.TH "gtrfs" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME gtrfs \- gtrfs: iterative refinement .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBcgtrfs\fP (trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)" .br .RI "\fBCGTRFS\fP " .ti -1c .RI "subroutine \fBdgtrfs\fP (trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)" .br .RI "\fBDGTRFS\fP " .ti -1c .RI "subroutine \fBsgtrfs\fP (trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)" .br .RI "\fBSGTRFS\fP " .ti -1c .RI "subroutine \fBzgtrfs\fP (trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)" .br .RI "\fBZGTRFS\fP " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "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)" .PP \fBCGTRFS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANS\fP .PP .nf 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) .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrix B\&. NRHS >= 0\&. .fi .PP .br \fIDL\fP .PP .nf DL is COMPLEX array, dimension (N-1) The (n-1) subdiagonal elements of A\&. .fi .PP .br \fID\fP .PP .nf D is COMPLEX array, dimension (N) The diagonal elements of A\&. .fi .PP .br \fIDU\fP .PP .nf DU is COMPLEX array, dimension (N-1) The (n-1) superdiagonal elements of A\&. .fi .PP .br \fIDLF\fP .PP .nf 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\&. .fi .PP .br \fIDF\fP .PP .nf DF is COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A\&. .fi .PP .br \fIDUF\fP .PP .nf DUF is COMPLEX array, dimension (N-1) The (n-1) elements of the first superdiagonal of U\&. .fi .PP .br \fIDU2\fP .PP .nf DU2 is COMPLEX array, dimension (N-2) The (n-2) elements of the second superdiagonal of U\&. .fi .PP .br \fIIPIV\fP .PP .nf 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\&. .fi .PP .br \fIB\fP .PP .nf B is COMPLEX array, dimension (LDB,NRHS) The right hand side matrix B\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,N)\&. .fi .PP .br \fIX\fP .PP .nf X is COMPLEX array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by CGTTRS\&. On exit, the improved solution matrix X\&. .fi .PP .br \fILDX\fP .PP .nf LDX is INTEGER The leading dimension of the array X\&. LDX >= max(1,N)\&. .fi .PP .br \fIFERR\fP .PP .nf 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\&. .fi .PP .br \fIBERR\fP .PP .nf 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)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX array, dimension (2*N) .fi .PP .br \fIRWORK\fP .PP .nf RWORK is REAL array, dimension (N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBInternal Parameters:\fP .RS 4 .PP .nf ITMAX is the maximum number of steps of iterative refinement\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBDGTRFS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANS\fP .PP .nf 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) .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrix B\&. NRHS >= 0\&. .fi .PP .br \fIDL\fP .PP .nf DL is DOUBLE PRECISION array, dimension (N-1) The (n-1) subdiagonal elements of A\&. .fi .PP .br \fID\fP .PP .nf D is DOUBLE PRECISION array, dimension (N) The diagonal elements of A\&. .fi .PP .br \fIDU\fP .PP .nf DU is DOUBLE PRECISION array, dimension (N-1) The (n-1) superdiagonal elements of A\&. .fi .PP .br \fIDLF\fP .PP .nf 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\&. .fi .PP .br \fIDF\fP .PP .nf DF is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A\&. .fi .PP .br \fIDUF\fP .PP .nf DUF is DOUBLE PRECISION array, dimension (N-1) The (n-1) elements of the first superdiagonal of U\&. .fi .PP .br \fIDU2\fP .PP .nf DU2 is DOUBLE PRECISION array, dimension (N-2) The (n-2) elements of the second superdiagonal of U\&. .fi .PP .br \fIIPIV\fP .PP .nf 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\&. .fi .PP .br \fIB\fP .PP .nf B is DOUBLE PRECISION array, dimension (LDB,NRHS) The right hand side matrix B\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,N)\&. .fi .PP .br \fIX\fP .PP .nf 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\&. .fi .PP .br \fILDX\fP .PP .nf LDX is INTEGER The leading dimension of the array X\&. LDX >= max(1,N)\&. .fi .PP .br \fIFERR\fP .PP .nf 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\&. .fi .PP .br \fIBERR\fP .PP .nf 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)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is DOUBLE PRECISION array, dimension (3*N) .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBInternal Parameters:\fP .RS 4 .PP .nf ITMAX is the maximum number of steps of iterative refinement\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBSGTRFS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANS\fP .PP .nf 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) .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrix B\&. NRHS >= 0\&. .fi .PP .br \fIDL\fP .PP .nf DL is REAL array, dimension (N-1) The (n-1) subdiagonal elements of A\&. .fi .PP .br \fID\fP .PP .nf D is REAL array, dimension (N) The diagonal elements of A\&. .fi .PP .br \fIDU\fP .PP .nf DU is REAL array, dimension (N-1) The (n-1) superdiagonal elements of A\&. .fi .PP .br \fIDLF\fP .PP .nf 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\&. .fi .PP .br \fIDF\fP .PP .nf DF is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A\&. .fi .PP .br \fIDUF\fP .PP .nf DUF is REAL array, dimension (N-1) The (n-1) elements of the first superdiagonal of U\&. .fi .PP .br \fIDU2\fP .PP .nf DU2 is REAL array, dimension (N-2) The (n-2) elements of the second superdiagonal of U\&. .fi .PP .br \fIIPIV\fP .PP .nf 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\&. .fi .PP .br \fIB\fP .PP .nf B is REAL array, dimension (LDB,NRHS) The right hand side matrix B\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,N)\&. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by SGTTRS\&. On exit, the improved solution matrix X\&. .fi .PP .br \fILDX\fP .PP .nf LDX is INTEGER The leading dimension of the array X\&. LDX >= max(1,N)\&. .fi .PP .br \fIFERR\fP .PP .nf 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\&. .fi .PP .br \fIBERR\fP .PP .nf 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)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is REAL array, dimension (3*N) .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBInternal Parameters:\fP .RS 4 .PP .nf ITMAX is the maximum number of steps of iterative refinement\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBZGTRFS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANS\fP .PP .nf 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) .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrix B\&. NRHS >= 0\&. .fi .PP .br \fIDL\fP .PP .nf DL is COMPLEX*16 array, dimension (N-1) The (n-1) subdiagonal elements of A\&. .fi .PP .br \fID\fP .PP .nf D is COMPLEX*16 array, dimension (N) The diagonal elements of A\&. .fi .PP .br \fIDU\fP .PP .nf DU is COMPLEX*16 array, dimension (N-1) The (n-1) superdiagonal elements of A\&. .fi .PP .br \fIDLF\fP .PP .nf 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\&. .fi .PP .br \fIDF\fP .PP .nf DF is COMPLEX*16 array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A\&. .fi .PP .br \fIDUF\fP .PP .nf DUF is COMPLEX*16 array, dimension (N-1) The (n-1) elements of the first superdiagonal of U\&. .fi .PP .br \fIDU2\fP .PP .nf DU2 is COMPLEX*16 array, dimension (N-2) The (n-2) elements of the second superdiagonal of U\&. .fi .PP .br \fIIPIV\fP .PP .nf 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\&. .fi .PP .br \fIB\fP .PP .nf B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,N)\&. .fi .PP .br \fIX\fP .PP .nf 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\&. .fi .PP .br \fILDX\fP .PP .nf LDX is INTEGER The leading dimension of the array X\&. LDX >= max(1,N)\&. .fi .PP .br \fIFERR\fP .PP .nf 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\&. .fi .PP .br \fIBERR\fP .PP .nf 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)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX*16 array, dimension (2*N) .fi .PP .br \fIRWORK\fP .PP .nf RWORK is DOUBLE PRECISION array, dimension (N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBInternal Parameters:\fP .RS 4 .PP .nf ITMAX is the maximum number of steps of iterative refinement\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.