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

📦 version 3.12.0, in opensuse-leap 16.0

🎀 NAME 🎀

okay so first things first, bestie — we're looking at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zporfsx.f, and this whole page lives in section (3) of the Library Functions Manual (/home/abui.../zporfsx.f(3)). It's a Fortran source file, it's part of LAPACK, and it's about to be your new fave. 💖

✨ SYNOPSIS ✨

Functions/Subroutines

so there's exactly one star of the show here, and her name is ZPORFSX:

subroutine ZPORFSX (uplo, equed, n, nrhs, a, lda, af, ldaf, s, b, ldb,
    x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp,
    nparams, params, work, rwork, info)

literally that's her, that's the whole signature. 🫶

💅 Function/Subroutine Documentation 💅

Purpose

okay so ZPORFSX is the glow-up routine for your solution. It improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite, and it provides error bounds and backward error estimates for the solution. Basically she takes your answer from "eh, probably fine" to "no cap, I can vouch for this." 🔥

And she doesn't stop at just a normwise error bound — in addition to the normwise error bound, the code provides a maximum componentwise error bound if possible. For the deets on what those bounds actually mean, go read the comments for ERR_BNDS_NORM and ERR_BNDS_COMP. 👑

One super important thing: the original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED and S below. If that's the case, the solution and error bounds returned are for the original unequilibrated system — so she un-does the scaling in her head and gives you the answer for the system you actually walked in with. 🦋

Also, some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. And if the defaults are acceptable to you? Users can pass NPARAMS = 0, which prevents the source code from accessing the PARAMS argument entirely — she just won't even look at it. iconic. 💀

Parameters

  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • EQUED — EQUED is CHARACTER*1 Specifies the form of equilibration that was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.
    • = 'N': No equilibration
    • = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). The right hand side B has been changed accordingly.
  • N — N is INTEGER 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 matrices B and X. NRHS >= 0.
  • A — A is COMPLEX*16 array, dimension (LDA,N) The Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
  • 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**H*U or A = L*L**H, as computed by ZPOTRF.
  • LDAF — LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).
  • S — S is DOUBLE PRECISION array, dimension (N) The scale factors for A. If EQUED = 'Y', A is multiplied on the left and right by diag(S). S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. If S is output, each element of S is a power of the radix. If S is input, each element of S 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.
bestie please read that last chunk twice. if you scale with something that isn't a power of the radix, you get rounding errors, and then you're refining against a matrix that isn't even the one you handed in — and the error estimates you get back may not be reliable. that's a real "😭 why is my answer wrong" moment, so don't do 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).
  • X — X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZGETRS. On exit, the improved solution matrix X.
she literally upgrades it in place. main character energy. 👑
  • LDX — LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).
  • RCOND — RCOND is 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… tiny RCOND = the matrix looks a mess. but it's not automatically a disaster — the error can still be small. don't panic-read a single number, k? 🌸
  • BERR — BERR is DOUBLE PRECISION array, dimension (NRHS) Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).
  • N_ERR_BNDS — N_ERR_BNDS is INTEGER Number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below.
  • 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) * dlamch('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) * dlamch('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) * dlamch('Epsilon') to determine if the error estimate is (the source text genuinely cuts off right there, bestie, not me 💀). 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.
See Lapack Working Note 165 for further details and extra cautions. 📚 (yes, extra cautions. go read it.)
  • ERR_BNDS_COMP —

🫶 ERR_BNDS_COMP

ERR_BNDS_COMP is DOUBLE PRECISION 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 defined as follows:

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

That's the componentwise relative error in the ith solution vector, bestie. 💅

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) * dlamch('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) * dlamch('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) * dlamch('Epsilon') to determine if the error estimate is . (yes, the source text literally trails off right there, no cap 💀) 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.
📚 See Lapack Working Note 165 for further details and extra cautions.

🌸 NPARAMS

NPARAMS is INTEGER.

Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used.

🎀 PARAMS

PARAMS is DOUBLE PRECISION array, dimension NPARAMS.

Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters.

  • PARAMS(LA_LINRX_ITREF_I = 1) — Whether to perform iterative refinement or not.
    • Default: 1.0D+0
    • = 0.0: No refinement is performed, and no error bounds are computed.
    • = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
    • (other values are reserved for future use)
  • PARAMS(LA_LINRX_ITHRESH_I = 2) — Maximum number of residual computations allowed for refinement.
    • Default: 10
    • 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.
  • PARAMS(LA_LINRX_CWISE_I = 3) — Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
    • Default: 1.0 (attempt componentwise convergence)

💜 WORK

WORK is COMPLEX*16 array, dimension (2*N).

💜 RWORK

RWORK is DOUBLE PRECISION array, dimension (2*N).

🔥 INFO

INFO is INTEGER.

  • = 0: Successful exit. The solution to every right-hand side is guaranteed.
  • < 0: If INFO = -i, the i-th argument had an illegal value
  • > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
  • = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.

👑 Author

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

Definition at line 389 of file zporfsx.f.

🌈 Author

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