gbrfs(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🌸 NAME 🌸

gbrfs — gbrfs: iterative refinement. This is the gbrfs(3) page from the Library Functions Manual, bestie, and it's literally all about refining your computed solution when the coefficient matrix is banded. 💅

✨ SYNOPSIS ✨

💖 Functions 💖

       subroutine cgbrfs (trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
           b, ldb, x, ldx, ferr, berr, work, rwork, info)
           CGBRFS
       subroutine dgbrfs (trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
           b, ldb, x, ldx, ferr, berr, work, iwork, info)
           DGBRFS
       subroutine sgbrfs (trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
           b, ldb, x, ldx, ferr, berr, work, iwork, info)
           SGBRFS
       subroutine zgbrfs (trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
           b, ldb, x, ldx, ferr, berr, work, rwork, info)
           ZGBRFS

🎀 DETAILED DESCRIPTION 🎀

No standalone extra prose here, bestie — the detailed description is lowkey just the runway to the function docs below. 🫶

🔥 FUNCTION DOCUMENTATION 🔥

All the function-level tea lives here. Four subroutines show up in the synopsis, and this part has the detailed docs for CGBRFS and DGBRFS. 💖

💅 cgbrfs — subroutine

   subroutine cgbrfs (character trans, integer n, integer kl, integer ku,
       integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex,
       dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv,
       complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, *
       ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr,
       complex, dimension( * ) work, real, dimension( * ) rwork, integer info)
       CGBRFS

🌟 Purpose

CGBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded. It also provides error bounds and backward error estimates for the solution. Obsessed with the completeness. ✨

💖 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)
  • 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 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).
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
  • AFB — AFB is COMPLEX array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by CGBTRF. 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 CGBTRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
  • B — B is COMPLEX 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 array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by CGBTRS. On exit, the improved solution matrix X.
  • LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • FERR — FERR is REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), 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). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
  • BERR — BERR is REAL array, dimension (NRHS). 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 COMPLEX array, dimension (2*N).
  • RWORK — RWORK is REAL array, dimension (N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

🌈 Internal Parameters

ITMAX is the maximum number of steps of iterative refinement. That's the cap, bestie. 🔥

👑 Author

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

Definition at line 203 of file cgbrfs.f.

💅 dgbrfs — subroutine

   subroutine dgbrfs (character trans, integer n, integer kl, integer ku,
       integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab,
       double precision, dimension( ldafb, * ) afb, integer ldafb, integer,
       dimension( * ) ipiv, double precision, dimension( ldb, * ) b, integer
       ldb, double precision, dimension( ldx, * ) x, integer ldx, double
       precision, dimension( * ) ferr, double precision, dimension( * ) berr,
       double precision, dimension( * ) work, integer, dimension( * ) iwork,
       integer info)
       DGBRFS

🌟 Purpose

DGBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded. It also provides error bounds and backward error estimates for the solution. Same energy, double precision. 💜

💖 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 = Transpose)
  • 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 DOUBLE PRECISION 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).
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
  • AFB — AFB is DOUBLE PRECISION array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as computed by DGBTRF. 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 DGBTRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
  • B — B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by DGBTRS. On exit, the improved solution matrix X.
  • LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • FERR — FERR is DOUBLE PRECISION array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), 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). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
  • BERR — BERR is DOUBLE PRECISION array, dimension (NRHS). 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 DOUBLE PRECISION array, dimension (3*N).
  • IWORK — IWORK is INTEGER array, dimension (N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

🌈 Internal Parameters

ITMAX is the maximum number of steps of iterative refinement. Same cap, bestie. 🔥

👑 Author

  • Univ. of Tennessee
  • Univ. of California Berkeley

Alright, picking up mid-glow-up in the dgbrfs author block — no re-intro, just the receipts:

  • Univ. of Colorado Denver
  • NAG Ltd.
  • Definition at line 202 of file dgbrfs.f.

💖 sgbrfs — single precision, main character energy 💖

subroutine sgbrfs (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, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SGBRFS

Purpose

SGBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded, and provides error bounds and backward error estimates for the solution. That’s the whole vibe: better solution, plus receipts on how wrong it might still be. 💅

Parameters

  • TRANS — 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 = Transpose)
  • N — INTEGER, the order of the matrix A. N >= 0.
  • KL — INTEGER, the number of subdiagonals within the band of A. KL >= 0.
  • KU — INTEGER, the number of superdiagonals within the band of A. KU >= 0.
  • NRHS — INTEGER, the number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • AB — REAL 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).
  • LDAB — INTEGER, the leading dimension of the array AB. LDAB >= KL+KU+1.
  • AFB — REAL array, dimension (LDAFB,N). 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.
  • LDAFB — INTEGER, the leading dimension of the array AFB. The page says LDAFB >= 2*KL*KU+1.
  • IPIV — INTEGER array, dimension (N). The pivot indices from SGBTRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
  • B — REAL array, dimension (LDB,NRHS). The right hand side matrix B.
  • LDB — INTEGER, the leading dimension of the array B. LDB >= max(1,N).
  • X — REAL array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by SGBTRS. On exit, the improved solution matrix X. ✨ glow-up complete ✨
  • LDX — INTEGER, the leading dimension of the array X. LDX >= max(1,N).
  • FERR — REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), 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). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
  • BERR — REAL array, dimension (NRHS). 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 — REAL array, dimension (3*N).
  • IWORK — INTEGER array, dimension (N).
  • INFO — INTEGER:
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

Internal Parameters

  • ITMAX is the maximum number of steps of iterative refinement.

Author

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

Definition at line 202 of file sgbrfs.f.

💜 zgbrfs — complex*16 glam edition 💜

subroutine zgbrfs (character trans, 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, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZGBRFS

Purpose

ZGBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded, and provides error bounds and backward error estimates for the solution. Same iconic behavior, now with COMPLEX*16 sparkle. 🦋

Parameters

  • TRANS — 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)
  • N — INTEGER, the order of the matrix A. N >= 0.
  • KL — INTEGER, the number of subdiagonals within the band of A. KL >= 0.
  • KU — INTEGER, the number of superdiagonals within the band of A. KU >= 0.
  • NRHS — INTEGER, the number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • AB — 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).
  • LDAB — INTEGER, the leading dimension of the array AB. LDAB >= KL+KU+1.
  • AFB — 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 — INTEGER, the leading dimension of the array AFB. The page says LDAFB >= 2*KL*KU+1.
  • IPIV — INTEGER array, dimension (N). The pivot indices from ZGBTRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).
  • B — COMPLEX*16 array, dimension (LDB,NRHS). The right hand side matrix B.
  • LDB — INTEGER, the leading dimension of the array B. LDB >= max(1,N).
  • X — COMPLEX*16 array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by ZGBTRS. On exit, the improved solution matrix X. 💫
  • LDX — INTEGER, the leading dimension of the array X. LDX >= max(1,N).
  • FERR — DOUBLE PRECISION array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), 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). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
  • BERR — DOUBLE PRECISION array, dimension (NRHS). 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 — COMPLEX*16 array, dimension (2*N).
  • RWORK — DOUBLE PRECISION array, dimension (N).
  • INFO — INTEGER:
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value

Internal Parameters

  • ITMAX is the maximum number of steps of iterative refinement.

Author

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

Definition at line 203 of file zgbrfs.f.

Author

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Okay bestie, that’s the whole gbrfs(3) saga 💖 The footer stamps this as LAPACK, Version 3.12.0, page gbrfs(3) — no extra date line in this chunk, just the version-and-page receipt. Slay, and may your bands always be factorized. 🎀✨

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