SRC/zgbrfsx.f(3)

📦

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okay bestie, welcome to SRC/zgbrfsx.f(3) from the Library Functions Manual 💅 this is the group chat where ZGBRFSX explains itself, and honestly it’s giving main-character linear algebra energy ✨

💖 NAME 💖

SRC/zgbrfsx.f

🦋 SYNOPSIS 🦋

🎀 Functions/Subroutines 🎀

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

🌸 Function/Subroutine Documentation 🌸

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 ✨

so ZGBRFSX is here to improve the computed solution to a system of linear equations, and it also hands you error bounds and backward error estimates for that solution 💖

bestie, it does normwise error bound and, if possible, also gives the maximum componentwise error bound. for the details of those error bounds, go look at the comments for ERR_BNDS_NORM and ERR_BNDS_COMP — iykyk 👑

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

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 🎀

TRANS — TRANS is CHARACTER*1. it specifies the form of the system of equations, and the options are literally:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate transpose)
EQUED — EQUED is CHARACTER*1. this tells you what kind of equilibration was done to A before calling this routine. it’s needed so the solution and error bounds can be computed correctly, no cap:
- = '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). The right hand side B has been changed accordingly.
N — N is INTEGER. 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 COMPLEX*16 array, dimension (LDAB,N). this is 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).
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 🔥
LDAFB — LDAFB is INTEGER. the leading dimension of the array AFB. LDAFB >= 2*KL*KU+1 🦋
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) 👑
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 😭 this is the dramatic part, bestie — respect the radix powers 💅

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 warning, same energy, now with columns 🧚

B — B is COMPLEX*16 array, dimension (LDB,NRHS). the right hand side matrix B 🫶
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 ✨
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 👀
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) 💀
N_ERR_BNDS — N_ERR_BNDS is INTEGER. number of error bounds to return for each right hand side and each type (normwise or componentwise). see ERR_BNDS_NORM and ERR_BNDS_COMP below 👑

💖 ERR_BNDS_NORM 💖

ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). for each right-hand side, this array contains 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) * 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 🎀

ERR_BNDS_COMP is a DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS).

For each right-hand side, this array is literally the info desk for various error bounds and condition numbers tied to the componentwise relative error, which is defined like this:

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

The array is indexed by the right-hand side i (because the componentwise relative error depends on it) and by the type of error info 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 … (the source literally drops the word after “is” here, bestie 😭). 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 DOUBLE PRECISION array, dimension NPARAMS.

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

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 COMPLEX*16 array, dimension (2*N).

RWORK 🌸

RWORK is 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 (the sequel) 🤖

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This page is LAPACK Version 3.12.0, SRC/zgbrfsx.f(3). Stay slayed and numerically stable, bestie 💖✨

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