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

📦 version 3.12.0, in opensuse-leap 16.0

🦋 /home/abui...extended.f(3) 🦋

okay so bestie, welcome to the Library Functions Manual page for /home/abui...extended.f(3) — yes that's literally the page name, the LAPACK build just names files like a caffeinated gremlin, we don't question it 💅

✨ NAME ✨

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

that's it. that's the name. it's giving "full filesystem path as a personality trait" and honestly? iconic. 🎀

🌈 SYNOPSIS 🌈

🧚 Functions/Subroutines

subroutine CLA_GBRFSX_EXTENDED (prec_type, trans_type, n, kl, ku, nrhs,
    ab, ldab, afb, ldafb, ipiv, colequ, c, b, ldb, y, ldy, berr_out,
    n_norms, err_bnds_norm, err_bnds_comp, res, ayb, dy, y_tail, rcond,
    ithresh, rthresh, dz_ub, ignore_cwise, info)

so CLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by doing extra-precise iterative refinement, and it also hands you back error bounds plus backward error estimates for that solution. slay. ✨

💖 Function/Subroutine Documentation 💖

🌸 subroutine CLA_GBRFSX_EXTENDED

full signature, character for character, because we do not play about types:

subroutine CLA_GBRFSX_EXTENDED (integer prec_type, integer trans_type,
    integer n, integer kl, integer ku, integer nrhs, complex, dimension(
    ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer
    ldafb, integer, dimension( * ) ipiv, logical colequ, real, dimension( *
    ) c, complex, dimension( ldb, * ) b, integer ldb, complex, dimension(
    ldy, * ) y, integer ldy, real, dimension( * ) berr_out, integer n_norms,
    real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * )
    err_bnds_comp, complex, dimension( * ) res, real, dimension(*) ayb,
    complex, dimension( * ) dy, complex, dimension( * ) y_tail, real rcond,
    integer ithresh, real rthresh, real dz_ub, logical ignore_cwise, integer
    info)

CLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement, and it provides error bounds and backward error estimates for the solution. (yes, same sentence twice, the source is just Like That, we respect the source 🙈)

🫶 Purpose:

!> CLA_GBRFSX_EXTENDED improves the computed solution to a system of !> linear equations by performing extra-precise iterative refinement !> and provides error bounds and backward error estimates for the solution. !> This subroutine is called by CGBRFSX to perform iterative refinement. !> 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. Note that this !> subroutine is only responsible for setting the second fields of !> ERR_BNDS_NORM and ERR_BNDS_COMP. !>

translation bestie: it's the refinement engine that CGBRFSX calls in, it gives you a normwise bound, and if it can, a max componentwise bound too — but it only fills in the second field of ERR_BNDS_NORM and ERR_BNDS_COMP, okay? that's the whole job. 👑

⭐ Parameters ⭐

🌟 PREC_TYPE

!>          PREC_TYPE is INTEGER
!>     Specifies the intermediate precision to be used in refinement.
!>     The value is defined by ILAPREC(P) where P is a CHARACTER and P
!>          = 'S':  Single
!>          = 'D':  Double
!>          = 'I':  Indigenous
!>          = 'X' or 'E':  Extra
!>

PREC_TYPE is INTEGER. it picks the intermediate precision used in the refinement, and the value comes from ILAPREC(P) where P is a CHARACTER:

  • 'S' — Single
  • 'D' — Double
  • 'I' — Indigenous
  • 'X' or 'E' — Extra

🌟 TRANS_TYPE

!>          TRANS_TYPE is INTEGER
!>     Specifies the transposition operation on A.
!>     The value is defined by ILATRANS(T) where T is a CHARACTER and T
!>          = 'N':  No transpose
!>          = 'T':  Transpose
!>          = 'C':  Conjugate transpose
!>

TRANS_TYPE is INTEGER and it says which transposition operation happens on A. the value is defined by ILATRANS(T) where T is a CHARACTER:

  • 'N' — No transpose
  • 'T' — Transpose
  • 'C' — Conjugate transpose

🌟 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. requirement: N >= 0. ✨

🌟 KL

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

KL is INTEGER — how many subdiagonals live inside the band of A. requirement: KL >= 0.

🌟 KU

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

KU is INTEGER — how many superdiagonals live inside the band of A. requirement: KU >= 0. (no period in the original, we keep it raw 😌)

🌟 NRHS

!>          NRHS is INTEGER
!>     The number of right-hand-sides, i.e., the number of columns of the
!>     matrix B.
!>

NRHS is INTEGER — the number of right-hand-sides, which is literally the number of columns of the matrix B.

🌟 AB

!>          AB is COMPLEX array, dimension (LDAB,N)
!>     On entry, the N-by-N matrix AB.
!>

AB is a COMPLEX array of dimension (LDAB,N). on entry, it's the N-by-N matrix AB. 🫶

🌟 LDAB

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

LDAB is INTEGER — the leading dimension of the array AB. requirement: LDAB >= max(1,N).

🌟 AFB

!>          AFB is COMPLEX array, dimension (LDAF,N)
!>     The factors L and U from the factorization
!>     A = P*L*U as computed by CGBTRF.
!>

AFB is a COMPLEX array of dimension (LDAF,N) — the factors L and U from the factorization A = P*L*U, as computed by CGBTRF.

🌟 LDAFB

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

LDAFB is INTEGER — the leading dimension of the array AF. requirement: LDAF >= max(1,N). (yes the source says AF and LDAF here. not me, not my typo, we're just the messenger 💀)

🌟 IPIV

!>          IPIV is INTEGER array, dimension (N)
!>     The pivot indices from the factorization A = P*L*U
!>     as computed by CGBTRF; row i of the matrix was interchanged
!>     with row IPIV(i).
!>

IPIV is an INTEGER array of dimension (N) — the pivot indices from the factorization A = P*L*U as computed by CGBTRF. row i of the matrix was interchanged with row IPIV(i). that's the whole tea. 🔥

🌟 COLEQU

!>          COLEQU is LOGICAL
!>     If .TRUE. then column equilibration was done to A before calling
!>     this routine. This is needed to compute the solution and error
!>     bounds correctly.
!>

COLEQU is LOGICAL. if .TRUE. then column equilibration was already done to A before this routine got called — and bestie, that matters, because it's needed to compute the solution and the error bounds correctly.

🌟 C

!>          C is REAL array, dimension (N)
!>     The column scale factors for A. If COLEQU = .FALSE., C
!>     is not accessed. 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.
!>

C is a REAL array of dimension (N) — the column scale factors for A.

  • if COLEQU = .FALSE., C is not accessed at all. ignored. ghosted. 🙈
  • if C is input, every element of C should be a power of the radix, and that's how you get a reliable solution and reliable error estimates.
  • scaling by powers of the radix does not cause rounding errors — unless the result underflows or overflows.
  • and if you do get rounding errors during scaling? you end up refining with a matrix that is not equivalent to the input matrix, and then your error estimates may not be reliable. so. don't do that. 😭

🌟 B

!>          B is COMPLEX array, dimension (LDB,NRHS)
!>     The right-hand-side matrix B.
!>

B is a COMPLEX array of dimension (LDB,NRHS) — the right-hand-side matrix B. short and sweet.

🌟 LDB

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

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

🌟 Y

!>          Y is COMPLEX array, dimension (LDY,NRHS)
!>     On entry, the solution matrix X, as computed by CGBTRS.
!>     On exit, the improved solution matrix Y.
!>

Y is a COMPLEX array of dimension (LDY,NRHS). on entry it's the solution matrix X, as computed by CGBTRS. on exit it's the improved solution matrix Y. glow-up arc, literally. ✨

🌟 LDY

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

LDY is INTEGER — the leading dimension of the array Y. requirement: LDY >= max(1,N).

🌟 BERR_OUT

!>          BERR_OUT is REAL array, dimension (NRHS)
!>     On exit, BERR_OUT(j) contains the componentwise relative backward
!>     error for right-hand-side j from the formula
!>         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
!>     where abs(Z) is the componentwise absolute value of the matrix
!>     or vector Z. This is computed by CLA_LIN_BERR.
!>

BERR_OUT is a REAL array of dimension (NRHS). on exit, BERR_OUT(j) holds the componentwise relative backward error for right-hand-side j, from this formula:

    max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )

where abs(Z) is the componentwise absolute value of the matrix or vector Z. this gets computed by CLA_LIN_BERR. 🍓

🌟 N_NORMS

!>          N_NORMS is INTEGER
!>     Determines which error bounds to return (see ERR_BNDS_NORM
!>     and ERR_BNDS_COMP).
!>     If N_NORMS >= 1 return normwise error bounds.
!>     If N_NORMS >= 2 return componentwise error bounds.
!>

N_NORMS is INTEGER — it decides which error bounds come back (peep ERR_BNDS_NORM and ERR_BNDS_COMP).

  • if N_NORMS >= 1 → return normwise error bounds.
  • if N_NORMS >= 2 → return componentwise error bounds.

🌟 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.
!>
!>     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.
!>
!>     This subroutine is only responsible for setting the second field
!>     above.
!>     See Lapack Working Note 165 for further details and extra
!>     cautions.
!>

ERR_BNDS_NORM is a REAL array of dimension (NRHS, N_ERR_BNDS). for each right-hand side it carries info about various error bounds and condition numbers tied to the normwise relative error — which is defined like this:

    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 info described below, and there are currently up to three pieces of info 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 these 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: the 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.
ngl bestie the source sentence just stops at "the error estimate is ." — that's not me being mysterious, that's literally what's written, and I refuse to invent a word for LAPACK. 💀

this subroutine is only responsible for setting the second field above. and see Lapack Working Note 165 for further details and extra cautions — fr fr, go read it. 👑

🌟 ERR_BNDS_COMP

  • ERR_BNDS_COMP — okay so this one is a REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, it carries info about 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))

Componentwise relative error in the ith solution vector, bestie 💅. The array is indexed by the right-hand side i (the componentwise relative error depends on it), and by the type of error information 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) * 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. 🧚

This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.

  • RES — COMPLEX array, dimension (N). Workspace to hold the intermediate residual. 🫧
  • AYB — REAL array, dimension (N). Workspace. 🧹
  • DY — COMPLEX array, dimension (N). Workspace to hold the intermediate solution. 💡
  • Y_TAIL — COMPLEX array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. 🎀
  • 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. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. 😭
  • ITHRESH — INTEGER. The maximum number of residual computations allowed for refinement. The default is 10. For '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. 🔥
  • RTHRESH — REAL. Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details. 🌊
  • DZ_UB — REAL. Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we define as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details. 🦋
  • IGNORE_CWISE — LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE.. 🙈
  • INFO — INTEGER.
    • = 0: Successful exit. ✅
    • < 0: if INFO = -i, the ith argument to CGBTRS had an illegal value. 💀

👑 Author

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

Definition at line 403 of file cla_gbrfsx_extended.f.

🌈 Author

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