SRC/dsysvxx.f(3)

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💅 SRC/dsysvxx.f — DSYSVXX, from the Library Functions Manual

Okay bestie, settle in, because this one is the linear-algebra girlie of the SY prefix family. We're in the SRC/dsysvxx.f page, and it's all about DSYSVXX 💖 — the double precision, symmetric, expert-driver routine that solves A * X = B and then hands you receipts about how wrong it might be. Iconic behaviour, truly.

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

subroutine dsysvxx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, equed,
    s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm,
    err_bnds_comp, nparams, params, work, iwork, info)
DSYSVXX — that's the name, that's the whole vibe, say it with your chest.

🫶 Function/Subroutine Documentation

🌸 subroutine dsysvxx

Full signature, character for character, because we do not paraphrase declarations in this house:

subroutine dsysvxx (character fact, character uplo, integer n, integer nrhs,
    double precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv,
    character equed, double precision, dimension( * ) s, double precision,
    dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, *
    ) x, integer ldx, double precision rcond, double precision rpvgrw,
    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)

DSYSVXX 👑

💜 Purpose:

So the tea is: DSYSVXX uses the diagonal pivoting factorization to compute the solution to a double precision system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. Slay. 💅

And if you ask nicely, both normwise and maximum componentwise error bounds are returned. Like, the routine doesn't just give you an answer, it gives you an answer with confidence intervals and a whole little report card. DSYSVXX will return a solution with a tiny guaranteed error (O(eps) where eps is the working machine precision) unless the matrix is very ill-conditioned, in which case — plot twist — a warning is returned. 😭 Relevant condition numbers also are calculated and returned, because we love transparency.

DSYSVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options, bestie, they're right down there. 👇

Lowkey important caveat, no cap: solving with refinement and using a factorization from a previous DSYSVXX call will also produce a solution with either O(eps) errors or warnings, but we cannot make that claim for general user-provided factorizations and equilibration factors if they differ from what DSYSVXX would itself produce. So like… don't hand it a random factorization from a mysterious source and expect the guarantee. That's the whole disclaimer. 🫶

🌈 Description:

Here's the step-by-step, and yes, we're doing all six steps, nobody's skipping the boring ones:

  1. If FACT = 'E', double precision scaling factors are computed to equilibrate the system:
diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  1. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = U * D * U**T,  if UPLO = 'U', or
A = L * D * L**T,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. Structure! We love structure. 🎀

  1. If some D(i,i)=0, so that D is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A (see argument RCOND). If the reciprocal of the condition number is less than machine precision, the routine still goes on to solve for X and compute error bounds as described below. She's persistent. 💅
  2. The system of equations is solved for X using the factored form of A.
  3. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), the routine will use iterative refinement to try to get a small error and error bounds. Refinement calculates the residual to at least twice the working precision. Twice! Overachiever energy. ⭐
  4. If equilibration was used, the matrix X is premultiplied by diag(R) so that it solves the original system before equilibration.
💀 And here's the little side note about the optional parameters: some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument. So you can just. Not deal with it. Iconic. 🙈

🎀 Parameters

🍓 FACT

FACT is CHARACTER*1. It specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.

  • = 'F' — On entry, AF and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by S. A, AF, and IPIV are not modified.
  • = 'N' — The matrix A will be copied to AF and factored.
  • = 'E' — The matrix A will be equilibrated if necessary, then copied to AF and factored.
Three moods, three letters. Pick your fighter. ✨

🦋 UPLO

UPLO is CHARACTER*1.

  • = 'U' — Upper triangle of A is stored;
  • = 'L' — Lower triangle of A is stored.

That's it. That's the option. Which triangle are we living in today? 🌸

👑 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 matrices B and X. NRHS >= 0.

💖 A

A is DOUBLE PRECISION array, dimension (LDA,N). 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.

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).

Read the "not referenced" parts again bestie, they matter — you're not storing a full matrix, you're storing half of one and vibing. 🫶

🌟 LDA

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

💜 AF

AF is DOUBLE PRECISION array, dimension (LDAF,N).

  • If FACT = 'F', then AF is an input argument and on entry 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.
  • If FACT = 'N', then AF is an output argument and on exit returns 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.

She switches roles depending on the FACT. Main character energy, honestly. ⭐

🎀 LDAF

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

🍓 IPIV

IPIV is INTEGER array, dimension (N).

  • If FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by DSYTRF. Specifically, the whole rulebook:
    • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
    • If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    • If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by DSYTRF.
Negative IPIV values are not a bug, they're a 2-by-2 block announcement. Do not freak out. 💀

🌈 EQUED

EQUED is CHARACTER*1. It specifies the form of equilibration that was done.

  • = 'N' — No equilibration (always true if FACT = 'N').
  • = 'Y' — Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S).

EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument. Yeah, same dual-life situation as AF and IPIV. 💅

🌸 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.
Okay so this is the part where you must listen, bestie 😭 — 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 if you hand it sloppy scale factors that aren't powers of the radix, the error bars you get back are lowkey lying to you. Not a slay.

🧚 B

B is DOUBLE PRECISION array, dimension (LDB,NRHS).

  • On entry, the N-by-NRHS right hand side matrix B.
  • On exit:
    • if EQUED = 'N', B is not modified;
    • if EQUED = 'Y', B is overwritten by diag(S)*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).

If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations.

⚠️ Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(S))*X. So the X you're holding is the solution to the equilibrated system, not the original one, unless you do that multiplication. Don't forget it, don't skip it, don't @ me. 🙈

🎀 LDX

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

💜 RCOND

And that, bestie, is exactly where part one stops — right on RCOND, mid-parameter list, which is honestly such a cliffhanger. 🫶 The rest of the parameters, plus all the INFO values, examples, and the SEE ALSO crew? Coming right up in the next part. No skips, no summaries, every single detail is coming with me. ✨

Lowkey picking right back up in the argument list, bestie — no need to re-intro, we’re mid-flow 💅

  • 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.
  • RPVGRW — DOUBLE PRECISION. Reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. If factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A.
  • BERR — 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 — 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 — 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 — 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 componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

                    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 . 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 — INTEGER. Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used.
  • PARAMS — 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 extra-precise refinement algorithm.
      • (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 — DOUBLE PRECISION array, dimension (4*N).
  • IWORK — INTEGER array, dimension (N).
  • INFO — 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 501 of file dsysvxx.f.

Author (Doxygen’s little encore) 🧚

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And the footer is serving LAPACK Version 3.12.0 + source SRC/dsysvxx.f(3) — that’s the source/version/date slot, and we are so done, bestie 💖✨

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