--- name: "gtrfs" 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/gtrfs" --- # gtrfs(3) gtrfs: iterative refinement ## NAME gtrfs - gtrfs: iterative refinement ## SYNOPSIS ### Functions subroutine **cgtrfs** (trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **CGTRFS** subroutine **dgtrfs** (trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)\ **DGTRFS** subroutine **sgtrfs** (trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)\ **SGTRFS** subroutine **zgtrfs** (trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **ZGTRFS** ## Detailed Description ## Function Documentation ### subroutine cgtrfs (character trans, integer n, integer nrhs, complex, dimension( \* ) dl, complex, dimension( \* ) d, complex, dimension( \* ) du, complex, dimension( \* ) dlf, complex, dimension( \* ) df, complex, dimension( \* ) duf, complex, dimension( \* ) du2, 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) **CGTRFS** **Purpose:** ``` !> !> CGTRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is tridiagonal, 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. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !> ``` *DL* ``` !> DL is COMPLEX array, dimension (N-1) !> The (n-1) subdiagonal elements of A. !> ``` *D* ``` !> D is COMPLEX array, dimension (N) !> The diagonal elements of A. !> ``` *DU* ``` !> DU is COMPLEX array, dimension (N-1) !> The (n-1) superdiagonal elements of A. !> ``` *DLF* ``` !> DLF is COMPLEX array, dimension (N-1) !> The (n-1) multipliers that define the matrix L from the !> LU factorization of A as computed by CGTTRF. !> ``` *DF* ``` !> DF is COMPLEX array, dimension (N) !> The n diagonal elements of the upper triangular matrix U from !> the LU factorization of A. !> ``` *DUF* ``` !> DUF is COMPLEX array, dimension (N-1) !> The (n-1) elements of the first superdiagonal of U. !> ``` *DU2* ``` !> DU2 is COMPLEX array, dimension (N-2) !> The (n-2) elements of the second superdiagonal of U. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> The pivot indices; for 1 <= i <= n, row i of the matrix was !> interchanged with row IPIV(i). IPIV(i) will always be either !> i or i+1; IPIV(i) = i indicates a row interchange was not !> required. !> ``` *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 CGTTRS. !> 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 **207** of file **cgtrfs.f**. ### subroutine dgtrfs (character trans, integer n, integer nrhs, double precision, dimension( \* ) dl, double precision, dimension( \* ) d, double precision, dimension( \* ) du, double precision, dimension( \* ) dlf, double precision, dimension( \* ) df, double precision, dimension( \* ) duf, double precision, dimension( \* ) du2, 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) **DGTRFS** **Purpose:** ``` !> !> DGTRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is tridiagonal, 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. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !> ``` *DL* ``` !> DL is DOUBLE PRECISION array, dimension (N-1) !> The (n-1) subdiagonal elements of A. !> ``` *D* ``` !> D is DOUBLE PRECISION array, dimension (N) !> The diagonal elements of A. !> ``` *DU* ``` !> DU is DOUBLE PRECISION array, dimension (N-1) !> The (n-1) superdiagonal elements of A. !> ``` *DLF* ``` !> DLF is DOUBLE PRECISION array, dimension (N-1) !> The (n-1) multipliers that define the matrix L from the !> LU factorization of A as computed by DGTTRF. !> ``` *DF* ``` !> DF is DOUBLE PRECISION array, dimension (N) !> The n diagonal elements of the upper triangular matrix U from !> the LU factorization of A. !> ``` *DUF* ``` !> DUF is DOUBLE PRECISION array, dimension (N-1) !> The (n-1) elements of the first superdiagonal of U. !> ``` *DU2* ``` !> DU2 is DOUBLE PRECISION array, dimension (N-2) !> The (n-2) elements of the second superdiagonal of U. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> The pivot indices; for 1 <= i <= n, row i of the matrix was !> interchanged with row IPIV(i). IPIV(i) will always be either !> i or i+1; IPIV(i) = i indicates a row interchange was not !> required. !> ``` *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 DGTTRS. !> 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 **206** of file **dgtrfs.f**. ### subroutine sgtrfs (character trans, integer n, integer nrhs, real, dimension( \* ) dl, real, dimension( \* ) d, real, dimension( \* ) du, real, dimension( \* ) dlf, real, dimension( \* ) df, real, dimension( \* ) duf, real, dimension( \* ) du2, 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) **SGTRFS** **Purpose:** ``` !> !> SGTRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is tridiagonal, 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. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !> ``` *DL* ``` !> DL is REAL array, dimension (N-1) !> The (n-1) subdiagonal elements of A. !> ``` *D* ``` !> D is REAL array, dimension (N) !> The diagonal elements of A. !> ``` *DU* ``` !> DU is REAL array, dimension (N-1) !> The (n-1) superdiagonal elements of A. !> ``` *DLF* ``` !> DLF is REAL array, dimension (N-1) !> The (n-1) multipliers that define the matrix L from the !> LU factorization of A as computed by SGTTRF. !> ``` *DF* ``` !> DF is REAL array, dimension (N) !> The n diagonal elements of the upper triangular matrix U from !> the LU factorization of A. !> ``` *DUF* ``` !> DUF is REAL array, dimension (N-1) !> The (n-1) elements of the first superdiagonal of U. !> ``` *DU2* ``` !> DU2 is REAL array, dimension (N-2) !> The (n-2) elements of the second superdiagonal of U. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> The pivot indices; for 1 <= i <= n, row i of the matrix was !> interchanged with row IPIV(i). IPIV(i) will always be either !> i or i+1; IPIV(i) = i indicates a row interchange was not !> required. !> ``` *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 SGTTRS. !> 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 **206** of file **sgtrfs.f**. ### subroutine zgtrfs (character trans, integer n, integer nrhs, complex\*16, dimension( \* ) dl, complex\*16, dimension( \* ) d, complex\*16, dimension( \* ) du, complex\*16, dimension( \* ) dlf, complex\*16, dimension( \* ) df, complex\*16, dimension( \* ) duf, complex\*16, dimension( \* ) du2, 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) **ZGTRFS** **Purpose:** ``` !> !> ZGTRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is tridiagonal, 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. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !> ``` *DL* ``` !> DL is COMPLEX*16 array, dimension (N-1) !> The (n-1) subdiagonal elements of A. !> ``` *D* ``` !> D is COMPLEX*16 array, dimension (N) !> The diagonal elements of A. !> ``` *DU* ``` !> DU is COMPLEX*16 array, dimension (N-1) !> The (n-1) superdiagonal elements of A. !> ``` *DLF* ``` !> DLF is COMPLEX*16 array, dimension (N-1) !> The (n-1) multipliers that define the matrix L from the !> LU factorization of A as computed by ZGTTRF. !> ``` *DF* ``` !> DF is COMPLEX*16 array, dimension (N) !> The n diagonal elements of the upper triangular matrix U from !> the LU factorization of A. !> ``` *DUF* ``` !> DUF is COMPLEX*16 array, dimension (N-1) !> The (n-1) elements of the first superdiagonal of U. !> ``` *DU2* ``` !> DU2 is COMPLEX*16 array, dimension (N-2) !> The (n-2) elements of the second superdiagonal of U. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> The pivot indices; for 1 <= i <= n, row i of the matrix was !> interchanged with row IPIV(i). IPIV(i) will always be either !> i or i+1; IPIV(i) = i indicates a row interchange was not !> required. !> ``` *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 ZGTTRS. !> 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 **207** of file **zgtrfs.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.