SLA_PORFSX_EXTENDED(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💖 SRC/sla_porfsx_extended.f(3) 💖

okay so real talk, bestie — this page is straight out of the Library Functions Manual, and it's a whole saga. Grab a lil snack 🍓, we're getting into extra-precise iterative refinement and it's honestly kind of iconic.

🌸 NAME 🌸

SRC/sla_porfsx_extended.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

  • subroutine sla_porfsx_extended (prec_type, uplo, n, nrhs, a, lda, af, ldaf, 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) — SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

🎀 Function/Subroutine Documentation 🎀

Like, here's the whole main character in full, no cuts, no edits:

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

SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

💜 Purpose

!> !> SLA_PORFSX_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 SPORFSX 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. !>

periodt. 💅

🦋 Parameters 🦋

⭐ 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 !>

the way that 'X' and 'E' are both doing the same vibe check… slay either way 👑

⭐ UPLO

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

⭐ N

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

⭐ NRHS

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

⭐ A

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

⭐ LDA

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

⭐ AF

!> AF is REAL array, dimension (LDAF,N) !> The triangular factor U or L from the Cholesky factorization !> A = U**T*U or A = L*L**T, as computed by SPOTRF.

the Cholesky factorization really said "I'll do it" and then did 💖

⭐ LDAF

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

⭐ 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.

bestie if you lie about this one, the error bounds just aren't going to be correct. not a vibe. 🫠

⭐ C

!> C is REAL 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.

lowkey the most unhinged paragraph on the page — rounding errors during scaling = refining with a matrix that is NOT equivalent to the input matrix = your error estimates are unreliable. 🙈 so keep those scales as powers of the radix fr fr.

⭐ B

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

⭐ LDB

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

⭐ Y

!> Y is REAL array, dimension (LDY,NRHS) !> On entry, the solution matrix X, as computed by SPOTRS. !> On exit, the improved solution matrix Y.

glow-up arc: comes in as X, leaves as an improved Y. 👑

⭐ LDY

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

⭐ BERR_OUT

!> BERR_OUT is REAL 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 SLA_LIN_BERR.

the formula, but make it couture:

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

and for the record: abs(Z) here is the componentwise absolute value of the matrix or vector Z, and yes SLA_LIN_BERR is the one doing the math. 🧮

⭐ 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.

⭐ ERR_BNDS_NORM

!> ERR_BNDS_NORM is REAL 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))

so like, literally this:

        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.

the three fields, laid out cute:

  1. err = 1 — boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  2. 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.
  3. 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 remember: this subroutine is only responsible for setting the second field above. Also go read Lapack Working Note 165 for further details and extra cautions — it's giving required reading. 📚

⭐ ERR_BNDS_COMP

…and that's where part one leaves us, bestie 🫶 the ERR_BNDS_COMP chapter is next, and it's so not the ending.

So ERR_BNDS_COMP is a REAL array, dimension (NRHS, N_ERR_BNDS) — literally the receipts for every right-hand side. For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

                   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

RES is a REAL array, dimension (N). Workspace to hold the intermediate residual. 🫶

AYB

AYB is a REAL array, dimension (N). Workspace. This can be the same workspace passed for Y_TAIL.

DY

DY is a REAL array, dimension (N). Workspace to hold the intermediate solution.

Y_TAIL

Y_TAIL is a REAL array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. ✨

RCOND

RCOND is REAL. 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

ITHRESH is 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

RTHRESH is REAL. 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

DZ_UB is REAL. 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

IGNORE_CWISE is LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE.. 🙈

INFO

INFO is INTEGER.

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

👑 Author

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

Definition at line 380 of file sla_porfsx_extended.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

💖 That’s the whole part 2, bestie. Signed off by LAPACK Version 3.12.0, source SRC/sla_po..._extended.f(3). Mwah! 💅✨

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