--- name: "trrfs" 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/trrfs" --- # trrfs(3) trrfs: triangular iterative refinement ## NAME trrfs - trrfs: triangular iterative refinement ## SYNOPSIS ### Functions subroutine **ctrrfs** (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **CTRRFS** subroutine **dtrrfs** (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, iwork, info)\ **DTRRFS** subroutine **strrfs** (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, iwork, info)\ **STRRFS** subroutine **ztrrfs** (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **ZTRRFS** ## Detailed Description ## Function Documentation ### subroutine ctrrfs (character uplo, character trans, character diag, integer n, integer nrhs, complex, dimension( lda, \* ) a, integer lda, 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) **CTRRFS** **Purpose:** ``` !> !> CTRRFS provides error bounds and backward error estimates for the !> solution to a system of linear equations with a triangular !> coefficient matrix. !> !> The solution matrix X must be computed by CTRTRS or some other !> means before entering this routine. CTRRFS does not do iterative !> refinement because doing so cannot improve the backward error. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': A is upper triangular; !> = 'L': A is lower triangular. !> ``` *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) !> ``` *DIAG* ``` !> DIAG is CHARACTER*1 !> = 'N': A is non-unit triangular; !> = 'U': A is unit triangular. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 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. !> ``` *A* ``` !> A is COMPLEX array, dimension (LDA,N) !> The triangular matrix A. If UPLO = 'U', the leading N-by-N !> upper triangular part of the array A contains the upper !> triangular matrix, and the strictly lower triangular part of !> A is not referenced. If UPLO = 'L', the leading N-by-N lower !> triangular part of the array A contains the lower triangular !> matrix, and the strictly upper triangular part of A is not !> referenced. If DIAG = 'U', the diagonal elements of A are !> also not referenced and are assumed to be 1. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !> ``` *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) !> The 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 !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **180** of file **ctrrfs.f**. ### subroutine dtrrfs (character uplo, character trans, character diag, integer n, integer nrhs, double precision, dimension( lda, \* ) a, integer lda, 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) **DTRRFS** **Purpose:** ``` !> !> DTRRFS provides error bounds and backward error estimates for the !> solution to a system of linear equations with a triangular !> coefficient matrix. !> !> The solution matrix X must be computed by DTRTRS or some other !> means before entering this routine. DTRRFS does not do iterative !> refinement because doing so cannot improve the backward error. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': A is upper triangular; !> = 'L': A is lower triangular. !> ``` *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) !> ``` *DIAG* ``` !> DIAG is CHARACTER*1 !> = 'N': A is non-unit triangular; !> = 'U': A is unit triangular. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 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. !> ``` *A* ``` !> A is DOUBLE PRECISION array, dimension (LDA,N) !> The triangular matrix A. If UPLO = 'U', the leading N-by-N !> upper triangular part of the array A contains the upper !> triangular matrix, and the strictly lower triangular part of !> A is not referenced. If UPLO = 'L', the leading N-by-N lower !> triangular part of the array A contains the lower triangular !> matrix, and the strictly upper triangular part of A is not !> referenced. If DIAG = 'U', the diagonal elements of A are !> also not referenced and are assumed to be 1. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !> ``` *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) !> The 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 !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **180** of file **dtrrfs.f**. ### subroutine strrfs (character uplo, character trans, character diag, integer n, integer nrhs, real, dimension( lda, \* ) a, integer lda, 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) **STRRFS** **Purpose:** ``` !> !> STRRFS provides error bounds and backward error estimates for the !> solution to a system of linear equations with a triangular !> coefficient matrix. !> !> The solution matrix X must be computed by STRTRS or some other !> means before entering this routine. STRRFS does not do iterative !> refinement because doing so cannot improve the backward error. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': A is upper triangular; !> = 'L': A is lower triangular. !> ``` *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) !> ``` *DIAG* ``` !> DIAG is CHARACTER*1 !> = 'N': A is non-unit triangular; !> = 'U': A is unit triangular. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 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. !> ``` *A* ``` !> A is REAL array, dimension (LDA,N) !> The triangular matrix A. If UPLO = 'U', the leading N-by-N !> upper triangular part of the array A contains the upper !> triangular matrix, and the strictly lower triangular part of !> A is not referenced. If UPLO = 'L', the leading N-by-N lower !> triangular part of the array A contains the lower triangular !> matrix, and the strictly upper triangular part of A is not !> referenced. If DIAG = 'U', the diagonal elements of A are !> also not referenced and are assumed to be 1. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !> ``` *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) !> The 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 !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **180** of file **strrfs.f**. ### subroutine ztrrfs (character uplo, character trans, character diag, integer n, integer nrhs, complex\*16, dimension( lda, \* ) a, integer lda, 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) **ZTRRFS** **Purpose:** ``` !> !> ZTRRFS provides error bounds and backward error estimates for the !> solution to a system of linear equations with a triangular !> coefficient matrix. !> !> The solution matrix X must be computed by ZTRTRS or some other !> means before entering this routine. ZTRRFS does not do iterative !> refinement because doing so cannot improve the backward error. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': A is upper triangular; !> = 'L': A is lower triangular. !> ``` *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) !> ``` *DIAG* ``` !> DIAG is CHARACTER*1 !> = 'N': A is non-unit triangular; !> = 'U': A is unit triangular. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 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. !> ``` *A* ``` !> A is COMPLEX*16 array, dimension (LDA,N) !> The triangular matrix A. If UPLO = 'U', the leading N-by-N !> upper triangular part of the array A contains the upper !> triangular matrix, and the strictly lower triangular part of !> A is not referenced. If UPLO = 'L', the leading N-by-N lower !> triangular part of the array A contains the lower triangular !> matrix, and the strictly upper triangular part of A is not !> referenced. If DIAG = 'U', the diagonal elements of A are !> also not referenced and are assumed to be 1. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !> ``` *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) !> The 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 !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **180** of file **ztrrfs.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.