realGBsolve(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

💖 realGBsolve(3) — the LAPACK band-solve girlies 💖

okay so bestie, welcome to the realGBsolve page — this is the LAPACK manual, and it's literally the whole squad of real solve driver functions for GB matrices. GB = general band. we love a narrow matrix. let's get into it 🦋

✨ NAME ✨

realGBsolve — real

like. that's it. that's the name. short, sweet, iconic 💅

✨ SYNOPSIS ✨

🌸 Functions 🌸

  • subroutine sgbsv (N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO) SGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)
  • subroutine sgbsvx (FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO) SGBSVX computes the solution to system of linear equations A * X = B for GB matrices
  • 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

✨ Detailed Description ✨

This is the group of real solve driver functions for GB matrices 💖

that's the whole vibe. three of them. all solving banded systems. all real. all slaying.

✨ Function Documentation ✨

🌈 subroutine sgbsv 🌈

subroutine sgbsv (integer N, integer KL, integer KU, integer NRHS, real,
    dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV,
    real, dimension( ldb, * ) B, integer LDB, integer INFO)

SGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)

🎀 Purpose:

SGBSV computes the solution to a real system of linear equations

A * X = B,

where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as

A = L * U,

where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals. The factored form of A is then used to solve the system of equations A * X = B.

so like. it factors, then it solves. no cap, that's the whole routine 💅

🦋 Parameters 🦋

  • 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 matrix B. NRHS >= 0.

  • AB

AB is REAL array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows:

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

On exit, details of the factorization: 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. See below for further details.

  • LDAB

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

  • IPIV

IPIV is INTEGER array, dimension (N) The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i).

  • B

B is REAL array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.

  • LDB

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

  • INFO

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed.
not me getting INFO > 0 😭 that means U is exactly singular and bestie, no solution was computed. it's giving singular. it's giving empty-handed. 💀

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

⭐ Further Details:

The band storage scheme is illustrated by the following example, when M = N = 6, KL = 2, KU = 1:

On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u14  u25  u36
    *    *    +    +    +    +       *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U because of fill-in resulting from the row interchanges.

like, the * girlies are just there for decoration, they're not being used, they're the audience 💅 and the + ones are empty seats that get filled by U's dramatic entrance (fill-in from row interchanges, iykyk).

🌈 subroutine sgbsvx 🌈

subroutine sgbsvx (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, dimension( * ) FERR, real, dimension( * )
    BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer
    INFO)

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

🎀 Purpose:

SGBSVX uses the LU factorization to compute the solution to a real system of linear equations

A * X = B, A**T * X = B, or A**H * X = B,

where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided.

okay so this one is the extra one. it solves AND it gives you receipts — error bounds and a condition estimate. she's thorough. she's giving main character energy ✨

🦋 Description:

The following steps are performed by this subroutine:

  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').

  1. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = L * U,

where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals.

  1. 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. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
  2. The system of equations is solved for X using the factored form of A.
  3. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
  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 before equilibration.
bestie. six whole steps. she's doing the MOST and honestly? we're obsessed. note that INFO = N+1 is a warning, not a fail — she keeps going, she just wanted you to know. 🫶

🦋 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, AFB 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. AB, AFB, and IPIV are not modified.

= 'N': The matrix A will be copied to AFB and factored.

= 'E': The matrix A will be equilibrated if necessary, then copied to AFB and factored.

  • TRANS

TRANS is CHARACTER*1 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 (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 A 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

…and that's where part 1 dips out, bestie — AFB keeps going in part 2 🌸

AFB

AFB</p> is a REAL array, dimension (LDAFB,N) — it's the band-matrix bestie holding all the factorization lore. And what it means depends on FACT, so let's vibe check each case:

  • If FACT = 'F', then AFB is an input argument and on entry it's carrying details of the LU factorization of the band matrix A, exactly as computed by SGBTRF. U gets 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 live in rows KL+KU+2 to 2*KL+KU+1. And if EQUED .ne. 'N', then AFB is the factored form of the equilibrated matrix A.
  • If FACT = 'N', then AFB is an output argument and on exit it returns details of the LU factorization of A. 🫶
  • If FACT = 'E', then AFB is an output argument and on exit it returns details of the LU factorization of the equilibrated matrix A (go peek at the description of AB for what that equilibrated matrix looks like).

LDAFB

LDAFB is INTEGER. It's the leading dimension of the array AFB, and the rule is LDAFB >= 2*KL+KU+1. Respect it, bestie. 💅

IPIV

IPIV is an INTEGER array, dimension (N). Pivot indices, depending on FACT:

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

EQUED

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).

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

R

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', then R is not accessed (she's off duty, don't touch her 🛑). R is an input argument if FACT = 'F'; otherwise, R is an output argument. And heads up: if FACT = 'F' and EQUED = 'R' or 'B', every single element of R must be positive.

C

C is a REAL array, dimension (N) — 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', then C is not accessed. C is an input argument if FACT = 'F'; otherwise, C is an output argument. Same energy as R: if FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive.

B

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

On entry, it's the right hand side matrix B. On exit, it depends:

  • 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.

LDB

LDB is INTEGER. It's the leading dimension of the array B, and LDB >= max(1,N). 👑

X

X is a REAL array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, 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(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. Leading dimension of the array X, and LDX >= max(1,N).

RCOND

RCOND is REAL. It's the estimate of the reciprocal condition number of the matrix A after equilibration (if any equilibration happened). Here's the tea: if RCOND is less than the machine precision (in particular, if RCOND = 0), then the matrix is singular to working precision. 😭 That situation gets flagged by a return code of INFO > 0.

FERR

FERR is a REAL array, dimension (NRHS). It's the estimated forward error bound for each solution vector X(j) (that's the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), then FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). This estimate is as reliable as the estimate for RCOND, and it is almost always a slight overestimate of the true error. (She's cautious, not messy. 🎀)

BERR

BERR is a REAL 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.

WORK

WORK is a REAL array, dimension (3*N).

On exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If WORK(1) 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 condition estimator RCOND, and the forward error bound FERR could all be unreliable. 💀 And if the factorization fails with 0<INFO<=N, then WORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A.

IWORK

IWORK is an INTEGER array, dimension (N).

INFO

INFO is INTEGER and it's the receipts:

  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, and i is
    • <= N: U(i,i) 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+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the

Author

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

subroutine sgbsvxx

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 ✨

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. (So like, don't hand her a random factorization and expect the same guarantees, bestie. 🙈)

📖 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').

  1. 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.

  1. 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.
  2. The system of equations is solved for X using the factored form of A.
  3. By default — like, unless you set PARAMS(LA_LINRX_ITREF_I) to zero, bestie — the routine is gonna do iterative refinement to chase down a tiny error and error bounds. Refinement computes the residual to at least twice the working precision. It's giving overachiever. ✨
  4. If equilibration happened, the matrix X gets premultiplied by diag(C) (when TRANS = 'N') or by diag(R) (when TRANS = 'T' or 'C'), so that it solves the original system before all that equilibration mess. 💅

Some optional parameters are bundled up in the PARAMS array — think of it like a little goodie bag 🎀. These settings decide how refinement gets performed, but honestly? The defaults are usually fine. If the defaults are acceptable to you, just pass NPARAMS = 0 and that stops the source code from ever touching the PARAMS argument. Iconic, low-effort, we love. 👑

🎀 Parameters 🎀

  • FACT

FACT is CHARACTER*1

This one says whether the factored form of matrix A is being supplied on entry, and if it isn't, whether A should be equilibrated before it gets factored. The options are:

  • = 'F': On entry, AF and IPIV contain the factored form of A. If EQUED is not 'N', then the matrix A has been equilibrated with the 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

Specifies the form of the system of equations, i.e. which flavour of A we're solving with:

  • = '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, a.k.a. the order of the matrix A. N >= 0. No negative girlies allowed. 🚫

  • 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 — that's 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 like this:

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 it contains the 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 (peep the description of A for what the equilibrated matrix looks like).

  • 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 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'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 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. 😬

  • C

C is REAL array, dimension (N)

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. 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.

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 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. (Yes, same energy as R. They're twins. 👯)

  • B

B is REAL 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 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

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

  • X

X is 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'.

  • LDX

LDX is INTEGER

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

  • 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 it happened). If this is less than the machine precision (and especially if it's zero), the matrix is singular to working precision. But like — note that the error may still be small even if this number is teeny tiny and the matrix looks ill-conditioned. So don't panic-freak. 🫶

  • RPVGRW

RPVGRW is REAL

Reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is what's used here. 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 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). 👀

  • 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. Slay, precision queen. ✨

  • 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 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.

Okay so the first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side, periodt. 💖

The second index in ERR_BNDS_NORM(:,err) contains the following three fields, and bestie they are so important:

  • err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — 'Guaranteed' 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 'guaranteed'. 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 information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

                Componentwise relative error in the ith solution vector:
                               abs(XTRUE(j,i) - X(j,i))
                        max_j ----------------------
                                    abs(X(j,i))

The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

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

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

  • err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — 'Guaranteed' 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 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions. 📝

🎀 NPARAMS 🎀

NPARAMS is INTEGER. Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. ✨

🎀 PARAMS 🎀

PARAMS is REAL array, dimension NPARAMS. Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters.

  • PARAMS(LA_LINRX_ITREF_I = 1) — Whether to perform iterative refinement or not.
    • Default: 1.0
    • = 0.0: No refinement is performed, and no error bounds are computed.
    • = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
    • (other values are reserved for future use)
  • PARAMS(LA_LINRX_ITHRESH_I = 2) — Maximum number of residual computations allowed for refinement.
    • Default: 10
    • Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy.
  • PARAMS(LA_LINRX_CWISE_I = 3) — Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
    • Default: 1.0 (attempt componentwise convergence)

💻 WORK 💻

WORK is REAL array, dimension (4*N). 🛠️

💻 IWORK 💻

IWORK is INTEGER array, dimension (N). 🛠️

🚨 INFO 🚨

INFO is INTEGER

  • = 0: Successful exit. The solution to every right-hand side is guaranteed. 🎉
  • < 0: If INFO = -i, the i-th argument had an illegal value. 🚩
  • > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. 💀
  • = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP. 🔍

👑 Author 👑

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

👑 Author (Doxygen's version) 👑

Generated automatically by Doxygen for LAPACK from the source code.

This iconic page is from LAPACK version 3.11.0, dated Sun Nov 27 2022, and it's the realGBsolve(3) man page. Byeeee bestie, slay responsibly! 💅✨🦋

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