimension (N) — the column scale factors for A. big important caveat energy:

  • if COLEQU = .FALSE., then C is not accessed. ghosted, respectfully 🙈
  • if C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates.
  • scaling by powers of the radix doesn't 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. ⚠️ so pls be careful, this is the risky part.

🌈 B

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

COMPLEX*16 array, dimension (LDB,NRHS) — the right-hand-side matrix B. the whole reason we're solving anything 🍓

🌈 LDB

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

INTEGER, leading dimension of the array B, rule: LDB >= max(1,N). you already know the pattern by now 💀

🌈 Y

!> Y is COMPLEX*16 array, dimension (LDY,NRHS) !> On entry, the solution matrix X, as computed by ZHETRS. !> On exit, the improved solution matrix Y.

COMPLEX*16 array, dimension (LDY,NRHS). on entry it's the solution matrix X, as computed by ZHETRS. on exit it's the improved solution matrix Y. glow-up arc, literally ✨

🌈 LDY

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

INTEGER, leading dimension of the array Y, rule: LDY >= max(1,N). consistent queen 💅

🌈 BERR_OUT

!> BERR_OUT is DOUBLE PRECISION array, dimension (NRHS) !> On exit, BERR_OUT(j) contains the componentwise relative backward !> error for right-hand-side j from the formula > max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) !> where abs(Z) is the componentwise absolute value of the matrix !> or vector Z. This is computed by ZLA_LIN_BERR.

DOUBLE PRECISION array, dimension (NRHS). on exit, BERR_OUT(j) holds the componentwise relative backward error for right-hand-side j, from this formula exactly:

max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )

where abs(Z) is the componentwise absolute value of the matrix or vector Z. and this is computed by ZLA_LIN_BERR. shoutout 🫶

🌈 N_NORMS

!> N_NORMS is INTEGER !> Determines which error bounds to return (see ERR_BNDS_NORM !> and ERR_BNDS_COMP). !> If N_NORMS >= 1 return normwise error bounds. !> If N_NORMS >= 2 return componentwise error bounds.

INTEGER, and it decides which error bounds to return (peep ERR_BNDS_NORM and ERR_BNDS_COMP). the rules:

  • if N_NORMS >= 1 → return normwise error bounds.
  • if N_NORMS >= 2 → return componentwise error bounds.

two-tier slider, very demure 🔥

🌈 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) * 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 . 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. !> !> This subroutine is only responsible for setting the second field !> above. !> See Lapack Working Note 165 for further details and extra !> cautions.

DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). for each right-hand side, this holds info about various error bounds and condition numbers for the normwise relative error — which is defined ~verbatim~ as:

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 info described below, and there are currently up to three pieces of info 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 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. ⚠️ (yes, read that twice, the trust is conditional)
    • 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 . 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.

and the crucial bit: this subroutine is only responsible for setting the second field above. just field two. that's all she signed up for 🎯

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

🌈 ERR_BNDS_COMP

ERR_BNDS_COMP — DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 🌈

okay so for each right-hand side, this array is literally packed with info about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

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

The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information 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 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 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.

This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.

RES — COMPLEX16 array, dimension (N)* 💅

Workspace to hold the intermediate residual.

AYB — DOUBLE PRECISION array, dimension (N) ✨

Workspace.

DY — COMPLEX16 array, dimension (N)* 🦋

Workspace to hold the intermediate solution.

Y_TAIL — COMPLEX16 array, dimension (N)* 🌸

Workspace to hold the trailing bits of the intermediate solution.

RCOND — 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.

ITHRESH — INTEGER 🔥

The maximum number of residual computations allowed for refinement. The default is 10. For '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.

RTHRESH — DOUBLE PRECISION 💜

Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details.

DZ_UB — DOUBLE PRECISION 🌈

Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we define as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details.

IGNORE_CWISE — LOGICAL 🧚

If .TRUE. then ignore componentwise convergence. Default value is .FALSE..

INFO — INTEGER 💀

  • = 0: Successful exit.
  • < 0: if INFO = -i, the ith argument to ZLA_HERFSX_EXTENDED had an illegal value

Author 👑

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

Definition at line 388 of file zla_herfsx_extended.f.

and that's a wrap on this absolute icon of a manpage! 💅 this page is from LAPACK Version 3.12.0, auto-generated by Doxygen for LAPACK from the source code, and the file is /home/abui...extended.f(3) — periodt. stay slay, bestie 💖✨

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