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

📦 version 3.12.0, in opensuse-leap 16.0

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

okay so bestie, this is the manual page for zgbrfsx.f — LAPACK's absolute icon of a band-matrix refinement routine 💅 she lives in the lapack-3.12.0/SRC/ folder and she is so good at what she does.

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

like, here's the whole cast, no cap:

subroutine ZGBRFSX (trans, equed, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
    ipiv, r, c, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
    err_bnds_comp, nparams, params, work, rwork, info)

ZGBRFSX — that's her. main character energy, fr fr 🌟

💖 Function/Subroutine Documentation 💖

⭐ the subroutine itself (full flex) ⭐

subroutine ZGBRFSX (character trans, character equed, integer n, integer kl,
    integer ku, integer nrhs, complex*16, dimension( ldab, * ) ab, integer
    ldab, complex*16, dimension( ldafb, * ) afb, integer ldafb, integer,
    dimension( * ) ipiv, double precision, dimension( * ) r, double
    precision, dimension( * ) c, complex*16, dimension( ldb, * ) b, integer
    ldb, complex*16, dimension( ldx , * ) x, integer ldx, double precision
    rcond, 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)

ZGBRFSX

🌸 Purpose 🌸

ZGBRFSX improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.

so basically: you hand her a solution you already computed, and she refines it and then hands you receipts — error bounds AND backward error estimates 😭👑

The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED, R and C below. In this case, the solution and error bounds returned are for the original unequilibrated system.

lowkey important: if you equilibrated before calling, that's fine, but what comes back is for the original unequilibrated system. she remembers where she came from 💜

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.

and like, the tuning knobs all live bundled up in PARAMS. defaults are usually exactly what you want, and if they are, pass NPARAMS = 0 and the source code literally won't even look at the PARAMS argument 🫶 no touching, no reading, nothing.

🌈 Parameters 🌈

💫 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 (Conjugate transpose)

pick your vibe check: no transpose, transpose, or conjugate transpose ✨

💫 EQUED

EQUED is CHARACTER*1 Specifies the form of equilibration that was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.
  • = 'N': No equilibration
  • = '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).

and for 'B' specifically: The right hand side B has been changed accordingly. so don't forget that part bestie 🙈

💫 N

N is INTEGER The order of the matrix A. N >= 0.

the size of the queen herself 👑

💫 KL

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

subdiagonals, count 'em 🍓

💫 KU

KU is INTEGER The number of superdiagonals within the band of A. KU >= 0.

superdiagonals, also count 'em 🌸

💫 NRHS

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

how many right hand sides we serving today 💅

💫 AB

AB is COMPLEX*16 array, dimension (LDAB,N) The original band matrix A, stored 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).

this indexing formula is the whole schtick, don't @ me — it's literally AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). 🧚

💫 LDAB

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

💫 AFB

AFB is COMPLEX*16 array, dimension (LDAFB,N) Details of the LU factorization of the band matrix A, as computed by ZGBTRF. 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.

so: U chills in rows 1 to KL+KU+1 with KL+KU superdiagonals, and the multipliers from the factorization are stacked in rows KL+KU+2 to 2*KL+KU+1. organized queen 💜

💫 LDAFB

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

(that's exactly how it's written in the source, bestie — 2*KL*KU+1, copy it as-is 👀)

💫 IPIV

IPIV is INTEGER array, dimension (N) The pivot indices from ZGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).

the swap lore 🌀 for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).

💫 R

R is DOUBLE PRECISION 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.

okay this one's a whole paragraph but it matters 😤 recap: if EQUED = 'R' or 'B', A gets multiplied on the left by diag(R); if EQUED = 'N' or 'C', R isn't accessed at all. input arg when FACT = 'F', output arg otherwise, and if FACT = 'F' with EQUED = 'R' or 'B', every element of R must be positive. powers of the radix = the safe zone; and heavy rounding during scaling can lowkey ruin your error estimates 💀

💫 C

C is DOUBLE PRECISION 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 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.

same energy as R but column-flavored: right-multiply by diag(C) when EQUED = 'C' or 'B', untouched when EQUED = 'N' or 'R', positive elements required when FACT = 'F' and EQUED = 'C' or 'B', and powers of the radix keep things honest 💖

💫 B

B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B.

the right hand side, the whole reason we're here 🎀

💫 LDB

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

💫 X

X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZGETRS. On exit, the improved solution matrix X.

entry = your ZGETRS solution. exit = the improved one, because she ate and left no crumbs 🔥

💫 LDX

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

💫 RCOND

RCOND is DOUBLE PRECISION Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned.

reciprocal Skeel condition number estimate after equilibration (if done). if it's below machine precision — especially if it's zero — the matrix is singular to working precision 😬 but plot twist: the error can still be small even when this number is tiny and the matrix looks ill-conditioned. don't panic-read the vibe check 🦋

💫 BERR

BERR is 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).

per-solution receipts: BERR gives you the componentwise relative backward error of each X(j) — aka the smallest relative change in any element of A or B that makes X(j) an exact solution. smallest change = biggest flex ✨

💫 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.

how many error bounds you want back per rhs, per type — normwise or componentwise. ERR_BNDS_NORM and ERR_BNDS_COMP are literally the next thing coming up 🫶

💫 ERR_BNDS_NORM

next up bestie, we're getting into ERR_BNDS_NORM in part 2 — she's literally the next heading and she has so much to say 💅

okay so ERR_BNDS_NORM is a DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 💖. For each right-hand side, this array is holding all the 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))

Normwise relative error in the ith solution vector, periodt ✨.

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

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

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

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

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

ERR_BNDS_COMP

ERR_BNDS_COMP is a DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 💅. For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

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

Componentwise relative error in the ith solution vector, no cap 🧚.

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

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

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

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

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

NPARAMS

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. 🎀

PARAMS

PARAMS is a DOUBLE PRECISION array, dimension NPARAMS. 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. 👑

  • 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 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 a COMPLEX*16 array, dimension (2*N). 🌸

RWORK

RWORK is a DOUBLE PRECISION array, dimension (2*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.

Definition at line 435 of file zgbrfsx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole vibe check, bestie — this iconic page is straight from LAPACK Version 3.12.0, file /home/abui.../zgbrfsx.f(3), no date listed but the energy is forever 💜✨

📦 where this page comes from