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

🦋 `sgbsvxx.f(3)` — Library Functions Manual 🦋

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sgbsvxx.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

       subroutine SGBSVXX (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
           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)

SGBSVXX computes the solution to system of linear equations A * X = B for GB matrices. 💅

💖 Function/Subroutine Documentation 💖

   subroutine SGBSVXX (character fact, character trans, integer n, integer kl,
       integer ku, integer nrhs, real, dimension( ldab, * ) ab, integer ldab,
       real, dimension( ldafb, * ) afb, integer ldafb, 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)

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

🫶 Purpose

Okay so bestie, here’s the tea: SGBSVXX 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. SGBSVXX 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. 🫶

SGBSVXX 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 SGBSVXX 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 SGBSVXX would itself produce. That’s the fine print, no cap. 💀

🔥 Description

Here’s the step-by-step glow-up:

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').

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.

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.
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(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. 💅

🎀 Parameters

Buckle up, bestie — here’s every parameter, no skips. 💖

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 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.

TRANS

TRANS is CHARACTER*1. It specifies the form of the system of equations:

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

N

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

KL

KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.

KU

KU is INTEGER. The number of superdiagonals within the band of A. KU >= 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.

AB

AB is REAL array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:

AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

If FACT = 'F' and EQUED is not 'N', then AB must have been equilibrated by the scaling factors in R and/or C. AB 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).

LDAB

LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.

AFB

AFB is REAL array, dimension (LDAFB,N). If FACT = 'F', then AFB is an input argument and on entry contains details of the LU factorization of the band matrix A, as computed by SGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB 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).

LDAFB

LDAFB is INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.

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.

EQUED

EQUED is CHARACTER*1. 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).

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

R

💖 `R`

R is a REAL array, dimension (N).

These are the row scale factors for A, bestie. If EQUED = 'R' or 'B', then A gets multiplied on the left by diag(R). If EQUED = 'N' or 'C', R isn't even looked at, ok? Not accessed, we simply don't go there.

So which team is R on?

R is an input argument if FACT = 'F'.
Otherwise, R is an output argument.

And if FACT = 'F' and EQUED = 'R' or 'B', then every single element of R must be positive — non-negotiable, no cap.

More deets:

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 tho? Because scaling by powers of the radix doesn't cause rounding errors unless the result underflows or overflows. But rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, and that produces error estimates that may not be reliable. Lowkey a whole situation. 😭

🌸 `C`

C is a REAL array, dimension (N).

Now for the column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C). If EQUED = 'N' or 'R', C is not accessed, periodt.

Same energy as R:

C is an input argument if FACT = 'F'.
Otherwise, C is an output argument.
If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive.

And the same radix rules, because consistency is iconic:

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.

Same warning as above, fr fr: scaling by powers of the radix doesn't cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. 🫠

🦋 `B`

B is a REAL array, dimension (LDB,NRHS).

On entry: it's the N-by-NRHS right hand side matrix B. Simple enough.

On exit though, it depends:

if EQUED = 'N', B is not modified — untouched, unbothered, thriving;
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.

🎀 `LDB`

LDB is INTEGER.

It's the leading dimension of the array B. The vibe check it has to pass:

LDB >= max(1,N)

⭐ `X`

X is a REAL array, dimension (LDX,NRHS).

If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. The moment we've all been waiting for. 👑

⚠️ But heads up: note that 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'.

🌈 `LDX`

LDX is INTEGER.

The leading dimension of the array X, and yeah it's the same deal:

LDX >= max(1,N)

💀 `RCOND`

RCOND is REAL.

It's the reciprocal scaled condition number — an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done, like, if you bothered).

🚨 If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. That's the bad place.

But wait, plot twist: note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. So don't spiral just because RCOND is giving tiny numbers. 🙈

🔥 `RPVGRW`

RPVGRW is REAL.

Reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor:

norm(A)/norm(U)

(The page literally just says "The norm is used" and leaves it at that — I'm not making up which one, ok, I'm just reporting what it says. 💅)

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

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

And in SGESVX, this quantity is returned in WORK(1). iykyk.

🍓 `BERR`

BERR is a REAL array, dimension (NRHS).

Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) — meaning the smallest relative change in any element of A or B that makes X(j) an exact solution. Little perfectionist behaviour, we stan. ✨

🧚 `N_ERR_BNDS`

N_ERR_BNDS is INTEGER.

It's 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, I gotchu. 👇

💜 `ERR_BNDS_NORM`

ERR_BNDS_NORM is a 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 as follows:

Normwise relative error in the *i*th 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... (the page literally cuts off there — some word is missing in the source, not me being lazy, 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, 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 a 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, which is defined as follows:

Componentwise relative error in the *i*th 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... (again, the source just stops 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*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 that important.)

🎀 `NPARAMS`

NPARAMS is INTEGER.

It specifies the number of parameters set in PARAMS. And if it's <= 0, the PARAMS array is never referenced and default values are used. Respect the boundaries, honestly.

🌸 `PARAMS`

Okay so the page cuts off right here mid-argument — PARAMS gets a heading and then... nothing. The source literally ends on this line, so the rest of this one is coming at you in the next part, bestie. Hold that thought. 💖

PARAMS is a REAL array, dimension NPARAMS. It's where you tell the routine how to behave — algorithm parameters, bestie, the whole vibe check of how the refinement goes down ✨ If any entry is < 0.0, that slot gets filled in with the default value used for that parameter. Only positions up to NPARAMS are actually accessed; anything higher-numbered just gets defaults anyway. 🙈
PARAMS(LA_LINRX_ITREF_I = 1) — whether to do iterative refinement or not, that's the whole question 💅
- Default: 1.0
- = 0.0: no refinement is performed, and no error bounds are computed. Straight up skipping the glow-up 😭
- = 1.0: use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION. Resourceful queen behavior 👑
- (other values are reserved for future use — so like, don't, iykyk)
PARAMS(LA_LINRX_ITHRESH_I = 2) — maximum number of residual computations allowed for refinement, aka the patience meter 🎀
- Default: 10
- Aggressive: set to 100 to permit convergence using approximate factorizations or factorizations other than LU. But heads up 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. Not me trusting the math less after that 💀
PARAMS(LA_LINRX_CWISE_I = 3) — a 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. Simple as that, no cap 🌸
- Default: 1.0 (attempt componentwise convergence — she's trying her best and honestly? iconic)

WORK

WORK is a REAL array, dimension (4*N). Just give her the space, she'll handle it 🫶

IWORK

IWORK is an INTEGER array, dimension (N). Integers only in this house, no floaters allowed 🧚

INFO

INFO is an INTEGER, and honestly it's the receipts of the whole operation. Read it, bestie, always read it 🔥
- = 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. Somebody passed something cursed 😭
- > 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 "we did the work but the answer is 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. Like, the whole tea is in those two arrays, go look 🍓

Author

Cred for this one goes to:

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

Definition at line 558 of file sgbsvxx.f. Yes, that far down. She's a long one 💅

Author

Generated automatically by Doxygen for LAPACK from the source code. The robots wrote the docs, we just made them cute 🎀

And that's the whole page, bestie — this was LAPACK Version 3.12.0, page /home/abui.../sgbsvxx.f(3), auto-generated from the source code and lovingly retold. Go forth and pass INFO checks, mwah 🫶💖

📦 where this page comes from

opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

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

🦋 sgbsvxx.f(3) — Library Functions Manual 🦋

🌸 NAME 🌸

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

💖 Function/Subroutine Documentation 💖

🫶 Purpose

🔥 Description

🎀 Parameters

FACT

TRANS

N

KL

KU

NRHS

AB

LDAB

AFB

LDAFB

IPIV

EQUED

R

💖 R

🌸 C

🦋 B

🎀 LDB

⭐ X

🌈 LDX

💀 RCOND

🔥 RPVGRW

🍓 BERR

🧚 N_ERR_BNDS

💜 ERR_BNDS_NORM

💅 ERR_BNDS_COMP

🎀 NPARAMS

🌸 PARAMS

WORK

IWORK

INFO

Author

Author

🦋 `sgbsvxx.f(3)` — Library Functions Manual 🦋

💖 `R`

🌸 `C`

🦋 `B`

🎀 `LDB`

⭐ `X`

🌈 `LDX`

💀 `RCOND`

🔥 `RPVGRW`

🍓 `BERR`

🧚 `N_ERR_BNDS`

💜 `ERR_BNDS_NORM`

💅 `ERR_BNDS_COMP`

🎀 `NPARAMS`

🌸 `PARAMS`