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

okay so this is /home/abui...extended.f(3) from the Library Functions Manual, and it's here to make extra-precise iterative refinement feel less like a mystery. The file it documents is literally:

💅 NAME

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

---

✨ SYNOPSIS

🧚 Functions/Subroutines

subroutine DLA_GERFSX_EXTENDED (prec_type, trans_type, n, nrhs, a, lda, af, ldaf, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, errs_n, errs_c, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)

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

---

🛠️ Function/Subroutine Documentation

subroutine DLA_GERFSX_EXTENDED (integer prec_type, integer trans_type, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, logical colequ, double precision, dimension( * ) c, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldy, * ) y, integer ldy, double precision, dimension( * ) berr_out, integer n_norms, double precision, dimension( nrhs, * ) errs_n, double precision, dimension( nrhs, * ) errs_c, double precision, dimension( * ) res, double precision, dimension( * ) ayb, double precision, dimension( * ) dy, double precision, dimension( * ) y_tail, double precision rcond, integer ithresh, double precision rthresh, double precision dz_ub, logical ignore_cwise, integer info)

DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. iconic behavior, honestly.

🎯 Purpose:

okay so the whole vibe is this: DLA_GERFSX_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 DGERFSX to perform iterative refinement. In addition to the normwise error bound, the code provides the maximum componentwise error bound if possible. Bestie, see the comments for ERRS_N and ERRS_C for the details of the error bounds. Note that this subroutine is only responsible for setting the second fields of ERRS_N and ERRS_C. periodt.

🌸 Parameters

PREC_TYPE

PREC_TYPE is INTEGER. It 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

TRANS_TYPE

TRANS_TYPE is INTEGER. It specifies the transposition operation on A. The value is defined by ILATRANS(T) where T is a CHARACTER and T:

• = 'N': No transpose
• = 'T': Transpose
• = 'C': Conjugate transpose

N

N is INTEGER. It's the number of linear equations, i.e., the order of the matrix A. N >= 0.

NRHS

NRHS is INTEGER. It's the number of right-hand-sides, i.e., the number of columns of the matrix B.

A

A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the N-by-N matrix A.

LDA

LDA is INTEGER. It's the leading dimension of the array A. LDA >= max(1,N).

AF

AF is DOUBLE PRECISION array, dimension (LDAF,N). The factors L and U from the factorization A = P*L*U as computed by DGETRF.

LDAF

LDAF is INTEGER. It's the leading dimension of the array AF. LDAF >= max(1,N).

IPIV

IPIV is INTEGER array, dimension (N). The pivot indices from the factorization A = P*L*U as computed by DGETRF; row i of the matrix was interchanged with row IPIV(i).

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 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. ⚠️ no cap, that last part matters.

B

B is DOUBLE PRECISION array, dimension (LDB,NRHS). The right-hand-side matrix B.

LDB

LDB is INTEGER. It's the leading dimension of the array B. LDB >= max(1,N).

Y

Y is DOUBLE PRECISION array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by DGETRS. On exit, the improved solution matrix Y.

LDY

LDY is INTEGER. It's the leading dimension of the array Y. LDY >= max(1,N).

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 DLA_LIN_BERR.

N_NORMS

N_NORMS is INTEGER. It determines which error bounds to return (see ERRS_N and ERRS_C).

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

ERRS_N

ERRS_N 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 ERRS_N(i,:) corresponds to the ith right-hand side.

The second index in ERRS_N(:,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.

ERRS_C

…and this is exactly where part 1 stops, bestie — ERRS_C is literally mid-moment, and part 2 has the rest. 💅

ERRS_C — DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

okay so for each right-hand side, this array is holding info about various error bounds and condition numbers that correspond to 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 (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 ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

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

The second index in ERRS_C(:,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 — DOUBLE PRECISION array, dimension (N)

Workspace to hold the intermediate residual. literally just vibing there holding the residue, bestie 💅

AYB — DOUBLE PRECISION array, dimension (N)

Workspace. This can be the same workspace passed for Y_TAIL. sharing is caring 🫶

DY — DOUBLE PRECISION array, dimension (N)

Workspace to hold the intermediate solution.

Y_TAIL — DOUBLE PRECISION array, dimension (N)

Workspace to hold the trailing bits of the intermediate solution. the little leftovers, iconic ✨

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. so like… don’t panic automatically, but also respect the warning 😭

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 ERRS_N and ERRS_C may no longer be trustworthy. periodt, that’s a real caveat 💀

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. aggressive era 🔥

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.. simple as that, bestie 🎀

INFO — INTEGER

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

---

🫶 Author 🫶

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

Definition at line 390 of file dla_gerfsx_extended.f.

Author (yes, again 😭)

Generated automatically by Doxygen for LAPACK from the source code.

---

okay that’s the whole parameter saga, bestie 💅 LAPACK Version 3.12.0, source /home/abui...extended.f(3) — that’s the source/version footer energy, and the date is the one sitting in the footer. no cap, bye 🫶✨