--- name: "pbrfs" 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/pbrfs" --- # pbrfs(3) pbrfs: iterative refinement ## NAME pbrfs - pbrfs: iterative refinement ## SYNOPSIS ### Functions subroutine **cpbrfs** (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **CPBRFS** subroutine **dpbrfs** (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info)\ **DPBRFS** subroutine **spbrfs** (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info)\ **SPBRFS** subroutine **zpbrfs** (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **ZPBRFS** ## Detailed Description ## Function Documentation ### subroutine cpbrfs (character uplo, integer n, integer kd, integer nrhs, complex, dimension( ldab, \* ) ab, integer ldab, complex, dimension( ldafb, \* ) afb, integer ldafb, 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) **CPBRFS** **Purpose:** ``` !> !> CPBRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is Hermitian positive definite !> and banded, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *KD* ``` !> KD is INTEGER !> The number of superdiagonals of the matrix A if UPLO = 'U', !> or the number of subdiagonals if UPLO = 'L'. KD >= 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 upper or lower triangle of the Hermitian band matrix A, !> stored in the first KD+1 rows of the array. The j-th column !> of A is stored in the j-th column of the array AB as follows: !> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; !> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). !> ``` *LDAB* ``` !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KD+1. !> ``` *AFB* ``` !> AFB is COMPLEX array, dimension (LDAFB,N) !> The triangular factor U or L from the Cholesky factorization !> A = U**H*U or A = L*L**H of the band matrix A as computed by !> CPBTRF, in the same storage format as A (see AB). !> ``` *LDAFB* ``` !> LDAFB is INTEGER !> The leading dimension of the array AFB. LDAFB >= KD+1. !> ``` *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 CPBTRS. !> 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 **187** of file **cpbrfs.f**. ### subroutine dpbrfs (character uplo, integer n, integer kd, integer nrhs, double precision, dimension( ldab, \* ) ab, integer ldab, double precision, dimension( ldafb, \* ) afb, integer ldafb, 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) **DPBRFS** **Purpose:** ``` !> !> DPBRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is symmetric positive definite !> and banded, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *KD* ``` !> KD is INTEGER !> The number of superdiagonals of the matrix A if UPLO = 'U', !> or the number of subdiagonals if UPLO = 'L'. KD >= 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 upper or lower triangle of the symmetric band matrix A, !> stored in the first KD+1 rows of the array. The j-th column !> of A is stored in the j-th column of the array AB as follows: !> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; !> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). !> ``` *LDAB* ``` !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KD+1. !> ``` *AFB* ``` !> AFB is DOUBLE PRECISION array, dimension (LDAFB,N) !> The triangular factor U or L from the Cholesky factorization !> A = U**T*U or A = L*L**T of the band matrix A as computed by !> DPBTRF, in the same storage format as A (see AB). !> ``` *LDAFB* ``` !> LDAFB is INTEGER !> The leading dimension of the array AFB. LDAFB >= KD+1. !> ``` *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 DPBTRS. !> 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 **187** of file **dpbrfs.f**. ### subroutine spbrfs (character uplo, integer n, integer kd, integer nrhs, real, dimension( ldab, \* ) ab, integer ldab, real, dimension( ldafb, \* ) afb, integer ldafb, 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) **SPBRFS** **Purpose:** ``` !> !> SPBRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is symmetric positive definite !> and banded, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *KD* ``` !> KD is INTEGER !> The number of superdiagonals of the matrix A if UPLO = 'U', !> or the number of subdiagonals if UPLO = 'L'. KD >= 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 upper or lower triangle of the symmetric band matrix A, !> stored in the first KD+1 rows of the array. The j-th column !> of A is stored in the j-th column of the array AB as follows: !> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; !> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). !> ``` *LDAB* ``` !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KD+1. !> ``` *AFB* ``` !> AFB is REAL array, dimension (LDAFB,N) !> The triangular factor U or L from the Cholesky factorization !> A = U**T*U or A = L*L**T of the band matrix A as computed by !> SPBTRF, in the same storage format as A (see AB). !> ``` *LDAFB* ``` !> LDAFB is INTEGER !> The leading dimension of the array AFB. LDAFB >= KD+1. !> ``` *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 SPBTRS. !> 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 **187** of file **spbrfs.f**. ### subroutine zpbrfs (character uplo, integer n, integer kd, integer nrhs, complex\*16, dimension( ldab, \* ) ab, integer ldab, complex\*16, dimension( ldafb, \* ) afb, integer ldafb, 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) **ZPBRFS** **Purpose:** ``` !> !> ZPBRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is Hermitian positive definite !> and banded, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *KD* ``` !> KD is INTEGER !> The number of superdiagonals of the matrix A if UPLO = 'U', !> or the number of subdiagonals if UPLO = 'L'. KD >= 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 upper or lower triangle of the Hermitian band matrix A, !> stored in the first KD+1 rows of the array. The j-th column !> of A is stored in the j-th column of the array AB as follows: !> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; !> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). !> ``` *LDAB* ``` !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KD+1. !> ``` *AFB* ``` !> AFB is COMPLEX*16 array, dimension (LDAFB,N) !> The triangular factor U or L from the Cholesky factorization !> A = U**H*U or A = L*L**H of the band matrix A as computed by !> ZPBTRF, in the same storage format as A (see AB). !> ``` *LDAFB* ``` !> LDAFB is INTEGER !> The leading dimension of the array AFB. LDAFB >= KD+1. !> ``` *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 ZPBTRS. !> 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 **187** of file **zpbrfs.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.