.TH "trrfs" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME trrfs \- trrfs: triangular iterative refinement .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBctrrfs\fP (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, rwork, info)" .br .RI "\fBCTRRFS\fP " .ti -1c .RI "subroutine \fBdtrrfs\fP (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, iwork, info)" .br .RI "\fBDTRRFS\fP " .ti -1c .RI "subroutine \fBstrrfs\fP (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, iwork, info)" .br .RI "\fBSTRRFS\fP " .ti -1c .RI "subroutine \fBztrrfs\fP (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, rwork, info)" .br .RI "\fBZTRRFS\fP " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "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)" .PP \fBCTRRFS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular\&. .fi .PP .br \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 \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular\&. .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 matrices B and X\&. NRHS >= 0\&. .fi .PP .br \fIA\fP .PP .nf 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\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .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) The 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 \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBDTRRFS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular\&. .fi .PP .br \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 \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular\&. .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 matrices B and X\&. NRHS >= 0\&. .fi .PP .br \fIA\fP .PP .nf 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\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .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) The 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 \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBSTRRFS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular\&. .fi .PP .br \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 \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular\&. .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 matrices B and X\&. NRHS >= 0\&. .fi .PP .br \fIA\fP .PP .nf 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\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .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) The 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 \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBZTRRFS\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular\&. .fi .PP .br \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 \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular\&. .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 matrices B and X\&. NRHS >= 0\&. .fi .PP .br \fIA\fP .PP .nf 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\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .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) The 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 \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\&.