hesvxx(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

💖 NAME 💖

okay bestie, welcome to hesvxx(3) — the manual page for the whole {he,sy}svxx crew. The NAME line is:

hesvxx — {he,sy}svxx: rook (v1, expert)

iconic. This is the one that covers the {he,sy}svxx family, and we are keeping every single detail exactly where it belongs.

✨ SYNOPSIS ✨

🫶 Functions 🫶

here's the function lineup, exactly as written, no cap:

subroutine chesvxx (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, rwork, info)
     CHESVXX computes the solution to system of linear equations A * X =
    B for HE matrices
subroutine csysvxx (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, rwork, info)
     CSYSVXX computes the solution to system of linear equations A * X =
    B for SY matrices
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
subroutine ssysvxx (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)
    SSYSVXX
subroutine zhesvxx (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, rwork, info)
     ZHESVXX computes the solution to system of linear equations A * X =
    B for HE matrices
subroutine zsysvxx (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, rwork, info)
     ZSYSVXX computes the solution to system of linear equations A * X =
    B for SY matrices

and yes, DSYSVXX and SSYSVXX are literally captioned with just their all-caps names in the original — keeping it exactly like that, bestie 💅.

🔥 Detailed Description 🔥

okay so this section is literally just the heading here — no extra paragraph, no side quest. Then we go straight into Function Documentation.

👑 Function Documentation 👑

🎀 subroutine chesvxx 🎀

subroutine chesvxx (character fact, character uplo, integer n, integer nrhs,
    complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, *
    ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real,
    dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex,
    dimension( ldx, * ) x, integer ldx, real rcond, real rpvgrw, real,
    dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * )
    err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer
    nparams, real, dimension( * ) params, complex, dimension( * ) work,
    real, dimension( * ) rwork, integer info)

CHESVXX computes the solution to system of linear equations A * X = B for HE matrices.

🎯 Purpose 🎯

okay so CHESVXX uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS matrices.

If requested, both normwise and maximum componentwise error bounds are returned. CHESVXX 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 a warning is returned. Relevant condition numbers also are calculated and returned.

CHESVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous CHESVXX 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 CHESVXX would itself produce.

📝 Description 📝

The following steps are performed:

If FACT = 'E', real 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.

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 Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

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.
The system of equations is solved for X using the factored form of A.
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.
If equilibration was used, the matrix X is premultiplied by diag(R) so that it solves the original system before equilibration.

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.

💅 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.

`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 matrices B and X. NRHS >= 0.

`A`

A is COMPLEX 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.

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

`LDA`

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

`AF`

AF is COMPLEX 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**H or A = L*D*L**H as computed by CHETRF.

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**H or A = L*D*L**H.

`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 CHETRF. 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 CHETRF.

`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.

`S`

S — this one's a REAL array, dimension (N), and it's holding the scale factors for A. If EQUED = 'Y', A gets multiplied on the left and the right by diag(S). Now here's the vibe switch, bestie: S is an input argument if FACT = 'F', otherwise S is an output argument. And if FACT = 'F' and EQUED = 'Y', every single element of S has to 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 so you get a reliable solution and reliable error estimates — scaling by powers of the radix doesn't cause rounding errors unless the result underflows or overflows. And if rounding errors do sneak in during scaling? You end up refining with a matrix that isn't equivalent to the input matrix, which produces error estimates that may not be reliable. 😭 Lowkey cursed, but okay.
B — COMPLEX array, dimension (LDB,NRHS). On entry, it's the N-by-NRHS right hand side matrix B. On exit though? Depends: if EQUED = 'N', B is not modified (untouched, unbothered, living its life ✨); if EQUED = 'Y', B is overwritten by diag(S)*B.
LDB — INTEGER. The leading dimension of the array B. Must satisfy LDB >= max(1,N). No notes, just rules. 💅
X — COMPLEX array, dimension (LDX,NRHS). If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. Big heads up though: A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(S))*X. Ship it 🫶
LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,N). Same energy as LDB, different letter.
RCOND — REAL. Reciprocal scaled condition number — an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if equilibration happened at all). If this is less than the machine precision (in particular, if it's zero), the matrix is singular to working precision. But — plot twist — the error may still be small even if this number is very small and the matrix appears ill-conditioned. So don't spiral. 🌸
RPVGRW — REAL. Reciprocal pivot growth. On exit, this holds the reciprocal pivot growth factor norm(A)/norm(U). The norm is used — yes, the source literally just says "The norm is used", no further gossip available 💀. If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. That also means the solution X, the estimated condition numbers, and the error bounds could all be unreliable. And if factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A.
BERR — REAL array, dimension (NRHS). Componentwise relative backward error — specifically 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). Tiny number, huge meaning. 🎀
N_ERR_BNDS — INTEGER. The 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, bestie, they're right there 👇
ERR_BNDS_NORM — REAL 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 normwise relative error, which is defined like this:

        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 *i*th right-hand side.

The second index in ERR_BNDS_NORM(:,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 normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is [the original page cuts off mid-sentence here 🙈]. 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 — REAL 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, 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 (the componentwise relative error depends on it, periodt) and the type of error information 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. And 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 *i*th 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 [again, cut off in the source, 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*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. 📚 (yes, same note, it's a whole thing)

NPARAMS — INTEGER. Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. Ignore it and it simply won't perceive you ✨
PARAMS — REAL array, dimension NPARAMS. Specifies algorithm parameters. If an entry is < 0.0, then that entry gets filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters. Here's the lineup:
- PARAMS(LA_LINRX_ITREF_I = 1) — whether to perform iterative refinement or not. Default: 1.0.
  - = 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. But 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 — COMPLEX array, dimension (5*N). Just vibes and workspace. 🧚
RWORK — REAL array, dimension (2*N). Same, but real.
INFO — and the deets on INFO are coming up next, bestie, hang tight 🫶

INFO (the exit-code tea ☕ — literally the vibe check of whether your solve is safe) is an INTEGER, and here's the whole spectrum:

= 0: Successful exit. The solution to every right-hand side is guaranteed. Slay, we love a guaranteed queen.
< 0: If INFO = -i, the i-th argument had an illegal value. Not cute — check your args, bestie.
> 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. Singular? The math said no. 💀
= 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. iykyk — the first warning gets the spotlight, but the others are still lurking.

Author ✍️

This routine is brought to you by:

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

Definition at line 505 of file chesvxx.f.

`csysvxx` 🦋

   subroutine csysvxx (character fact, character uplo, integer n, integer nrhs,
       complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, *
       ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real,
       dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex,
       dimension( ldx, * ) x, integer ldx, real rcond, real rpvgrw, real,
       dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * )
       err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer
       nparams, real, dimension( * ) params, complex, dimension( * ) work,
       real, dimension( * ) rwork, integer info)

And what does it do? The line says: CSYSVXX computes the solution to system of linear equations A * X = B for SY matrices. SY = symmetric, bestie — not “sister”. 💅

Purpose: ✨

CSYSVXX uses the diagonal pivoting factorization to compute the solution to a complex 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.

If requested, both normwise and maximum componentwise error bounds are returned. CSYSVXX 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 a warning is returned. Relevant condition numbers also are calculated and returned.

CSYSVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous CSYSVXX 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 CSYSVXX would itself produce. Translation: don't blame the routine if you hand it a weird custom factorization, okay? 🫶

Description: 🧚

The following steps are performed:

If FACT = 'E', real 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.

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.

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.
The system of equations is solved for X using the factored form of A.
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.
If equilibration was used, the matrix X is premultiplied by diag(R) so that it solves the original system before equilibration.

Parameters: 💅

FACT — is CHARACTER*1. 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.

UPLO — is CHARACTER*1.

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

N — is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.

NRHS — is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

A — is COMPLEX 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).

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

AF — is COMPLEX 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 CSYTRF.

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.

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

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 CSYTRF. 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 CSYTRF.

EQUED — is CHARACTER*1. 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.

S —

`S`

S is a REAL array, dimension (N) — the scale factors for A, bestie 💅

if EQUED = 'Y', A gets 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', every 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 keep your solution and error estimates reliable 👑
scaling by powers of the radix causes no rounding errors unless the result underflows or overflows, which, iconic behaviour honestly
rounding errors during scaling mean you're refining with a matrix that isn't equivalent to the input matrix, and that produces error estimates that may not be reliable 😭

`B`

B is COMPLEX 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). no notes, she said what she said 🫶

`X`

X is COMPLEX array, dimension (LDX,NRHS) ⭐

If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. Real talk though: A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(S))*X. iykyk 💜

`LDX`

LDX is INTEGER — the leading dimension of the array X. LDX >= max(1,N). simple as 🎀

`RCOND`

RCOND is REAL — 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's zero), the matrix is singular to working precision — red flag, but hear me out: the error may still be small even if this number is very small and the matrix appears ill-conditioned 💀

`RPVGRW`

RPVGRW is REAL — reciprocal pivot growth 🌱

On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. (yes the manpage literally says that, we move 😭)

If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor — which also means your solution X, your estimated condition numbers and your error bounds could all be unreliable, which is giving house of cards 🫠

If factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A.

`BERR`

BERR is REAL 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. queen behaviour 👑

`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 REAL 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 normwise relative error, defined like so:

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 *i*th 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 . (and yeah, the sentence just ends there in the original — the word went missing, we simply vibe with the mystery 💀) 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 is REAL 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, defined as:

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 *i*th 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) * 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 . (same dropped word, same 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. lowkey the easiest way to live ✨

`PARAMS`

PARAMS is REAL array, dimension NPARAMS 💅

Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with the 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: 1.0
       = 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)

⚠️ real one: that Aggressive: 100 option is giving main character energy, but if your factorization isn't Gaussian elimination then err_bnds_norm and err_bnds_comp may no longer be trustworthy. proceed with awareness, bestie 😭

`WORK`

WORK is COMPLEX array, dimension (2*N) 🧚

`RWORK`

RWORK is REAL array, dimension (2*N) 🍓

and next up we've got INFO, who has so much to say about exit codes — stay with me 💜

INFO is INTEGER — the final vibe check, and here’s how to read it, bestie:
- = 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 505 of file csysvxx.f.

DSYSVXX 💜

   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)

Purpose ✨

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.

If requested, both normwise and maximum componentwise error bounds are returned. 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 a warning is returned. Relevant condition numbers also are calculated and returned.

DSYSVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. 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.

Description 🦋

The following steps are performed:

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.

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.

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.
The system of equations is solved for X using the factored form of A.
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.
If equilibration was used, the matrix X is premultiplied by diag(R) so that it solves the original system before equilibration.

Parameters 💅

FACT — CHARACTER*1. 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.
UPLO — CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
A — 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).
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — 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.
LDAF — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — 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. 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.
EQUED — CHARACTER*1. 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.
S

S — DOUBLE PRECISION array, dimension (N). The scale factors for A, bestie. If EQUED = 'Y', then A gets multiplied on the left and the right by diag(S). Whether S is input or output depends on FACT: it's an input argument if FACT = 'F'; otherwise it's an output argument. If FACT = 'F' and EQUED = 'Y', every element of S has to be positive, no exceptions 💅.

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 make sure you get a reliable solution and reliable error estimates. Scaling by powers of the radix doesn't cause rounding errors unless the result underflows or overflows 🫶.

Careful, babe: rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, which produces error estimates that may not be reliable. 😭

B — DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, this is 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 — INTEGER. The leading dimension of the array B. Requirement: LDB >= max(1,N). That's it, that's the whole vibe check 🔥

X — DOUBLE PRECISION array, dimension (LDX,NRHS). If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations 👑. Heads up though: A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(S))*X — so like, don't come for X directly, iykyk 🌸

LDX — INTEGER. The leading dimension of the array X. Requirement: LDX >= max(1,N). Periodt 💅

RCOND — DOUBLE PRECISION. The reciprocal scaled condition number 🦋. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if equilibration was done). If this is less than the machine precision — and especially if it is zero — then the matrix is singular to working precision 💀.

But ngl, the error may still be small even if this number is very very small and the matrix appears ill-conditioned. So don't panic-spiral just because RCOND is tiny 🙈

RPVGRW — DOUBLE PRECISION. Reciprocal pivot growth 🌱. On exit, this holds the reciprocal pivot growth factor norm(A)/norm(U). (The norm is used — the original doc is a lil shy about which norm here, so we're reporting it exactly as written 😭.)

If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor 😬. That also means the solution X, the estimated condition numbers, and the error bounds could all be unreliable.

If the 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. Iconic little number, honestly 🎀

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 info about various error bounds and condition numbers corresponding to the normwise relative error, which is defined like this:

     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 doc trails off right there, ngl 😭). 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, that's a real homework assignment, go read it bestie 💜

ERR_BNDS_COMP — 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, defined like this:

     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 (which is what the componentwise relative error depends on), and by 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 (again, doc just stops mid-sentence, we don't make the rules 😭). 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.

Same deal: 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. Easy 🍓

PARAMS — DOUBLE PRECISION array, dimension (NPARAMS). Specifies algorithm parameters. If an entry is < 0.0, then that entry gets filled with the 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.

But fr fr, 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). Just vibes and scratch space 🧚

IWORK — INTEGER array, dimension (N). Integer scratch space, matching energy 💫

INFO —

INFO is INTEGER 💅 — okay so this is the receipts section, bestie, and you need to read it like your group chat after a dramatic exam:
- = 0: Successful exit. The solution to every right-hand side is guaranteed. slay. ✨
- < 0: If INFO = -i, the i-th argument had an illegal value. oop. 💀
- > 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. it's giving singular matrix energy. 😭
- = 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. literally the fine print but make it fashion. 🎀

👑 Author

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

📍 Definition

Definition at line 501 of file dsysvxx.f.

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

💅 `SSYSVXX`

✨ Purpose

SSYSVXX uses the diagonal pivoting factorization to compute the solution to a real 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. main character solver behavior, honestly. 🌈

If requested, both normwise and maximum componentwise error bounds are returned. SSYSVXX 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 a warning is returned. Relevant condition numbers also are calculated and returned. we love a routine that tells you when things are sketchy. 🫶

SSYSVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous SSYSVXX 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 SSYSVXX would itself produce. no cap: custom inputs are on you. ⚠️

🧚 Description

The following steps are performed:

If FACT = 'E', real 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. scaling era. 🧚

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. block diagonal? iconic structure, honestly. 💜

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. singular D is a whole situation. 💀
The system of equations is solved for X using the factored form of A. solving, periodt. ✨
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. iterative refinement ate, lowkey. 🔥
If equilibration was used, the matrix X is premultiplied by diag(R) so that it solves the original system before equilibration. putting it back, bestie. 🎀

💖 Parameters

FACT — FACT is CHARACTER*1. 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. frozen. untouched. we respect it. 🫶
- = 'N': The matrix A will be copied to AF and factored. copy + factor combo. ✨
- = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored. equilibrate first, then serve. 💅
UPLO — UPLO is CHARACTER*1.
- = 'U': Upper triangle of A is stored; upper triangle supremacy. 🦋
- = 'L': Lower triangle of A is stored. lower triangle gang. 💜
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. zero is allowed, bestie. 🍓
NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. more RHS, more columns, more vibes. 🌸
A — A is REAL 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. only one triangle gets read. the other is not invited. 🙈 On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S). overwritten, so don't act surprised. 😭
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). respect the leading dimension, fr fr. 🫶
AF — AF is REAL 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 SSYTRF. precomputed factor? we love a shortcut. ✨ 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. output mode: AF gets the goods. 🎀
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N). another leading dimension check, bestie. 💜
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 SSYTRF. 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. pivot lore. read it slowly, it's giving lore. 🔥 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 SSYTRF. same details, but it's output era now. 💅
EQUED — EQUED is CHARACTER*1. Specifies the form of equilibration that was done.
- = 'N': No equilibration (always true if FACT = 'N'). no scaling, no drama. 🍓
- = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). row AND column? balanced. iconic. 🌈 EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument. depends on FACT, bestie. 🎀
S — S is REAL 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, this is the caution paragraph. read it carefully. ⚠️
B —

B — this one's a REAL array, dimension (LDB,NRHS) 🫶 On entry, it's your N-by-NRHS right hand side matrix B. On exit? Depends on the vibe:

if EQUED = 'N', B is literally not modified, untouched, left alone 💅
if EQUED = 'Y', B gets overwritten by diag(S)*B (yes, your input becomes a whole different matrix, don't be shook)

LDB — an INTEGER, bestie. It's the leading dimension of the array B, and it better be LDB >= max(1,N). No exceptions, we're not doing that today.

X — a REAL array, dimension (LDX,NRHS) ✨ If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. Big note though: A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(S))*X. So like, the X you get back needs that little extra step to actually be the answer, iykyk 🦋

LDX — an INTEGER. Leading dimension of the array X, and it's LDX >= max(1,N). Same energy as LDB, same non-negotiable gatekeeping 👑

RCOND — a REAL, and it's the reciprocal scaled condition number. Specifically it's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if that was done). Here's the tea: if this is less than the machine precision (in particular, if it's zero), the matrix is singular to working precision 😭 But — and this is the plot twist — the error may still be small even if this number is very small and the matrix appears ill-conditioned. So don't panic-jump to conclusions.

RPVGRW — a REAL, the reciprocal pivot growth 🧚 On exit, it 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 — it's giving unstable, and not in a fun way. That also means your solution X, your estimated condition numbers, and your error bounds could all be unreliable. And if the factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A. Noted. Filed. Remembered. 📌

BERR — a REAL array, dimension (NRHS), holding the componentwise relative backward error. So for each solution vector X(j), this is the componentwise relative backward error of that vector — i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution. Tiny change, exact solution, that's the whole flex 💀

N_ERR_BNDS — an INTEGER. The number of error bounds to return for each right hand side and each type (normwise or componentwise). Check ERR_BNDS_NORM and ERR_BNDS_COMP below, they're the main characters here ⭐

ERR_BNDS_NORM — a REAL array, dimension (NRHS, N_ERR_BNDS) 🌈 For each right-hand side, this array is packed with info about various error bounds and condition numbers for the normwise relative error, which is defined like this:

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

That's the normwise relative error in the ith solution vector, bestie, right there ^^. 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 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. Only then. Don't be trusting it otherwise 🙈
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.

And yeah, that sentence really does just trail off into nothing in the source — that's not me being lazy, that's just what we were handed 💀

See Lapack Working Note 165 for further details and extra cautions 📚

ERR_BNDS_COMP — a REAL array, dimension (NRHS, N_ERR_BNDS) 🌸 Same deal for each right-hand side, except this one's all about the error bounds and condition numbers corresponding to the componentwise relative error, defined like so:

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

That's the componentwise relative error in the ith solution vector ✨ 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. And here's the important bit: if componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed at all. 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 (yes, same rule, it's a package deal 🫶).
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.

Same dangling sentence situation, same source truncation, we move 💀

See Lapack Working Note 165 for further details and extra cautions 📚 (yes, again — it's that serious)

NPARAMS — an INTEGER. Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. Never referenced. Not even glanced at. 👀

PARAMS — a REAL array, dimension NPARAMS, and this is where you get to be a control freak (affectionate) 💅 It specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters. The lineup:

PARAMS(LA_LINRX_ITREF_I = 1) — whether to perform iterative refinement or not.
- Default: 1.0
- = 0.0: no refinement is performed, and no error bounds are computed. Straight up no notes.
- = 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 — so like, don't invent your own 🎀)
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. ⚠️ But 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, aggressive mode = sketchy mode, choose wisely.
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 — a REAL array, dimension (4*N). That's it, that's the whole description, it's workspace, give it 4*N and let it cook 🍓

IWORK — an INTEGER array, dimension (N). Short, sweet, integer-flavored 💜

INFO — okay so INFO is about to get its own whole moment and we are not rushing it, so hold that thought bestie ✨

INFO — okay bestie, this is the INTEGER exit-status tea ☕ and it's lowkey the vibe check for the whole run 👑
- = 0: successful exit. The solution to every right-hand side is guaranteed. slay ✨
- < 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 504 of file ssysvxx.f.

🦋 subroutine `zhesvxx`

   subroutine zhesvxx (character fact, character uplo, integer n, integer nrhs,
       complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
       ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character
       equed, double precision, dimension( * ) s, complex*16, dimension( ldb, *
       ) b, integer ldb, complex*16, 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, complex*16,
       dimension( * ) work, double precision, dimension( * ) rwork, integer
       info)

ZHESVXX computes the solution to system of linear equations A * X = B for HE matrices 💅 periodt.

✨ Purpose

ZHESVXX uses the diagonal pivoting factorization to compute the solution to a complex*16 system of linear equations A * X = B, where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS matrices. 🦋

If requested, both normwise and maximum componentwise error bounds are returned. ZHESVXX 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 a warning is returned. Relevant condition numbers also are calculated and returned. 🔥

ZHESVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous ZHESVXX 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 ZHESVXX would itself produce. 🫶

🌸 Description

The following steps are performed:

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. ✨

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 Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. 💜

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. 🔥
The system of equations is solved for X using the factored form of A. 🎀
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. 🌟
If equilibration was used, the matrix X is premultiplied by diag(R) so that it solves the original system before equilibration. 🦋

🎀 Parameters

FACT — FACT is CHARACTER*1. 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.
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 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.

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

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is COMPLEX*16 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**H or A = L*D*L**H as computed by ZHETRF.

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 ZHETRF. 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 ZHETRF.

EQUED — EQUED is CHARACTER*1. 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.

S
S — S is a DOUBLE PRECISION array, dimension (N). These are the scale factors for A, bestie ✨ If EQUED = 'Y', A gets multiplied on the left and the right by diag(S). S is an input argument when FACT = 'F' — otherwise it's an output argument. And if FACT = 'F' with EQUED = 'Y', every single element of S has to be positive, no exceptions 🫶 If S is output, each element is a power of the radix. If S is input, each element should be a power of the radix so you get a reliable solution and reliable error estimates — scaling by powers of the radix doesn't cause rounding errors unless the result underflows or overflows. Lowkey important: rounding errors during scaling mean you're refining with a matrix that is not equivalent to the input matrix, and that produces error estimates that may not be reliable 😭
B — B is a COMPLEX*16 array, dimension (LDB,NRHS). On entry it's the N-by-NRHS right hand side matrix B. On exit? Depends on the vibe check:
- if EQUED = 'N', B is not modified;
- if EQUED = 'Y', B gets overwritten by diag(S)*B 💅
LDB — LDB is INTEGER. It's the leading dimension of the array B, and it must satisfy LDB >= max(1,N) — the way that math is just mathing 🎀
X — X is a COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. That "original" part matters, so heads up: A and B are both modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(S))*X 👑
LDX — LDX is INTEGER, the leading dimension of the array X. Gotta satisfy LDX >= max(1,N). Same energy as LDB, different letter 🦋
RCOND — RCOND is DOUBLE PRECISION. This is the reciprocal scaled condition number — an estimate of the reciprocal Skeel condition number of matrix A after equilibration (if any was done). If it comes back less than the machine precision (and especially if it's zero), the matrix is singular to working precision ⚠️ But — and this is the plot twist — the error may still be small even when this number is very small and the matrix looks ill-conditioned, so don't panic-spiral just yet 🙈
RPVGRW — RPVGRW is DOUBLE PRECISION, the reciprocal pivot growth. On exit it holds 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 💀 Which also means the solution X, the estimated condition numbers, and the error bounds could all be unreliable. And if the factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A. Rude, but informative.
BERR — BERR is a DOUBLE PRECISION array, dimension (NRHS). It's 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. Cute little minimum, extremely serious 🍓
N_ERR_BNDS — N_ERR_BNDS is INTEGER, the number of error bounds to return for each right hand side and each type (normwise or componentwise). Peep ERR_BNDS_NORM and ERR_BNDS_COMP below 👇
ERR_BNDS_NORM — ERR_BNDS_NORM is a DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side it holds info about various error bounds and condition numbers tied to the normwise relative error, defined like this:

     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 are currently up to three pieces of information returned. The first index in ERR_BNDS_NORM(i,:) corresponds to the *i*th right-hand side. The second index in ERR_BNDS_NORM(:,err) contains these 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 — genuinely, go read it, iykyk 📖

ERR_BNDS_COMP — ERR_BNDS_COMP is a DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). Same deal, but for the componentwise relative error, defined as:

     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 (because the componentwise relative error depends on it) and by the type of error information described below. There are currently up to three pieces of information returned per right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. And 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 *i*th right-hand side. The second index in ERR_BNDS_COMP(:,err) contains:

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 🌈

Again: see Lapack Working Note 165 for further details and extra cautions. Not optional reading, bestie 🧚

NPARAMS — NPARAMS is INTEGER. It specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. Ghosted. Not even looked at 👻
PARAMS — PARAMS is a DOUBLE PRECISION array, dimension NPARAMS. It specifies algorithm parameters. If an entry is < 0.0, that entry gets filled with the 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. But fr fr, 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 a COMPLEX*16 array, dimension (5*N). It works. That's the whole personality 🛠️
RWORK — RWORK is a DOUBLE PRECISION array, dimension (2*N). The real one of the two, literally 💜

and yes, INFO is right there next, we're getting to it — hold that thought 🦋

INFO — INTEGER. Bestie, this is the receipts section for what just happened, and the tea is:
- = 0 — Successful exit. The solution to every right-hand side is guaranteed. Slay. ✨
- < 0 — If INFO = -i, the i-th argument had an illegal value. Not iconic. 💀
- > 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. Big yikes.
- = 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 📌

Definition at line 502 of file zhesvxx.f.

💅 subroutine zsysvxx 💅

   subroutine zsysvxx (character fact, character uplo, integer n, integer nrhs,
       complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
       ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character
       equed, double precision, dimension( * ) s, complex*16, dimension( ldb, *
       ) b, integer ldb, complex*16, 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, complex*16,
       dimension( * ) work, double precision, dimension( * ) rwork, integer
       info)

ZSYSVXX computes the solution to system of linear equations A * X = B for SY matrices.

✨ Purpose ✨

Okay so, ZSYSVXX uses the diagonal pivoting factorization to compute the solution to a complex*16 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. Obsessed with the efficiency. 💖

If requested, both normwise and maximum componentwise error bounds are returned. ZSYSVXX 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 a warning is returned. Relevant condition numbers also are calculated and returned. Numbers? Calculated. Bounds? Returned. We love a thorough queen. 👑

ZSYSVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous ZSYSVXX 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 ZSYSVXX would itself produce. Read that again, bestie — user-provided stuff is not automatically getting the same guarantee. 🫶

🌈 Description 🌈

The following steps are performed:

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.

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. Block diagonal D has main character energy, honestly. ✨

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. It does not give up. It keeps going. Resilient icon behavior. 💅
The system of equations is solved for X using the factored form of A.
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. Double precision residual? She said “do it properly.” 🔥
If equilibration was used, the matrix X is premultiplied by diag(R) so that it solves the original system before equilibration.

🎀 Parameters 🎀

FACT — CHARACTER*1. 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.
UPLO — CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
A — COMPLEX*16 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).

LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — COMPLEX*16 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 ZSYTRF.

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

EQUED — CHARACTER*1. 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.

S —

`S`

S is DOUBLE PRECISION array, dimension (N) — the scale factors for A 💅

If EQUED = 'Y', then A gets multiplied on the left and right by diag(S). S is an input argument when FACT = 'F'; otherwise S is an output argument. And if FACT = 'F' and EQUED = 'Y', every single element of S must be positive, no cap.

If S is an output, each element of S is a power of the radix. If S is an input, each element of S should be a power of the radix so you actually get a reliable solution and reliable error estimates 🫶 Scaling by powers of the radix doesn't cause rounding errors unless the result underflows or overflows.

Okay so this part matters: rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, which produces error estimates that may not be reliable. It's giving house of cards. Be careful. 😭

`B`

B is COMPLEX*16 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)

That's the vibe check, bestie. Respect it. 🎀

`X`

X is COMPLEX*16 array, dimension (LDX,NRHS) 🦋

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

Heads up though — 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. That distinction is literally the whole plot. 💀

`LDX`

LDX is INTEGER — the leading dimension of the array X.

LDX >= max(1,N)

Same energy as LDB, different letter. Slay. 💜

`RCOND`

RCOND is DOUBLE PRECISION — the reciprocal scaled condition number 🔥

It's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if equilibration was even done). If this is less than the machine precision — in particular, if it is zero — then the matrix is singular to working precision. Full stop.

But plot twist: the error may still be small even if this number is very small and the matrix appears ill-conditioned. Looks can be deceiving, iykyk. 🙈

`RPVGRW`

RPVGRW is DOUBLE PRECISION — reciprocal pivot growth 🌸

On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U).

Ngl, the source text here literally just says "the norm is used" — the actual norm name got eaten somewhere in transmission, so that's all we've got. 🫠

If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. That also means the solution X, the estimated condition numbers, and the error bounds could all be unreliable. Rough.

If factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A.

`BERR`

BERR is DOUBLE PRECISION array, dimension (NRHS) — componentwise relative backward error, bestie 💖

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. Tiny little nudge, huge consequences. Iconic.

`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 info about various error bounds and condition numbers corresponding to the normwise relative error, defined like this:

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 *i*th 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 cuts off right there, no shade to it, that's just how it arrived). 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 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, defined like so:

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 (which is what the componentwise relative error depends on) and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side.

Important vibes: if componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. And 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 *i*th 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) * 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 (again, the source text stops mid-sentence here — not me, I promise). 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. 📚 (yes, same note, she's the main character here)

`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. Respect the defaults, they're lowkey serving. 💅

`PARAMS`

PARAMS is DOUBLE PRECISION array, dimension NPARAMS — specifies algorithm parameters 🎀

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

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.

Real talk: 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) — that's her, the scratch space queen 👑

`RWORK`

RWORK is DOUBLE PRECISION array, dimension (2*N) — same energy, double precision edition 💜

`INFO`

INFO is INTEGER — and the source text cuts off right here, mid-entry, because the transmission ran out of road. 🫠 No more details made it through in this chunk — bestie, we'll see what the next part of the page has to say.

INFO is INTEGER — okay bestie, here's the exit-status tea: 🍵✨

= 0: successful exit, and every right-hand side's solution is guaranteed. 💖
< 0: if INFO = -i, that means the i-th argument had an illegal value. 😭
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization finished, but factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. 💀
= N+J: okay so the Jth right-hand side's solution is not guaranteed. Other right-hand sides K with K > J might also be not guaranteed, but only the first such right-hand side gets reported. If a small componentwise error is not requested (PARAMS(3) = 0.0), then the Jth right-hand side is the first one with a normwise error bound that's not guaranteed — literally 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's not guaranteed — the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0. Peep the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1) for the full story. Wanna know about all 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 502 of file zsysvxx.f.

🤖 Author 🤖

Generated automatically by Doxygen for LAPACK from the source code.

And that's the whole hesvxx(3) vibe from LAPACK version 3.12.0 — footer source/version check: iconic, no cap. Bye bestie! 💅✨🫶

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

hesvxx(3)

💖 NAME 💖

✨ SYNOPSIS ✨

🫶 Functions 🫶

🔥 Detailed Description 🔥

👑 Function Documentation 👑

🎀 subroutine chesvxx 🎀

🎯 Purpose 🎯

📝 Description 📝

💅 Parameters 💅

FACT

UPLO

N

NRHS

A

LDA

AF

LDAF

IPIV

EQUED

S

Author ✍️

csysvxx 🦋

S

B

LDB

X

LDX

RCOND

RPVGRW

BERR

N_ERR_BNDS

ERR_BNDS_NORM

ERR_BNDS_COMP

NPARAMS

PARAMS

WORK

RWORK

Author 🫶

DSYSVXX 💜

Purpose ✨

Description 🦋

Parameters 💅

👑 Author

📍 Definition

💅 SSYSVXX

✨ Purpose

🧚 Description

💖 Parameters

🎀 Author

🦋 subroutine zhesvxx

✨ Purpose

🌸 Description

🎀 Parameters

🫶 Author 🫶

📌 Definition at line 📌

💅 subroutine zsysvxx 💅

✨ Purpose ✨

🌈 Description 🌈

🎀 Parameters 🎀

S

B

LDB

X

LDX

RCOND

RPVGRW

BERR

N_ERR_BNDS

ERR_BNDS_NORM

ERR_BNDS_COMP

NPARAMS

PARAMS

WORK

RWORK

INFO

👑 Author 👑

🤖 Author 🤖

`FACT`

`UPLO`

`N`

`NRHS`

`A`

`LDA`

`AF`

`LDAF`

`IPIV`

`EQUED`

`S`

`csysvxx` 🦋

`S`

`B`

`LDB`

`X`

`LDX`

`RCOND`

`RPVGRW`

`BERR`

`N_ERR_BNDS`

`ERR_BNDS_NORM`

`ERR_BNDS_COMP`

`NPARAMS`

`PARAMS`

`WORK`

`RWORK`

💅 `SSYSVXX`

🦋 subroutine `zhesvxx`

`S`

`B`

`LDB`

`X`

`LDX`

`RCOND`

`RPVGRW`

`BERR`

`N_ERR_BNDS`

`ERR_BNDS_NORM`

`ERR_BNDS_COMP`

`NPARAMS`

`PARAMS`

`WORK`

`RWORK`

`INFO`