ZLA_PORFSX_EXTENDED(3)

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version 3.12.1 · 7 versions
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💅 NAME 💅

SRC/zla_porfsx_extended.f(3) — from the Library Functions Manual. Hiii bestie, this is part 1 and it's giving legendary LAPACK energy. 💖

✨ SYNOPSIS ✨

Functions/Subroutines

       subroutine zla_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)

ZLA_PORFSX_EXTENDED is the extra-precise iterative-refinement glow-up for a computed solution to a system of linear equations when the matrix is symmetric or Hermitian positive-definite. It also hands you error bounds and backward error estimates for the solution. No cap, it's doing a lot. ✨

🌸 Function/Subroutine Documentation 🌸

       subroutine zla_porfsx_extended (integer prec_type, character uplo, integer
           n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda,
           complex*16, dimension( ldaf, * ) af, integer ldaf, 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_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. It's literally the aftercare routine for your linear solve. 🫶

🎀 Purpose 🎀

Okay so ZLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations by performing extra-precise iterative refinement, and it provides error bounds and backward error estimates for the solution.

This subroutine is called by ZPORFSX to perform iterative refinement. In addition to the normwise error bound, the code provides the maximum componentwise error bound if possible. Peep the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the deets on those error bounds.

Note that this subroutine is only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP. That's its whole job, bestie. 💅

🧚 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

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*16 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*16 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 ZPOTRF.

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.

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.

This one is serious: the scale factors gotta be powers of the radix if you want reliable results. Don't play with it. 😭

B 🧚

B is COMPLEX*16 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*16 array, dimension (LDY,NRHS).

On entry, the solution matrix X, as computed by ZPOTRS. 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 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.

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 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.

That err = 2 field is the moment this routine is actually responsible for. Keep your eyes on the second field, bestie. 👀

ERR_BNDS_COMP

  • ERR_BNDS_COMP — DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array is literally carrying all the tea about various error bounds and condition numbers for the componentwise relative error, which is 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 info described below. There are currently up to three pieces of info 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 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.
  • 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*16 array, dimension (N). Workspace to hold the intermediate residual. Just a lil scratchpad, no cap.
  • AYB — DOUBLE PRECISION array, dimension (N). Workspace. That's it, that's the whole job.
  • DY — COMPLEX*16 PRECISION array, dimension (N). Workspace to hold the intermediate solution. (Yes, the source literally says COMPLEX*16 PRECISION — we're not fixing it, we're respecting the lore 💀)
  • Y_TAIL — COMPLEX*16 array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. Trailing bits, get it? 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 don't panic immediately, bestie.
  • 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. So like, proceed with caution, okay?
  • 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. Stable queen energy only.
  • IGNORE_CWISE — LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE..
  • INFO — INTEGER.
    • = 0: Successful exit. We love a zero. 👑
    • < 0: if INFO = -i, the ith argument to ZPOTRS had an illegal value.

Author 👑

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

Definition at line 380 of file zla_porfsx_extended.f.

Author (Doxygen's version) 🎀

Generated automatically by Doxygen for LAPACK from the source code.

That's the whole tea, bestie! 🫶 This iconic page comes from SRC/zla_po..._extended.f(3), LAPACK Version 3.12.0. No date on the footer, but honestly time is a construct anyway. ⏰💀 Go forth and slay your linear algebra. 💅✨

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