--- name: "gbrfs" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/gbrfs" --- # gbrfs(3) gbrfs: iterative refinement ## NAME gbrfs - gbrfs: iterative refinement ## 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 ## Function Documentation ### 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, and provides !> error bounds and backward error estimates for the solution. !> ``` **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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **203** of file **cgbrfs.f**. ### 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, and provides !> error bounds and backward error estimates for the solution. !> ``` **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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **202** of file **dgbrfs.f**. ### 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. !> ``` **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 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* ``` !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KL+KU+1. !> ``` *AFB* ``` !> AFB is 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* ``` !> 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 SGBTRF; for 1<=i<=N, row i of the !> matrix was interchanged with row IPIV(i). !> ``` *B* ``` !> B is REAL 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 REAL array, dimension (LDX,NRHS) !> On entry, the solution matrix X, as computed by SGBTRS. !> 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 REAL 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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **202** of file **sgbrfs.f**. ### 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. !> ``` **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*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* ``` !> 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 ZGBTRF; for 1<=i<=N, row i of the !> matrix was interchanged with row IPIV(i). !> ``` *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 ZGBTRS. !> 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 COMPLEX*16 array, dimension (2*N) !> ``` *RWORK* ``` !> RWORK is DOUBLE PRECISION 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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **203** of file **zgbrfs.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.