SRC/sgesvxx.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💖 NAME

okay so bestie, welcome to SRC/sgesvxx.f — that's the file, that's the whole name section, and it lives in the Library Functions Manual, section 3. It's a Fortran thing. It's iconic. Let's get into it. 🫶

✨ SYNOPSIS

🌸 Functions/Subroutines

  • subroutine sgesvxx (fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info) — SGESVXX computes the solution to system of linear equations A * X = B for GE matrices

🦋 Function/Subroutine Documentation

okay so the full signature, in all its glory — look at this absolute unit of an argument list, no cap:

subroutine sgesvxx (character fact, character trans, integer n, integer
    nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, *
    ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real,
    dimension( * ) r, real, dimension( * ) c, 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)

SGESVXX computes the solution to system of linear equations A * X = B for GE matrices 💅

🌟 Purpose:

!> !> SGESVXX uses the LU factorization to compute the solution to a !> real system of linear equations A * X = B, where A is an !> N-by-N matrix and X and B are N-by-NRHS matrices. !> !> If requested, both normwise and maximum componentwise error bounds !> are returned. SGESVXX 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. !> !> SGESVXX 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 !> SGESVXX 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 SGESVXX would itself produce. !>

so like, the vibe is: it does LU factorization, it gives you a solution with a tiny guaranteed error — we're talking O(eps), where eps is the working machine precision — unless your matrix is very ill-conditioned, and then it just hands you a warning instead, bestie. 😭 Both normwise and maximum componentwise error bounds, if you ask for them. And the relevant condition numbers come back too, which is lowkey generous of it.

It also accepts user-provided factorizations and equilibration factors — see the FACT and EQUED definitions below. Refining a solution, or reusing a factorization from a previous SGESVXX call, will also give you O(eps) errors or warnings — but ⚠️ warning time: we cannot make that claim for general user-provided factorizations and equilibration factors if they differ from what SGESVXX would itself produce.

💫 Description:

!> !> The following steps are performed: !> !> 1. If FACT = 'E', real scaling factors are computed to equilibrate !> the system: !> !> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B !> TRANS = 'T': (diag(R)*A*diag(C))T *inv(diag(R))*X = diag(C)*B !> TRANS = 'C': (diag(R)*A*diag(C))H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') !> or diag(C)*B (if TRANS = 'T' or 'C'). !> !> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor !> the matrix A (after equilibration if FACT = 'E') as !> !> A = P * L * U, !> !> where P is a permutation matrix, L is a unit lower triangular !> matrix, and U is upper triangular. !> !> 3. If some U(i,i)=0, so that U 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. !> !> 4. The system of equations is solved for X using the factored form !> of A. !> !> 5. 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. !> !> 6. If equilibration was used, the matrix X is premultiplied by !> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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. !>

✨ so here's the little six-step choreography it runs through, bestie:

  1. If FACT = 'E', real scaling factors get computed to equilibrate the system:
TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B

Whether the system actually gets equilibrated depends on the scaling of the matrix A — vibe check dependent. But if equilibration is used, A is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C'). She rewrites things, that's just her process. 💅

  1. If FACT = 'N' or 'E', the LU decomposition factors A (after equilibration, if FACT = 'E') as
A = P * L * U,

where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular. the holy trinity, honestly. 🔥

  1. If some U(i,i)=0, so U is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of A (see argument RCOND). And if the reciprocal of the condition number is less than machine precision, the routine still goes on to solve for X and compute error bounds, as described below. she doesn't give up. main character energy.
  2. The system of equations is solved for X using the factored form of A. ✅
  3. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), the routine uses iterative refinement to try to get a small error and error bounds. Refinement calculates the residual to at least twice the working precision, which, slay.
  4. If equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so that it solves the original system from before equilibration. putting the outfit back on after the fitting, iykyk. 🎀
also!! some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. And if the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument. efficiency queen. 👑

🍓 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 R and C. !> 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. !>

FACT is CHARACTER*1 and it decides whether you're handing over the factored form of A on entry — and if not, whether A should get equilibrated before it's 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 R and C. And 🫶 A, AF, and IPIV are not modified — she's respectful like that.
  • = '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.

TRANS

!> TRANS is CHARACTER*1 !> Specifies the form of the system of equations: !> = 'N': A * X = B (No transpose) !> = 'T': AT * X = B (Transpose) !> = 'C': AH * X = B (Conjugate Transpose = Transpose) !>

TRANS is CHARACTER*1 and it says which form the system of equations is in:

  • = 'N': A * X = B (No transpose)
  • = 'T': A**T * X = B (Transpose)
  • = 'C': A**H * X = B (Conjugate Transpose = Transpose)

(and yeah, for real matrices 'C' and 'T' are literally the same thing, she's just being thorough 💀)

N

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

N is INTEGER — the number of linear equations, aka the order of the matrix A. Constraint: N >= 0. no negatives, bestie, she's not built for that. 🙈

NRHS

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

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

A

!> A is REAL array, dimension (LDA,N) !> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is !> not 'N', then A must have been equilibrated by the scaling !> factors in R and/or C. A is not modified if FACT = 'F' or !> 'N', or if FACT = 'E' and EQUED = 'N' on exit. !> !> On exit, if EQUED .ne. 'N', A is scaled as follows: !> EQUED = 'R': A := diag(R) * A !> EQUED = 'C': A := A * diag(C) !> EQUED = 'B': A := diag(R) * A * diag(C). !>

A is a REAL array, dimension (LDA,N). On entry: the N-by-N matrix A. If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. And A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled like so:

EQUED = 'R':  A := diag(R) * A
EQUED = 'C':  A := A * diag(C)
EQUED = 'B':  A := diag(R) * A * diag(C).

she changed clothes, but it's the same girl. 💜

LDA

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

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

AF

!> AF is REAL array, dimension (LDAF,N) !> If FACT = 'F', then AF is an input argument and on entry !> contains the factors L and U from the factorization !> A = P*L*U as computed by SGETRF. If EQUED .ne. 'N', then !> AF is the factored form of the equilibrated matrix A. !> !> If FACT = 'N', then AF is an output argument and on exit !> returns the factors L and U from the factorization A = P*L*U !> of the original matrix A. !> !> If FACT = 'E', then AF is an output argument and on exit !> returns the factors L and U from the factorization A = P*L*U !> of the equilibrated matrix A (see the description of A for !> the form of the equilibrated matrix). !>

AF is a REAL array, dimension (LDAF,N), and honestly its whole personality depends on FACT:

  • If FACT = 'F', then AF is an input argument, and on entry it contains the factors L and U from the factorization A = P*L*U as computed by SGETRF. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A.
  • If FACT = 'N', then AF is an output argument, and on exit it returns the factors L and U from the factorization A = P*L*U of the original matrix A.
  • If FACT = 'E', then AF is an output argument, and on exit it returns the factors L and U from the factorization A = P*L*U of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).

LDAF

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

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

IPIV

!> IPIV is INTEGER array, dimension (N) !> If FACT = 'F', then IPIV is an input argument and on entry !> contains the pivot indices from the factorization A = P*L*U !> as computed by SGETRF; row i of the matrix was interchanged !> with row IPIV(i). !> !> If FACT = 'N', then IPIV is an output argument and on exit !> contains the pivot indices from the factorization A = P*L*U !> of the original matrix A. !> !> If FACT = 'E', then IPIV is an output argument and on exit !> contains the pivot indices from the factorization A = P*L*U !> of the equilibrated matrix A. !>

IPIV is an INTEGER array, dimension (N) — the pivot index bestie:

  • If FACT = 'F', then IPIV is an input argument, and on entry it contains the pivot indices from the factorization A = P*L*U as computed by SGETRF; row i of the matrix was interchanged with row IPIV(i).
  • If FACT = 'N', then IPIV is an output argument, and on exit it contains the pivot indices from the factorization A = P*L*U of the original matrix A.
  • If FACT = 'E', then IPIV is an output argument, and on exit it contains the pivot indices from the factorization A = P*L*U of the equilibrated matrix A.

EQUED

!> EQUED is CHARACTER*1 !> Specifies the form of equilibration that was done. !> = 'N': No equilibration (always true if FACT = 'N'). !> = 'R': Row equilibration, i.e., A has been premultiplied by !> diag(R). !> = 'C': Column equilibration, i.e., A has been postmultiplied !> by diag(C). !> = 'B': Both row and column equilibration, i.e., A has been !> replaced by diag(R) * A * diag(C). !> EQUED is an input argument if FACT = 'F'; otherwise, it is an !> output argument. !>

EQUED is CHARACTER*1 and it specifies the form of equilibration that was done:

  • = 'N': no equilibration (always true if FACT = 'N').
  • = 'R': row equilibration, i.e. A has been premultiplied by diag(R).
  • = 'C': column equilibration, i.e. A has been postmultiplied by diag(C).
  • = 'B': both row and column equilibration, i.e. A has been replaced by diag(R) * A * diag(C). full glow-up ✨

EQUED is an input argument if FACT = 'F'; otherwise, it's an output argument.

R

!> R is REAL array, dimension (N) !> The row scale factors for A. If EQUED = 'R' or 'B', A is !> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R !> is not accessed. R is an input argument if FACT = 'F'; !> otherwise, R is an output argument. If FACT = 'F' and !> EQUED = 'R' or 'B', each element of R must be positive. !> If R is output, each element of R is a power of the radix. !> If R is input, each element of R 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. !>

R is a REAL array, dimension (N) — the row scale factors for A. If EQUED = 'R' or 'B', A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not accessed (she's off duty). R is an input argument if FACT = 'F'; otherwise, R is an output argument. And if FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive.

If R is output, each element of R is a power of the radix. If R is input, each element of R should be a power of the radix to ensure a reliable solution and error estimates. Why? because scaling by powers of the radix doesn't cause rounding errors unless the result underflows or overflows. 🫠 And rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix — producing error estimates that may not be reliable. so like, be careful bestie, that's a real one.

C

<!-- continued in part 2 -->

  • C — REAL array, dimension (N). These are the column scale factors for A, bestie. 💖 If EQUED = 'C' or 'B', then A gets multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is literally not even accessed, she's off duty.

C is an input argument if FACT = 'F'; otherwise C is an output argument. And if FACT = 'F' together with EQUED = 'C' or 'B', every single element of C must be positive — no cap.

If C is output, each element of C is a power of the radix. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. 😭 so like, respect the radix thing.

  • B — REAL 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 TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B;
    • if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.

She gets a whole glow-up on exit, okay? 🦋

  • LDB — INTEGER. The leading dimension of the array B. Gotta be LDB >= max(1,N). ⭐
  • X — REAL 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(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. So like, don't be shocked when your inputs aren't the same anymore fr fr. 🫶

  • LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,N). periodt. 💅
  • RCOND — 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 is 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 panic immediately.
  • RPVGRW — REAL. Reciprocal pivot growth. 😤 On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. If factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A. In SGESVX, this quantity is returned in WORK(1). iykyk 💀
  • BERR — 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). She's doing the most, and we love her for it.
  • 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. 👇
  • ERR_BNDS_NORM — REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows: 🧚

Normwise relative error in the ith solution vector:

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

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

  • err = 1 — boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). ✅
  • err = 2 — error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. 🙈
  • err = 3 — Reciprocal condition number: estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is . These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.

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 information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows: ✨

Componentwise relative error in the ith solution vector:

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

The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. Important vibe check: 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) * slamch('Epsilon'). ✅
  • err = 2 — error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. 🙈
  • err = 3 — Reciprocal condition number: estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is . These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

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. iconic lazy-girl behaviour honestly. 💅
  • PARAMS — 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. So like, you don't have to fill in everything, she'll handle it.
    • 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. 😬 like, proceed but with awareness.
    • 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 — and this is exactly where this part runs out of room, bestie — WORK is up next, she just hasn't had her turn to speak yet 😭✨
  • WORK — a REAL array, dimension (4*N) 💖 that’s the workspace, bestie, and it is literally 4*N, no cap.
  • IWORK — an INTEGER array, dimension (N) ✨ tiny but important.
  • INFO — INTEGER, and this is the main-character status report 👑:
    • = 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 538 of file sgesvxx.f. 📌

Okay so this page? Generated automatically by Doxygen for LAPACK from the source code, bestie — LAPACK version 3.12.0, from SRC/sgesvxx.f(3). The footer doesn’t list a date, so I’m not inventing one, periodt. 💅✨

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