CLA_SYRFSX_EXTENDED.F(3)

💖 NAME: SRC/cla_syrfsx_extended.f 💖

Library Functions Manual — okay so this is the page for SRC/cla_syrfsx_extended.f, and it's giving very serious numerical refinement energy. Bestie, buckle up 💅✨

✨ SYNOPSIS ✨

🌈 Functions/Subroutines 🌈

There's one function/subroutine on deck, and she's iconic:

subroutine cla_syrfsx_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)
    CLA_SYRFSX_EXTENDED improves the computed solution to a system of
    linear equations for symmetric indefinite matrices by performing
    extra-precise iterative refinement and provides error bounds and
    backward error estimates for the solution.

🦋 Function/Subroutine Documentation 🦋

Here's the full signature, exactly as she appears, no cap:

subroutine cla_syrfsx_extended (integer prec_type, character uplo, integer
    n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex,
    dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv,
    logical colequ, real, dimension( * ) c, complex, dimension( ldb, * ) b,
    integer ldb, complex, dimension( ldy, * ) y, integer ldy, real,
    dimension( * ) berr_out, integer n_norms, real, dimension( nrhs, * )
    err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, complex,
    dimension( * ) res, real, dimension( * ) ayb, complex, dimension( * )
    dy, complex, dimension( * ) y_tail, real rcond, integer ithresh, real
    rthresh, real dz_ub, logical ignore_cwise, integer info)
    CLA_SYRFSX_EXTENDED improves the computed solution to a system of linear
    equations for symmetric indefinite matrices by performing extra-precise
    iterative refinement and provides error bounds and backward error
    estimates for the solution.

💖 Purpose 💖

Okay so the whole point: CLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations by doing extra-precise iterative refinement, and it also provides error bounds and backward error estimates for the solution. This subroutine is called by CSYRFSX to perform iterative refinement. 🫶

In addition to the normwise error bound, the code provides the maximum componentwise error bound if possible. See the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the details of the error bounds. And note — important — this subroutine is only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP, bestie. Not the first fields. Don't get it twisted 💅

🎀 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
• 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 COMPLEX 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 COMPLEX array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.
• 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 CSYTRF.
• 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.
• 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. so yeah, respect the radix-power rule, okay? 😭
• B — B is COMPLEX 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 COMPLEX array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by CSYTRS. On exit, the improved solution matrix 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 CLA_LIN_BERR.

• 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))

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 . (yes, the source page literally has a blank word there, bestie — I'm not inventing it 😭) 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.

• ERR_BNDS_COMP — okay the page cuts off right here in part 1; part 2 picks up this parameter, no cap 👀💅

ERR_BNDS_COMP is a 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 componentwise relative error, which is literally defined as:

                   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 — COMPLEX array, dimension (N). Workspace to hold the intermediate residual.

AYB — REAL array, dimension (N). Workspace.

DY — COMPLEX array, dimension (N). Workspace to hold the intermediate solution.

Y_TAIL — COMPLEX array, dimension (N). Workspace to hold the trailing bits of the intermediate solution.

RCOND — 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 — 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 — 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 — 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 — LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE..

INFO — INTEGER.

• = 0: Successful exit.
• < 0: if INFO = -i, the ith argument to CLA_SYRFSX_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 cla_syrfsx_extended.f.

🪄 Author, but make it auto-generated ✨

Generated automatically by Doxygen for LAPACK from the source code.

That’s the whole vibe, bestie — LAPACK Version 3.12.0, file SRC/cla_sy..._extended.f(3), auto-generated by Doxygen for LAPACK from the source code. 💅✨🫶