SRC/dsyrfsx.f(3)

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version 3.12.1 · 7 versions
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okay bestie, this is SRC/dsyrfsx.f from the Library Functions Manual — part 1 of the page, and it’s giving “symmetric-indefinite solver refinement” main character energy. 💅

🎀 NAME 🎀

SRC/dsyrfsx.f — that’s the name section, bestie. Short, iconic, no notes.

⭐ SYNOPSIS ⭐

Functions/Subroutines

Here’s the callable tea ☕:

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

🦋 FUNCTION/SUBROUTINE DOCUMENTATION 🦋

okay so here’s the full signature, character-for-character:

   subroutine dsyrfsx (character uplo, character equed, integer n, integer
       nrhs, double precision, dimension( lda, * ) a, integer lda, double
       precision, dimension( ldaf, * ) af, integer ldaf, integer, dimension( *
       ) ipiv, double precision, dimension( * ) s, double precision, dimension(
       ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x,
       integer ldx, double precision rcond, double precision, dimension( * )
       berr, integer n_err_bnds, double precision, dimension( nrhs, * )
       err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp,
       integer nparams, double precision, dimension( * ) params, double
       precision, dimension( * ) work, integer, dimension( * ) iwork, integer
       info)
       DSYRFSX

💖 PURPOSE 💖

okay so DSYRFSX is the glow-up routine for a computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite. It also hands you error bounds and backward error estimates for the solution. ✨

On top of the normwise error bound, this code also gives you a maximum componentwise error bound if possible. For the full deets on the error bounds, peek at the comments for ERR_BNDS_NORM and ERR_BNDS_COMP. 🫶

The original system of linear equations may have already been equilibrated before this routine got called, and EQUED + S below tell you how that went. If that’s the case, the solution and error bounds you get back are for the original unequilibrated system.

Some optional parameters are bundled into the PARAMS array. Those settings decide how refinement happens, but honestly the defaults are often acceptable. If default behavior is fine, users can pass NPARAMS = 0, which prevents the source code from accessing the PARAMS argument. slay.

🌸 PARAMETERS 🌸

UPLO 💅

UPLO is CHARACTER*1, and it says which triangle of A got stored:

  • 'U': Upper triangle of A is stored.
  • 'L': Lower triangle of A is stored.

EQUED 🦋

EQUED is CHARACTER*1. It specifies the form of equilibration that was done to A before calling this routine. You need it 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. It’s 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 DOUBLE PRECISION array, dimension (LDA,N). It’s the symmetric 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 DOUBLE PRECISION array, dimension (LDAF,N). It’s the factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.

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 DSYTRF.

S 🍓

S is DOUBLE PRECISION array, dimension (N). These are 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.

B 👑

B is DOUBLE PRECISION array, dimension (LDB,NRHS). It’s the right hand side matrix B.

LDB 💖

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

X 🌟

X is DOUBLE PRECISION array, dimension (LDX,NRHS).

On entry, the solution matrix X, as computed by DGETRS. On exit, the improved solution matrix X.

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.

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 . 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.

ERR_BNDS_COMP

💖 ERR_BNDS_COMP

okay bestie, so ERR_BNDS_COMP is a 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))

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 . (the source sentence just ends there — mysterious main character energy 😭) 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 DOUBLE PRECISION array, dimension (4*N).

🍓 IWORK

IWORK is INTEGER array, dimension (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 398 of file dsyrfsx.f.

💜 Author

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