*> \brief \b SGTRFS * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGTRFS + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, * IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, * INFO ) * * .. Scalar Arguments .. * CHARACTER TRANS * INTEGER INFO, LDB, LDX, N, NRHS * .. * .. Array Arguments .. * INTEGER IPIV( * ), IWORK( * ) * REAL B( LDB, * ), BERR( * ), D( * ), DF( * ), * \$ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ), * \$ FERR( * ), WORK( * ), X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] TRANS *> \verbatim *> 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) *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of right hand sides, i.e., the number of columns *> of the matrix B. NRHS >= 0. *> \endverbatim *> *> \param[in] DL *> \verbatim *> DL is REAL array, dimension (N-1) *> The (n-1) subdiagonal elements of A. *> \endverbatim *> *> \param[in] D *> \verbatim *> D is REAL array, dimension (N) *> The diagonal elements of A. *> \endverbatim *> *> \param[in] DU *> \verbatim *> DU is REAL array, dimension (N-1) *> The (n-1) superdiagonal elements of A. *> \endverbatim *> *> \param[in] DLF *> \verbatim *> 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. *> \endverbatim *> *> \param[in] DF *> \verbatim *> DF is REAL array, dimension (N) *> The n diagonal elements of the upper triangular matrix U from *> the LU factorization of A. *> \endverbatim *> *> \param[in] DUF *> \verbatim *> DUF is REAL array, dimension (N-1) *> The (n-1) elements of the first superdiagonal of U. *> \endverbatim *> *> \param[in] DU2 *> \verbatim *> DU2 is REAL array, dimension (N-2) *> The (n-2) elements of the second superdiagonal of U. *> \endverbatim *> *> \param[in] IPIV *> \verbatim *> 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. *> \endverbatim *> *> \param[in] B *> \verbatim *> B is REAL array, dimension (LDB,NRHS) *> The right hand side matrix B. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,N). *> \endverbatim *> *> \param[in,out] X *> \verbatim *> X is REAL array, dimension (LDX,NRHS) *> On entry, the solution matrix X, as computed by SGTTRS. *> On exit, the improved solution matrix X. *> \endverbatim *> *> \param[in] LDX *> \verbatim *> LDX is INTEGER *> The leading dimension of the array X. LDX >= max(1,N). *> \endverbatim *> *> \param[out] FERR *> \verbatim *> 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. *> \endverbatim *> *> \param[out] BERR *> \verbatim *> 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). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (3*N) *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> \endverbatim * *> \par Internal Parameters: * ========================= *> *> \verbatim *> ITMAX is the maximum number of steps of iterative refinement. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup realGTcomputational * * ===================================================================== SUBROUTINE SGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, \$ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, \$ INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, LDB, LDX, N, NRHS * .. * .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) REAL B( LDB, * ), BERR( * ), D( * ), DF( * ), \$ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ), \$ FERR( * ), WORK( * ), X( LDX, * ) * .. * * ===================================================================== * * .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) REAL TWO PARAMETER ( TWO = 2.0E+0 ) REAL THREE PARAMETER ( THREE = 3.0E+0 ) * .. * .. Local Scalars .. LOGICAL NOTRAN CHARACTER TRANSN, TRANST INTEGER COUNT, I, J, KASE, NZ REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Subroutines .. EXTERNAL SAXPY, SCOPY, SGTTRS, SLACN2, SLAGTM, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX * .. * .. External Functions .. LOGICAL LSAME REAL SLAMCH EXTERNAL LSAME, SLAMCH * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. \$ LSAME( TRANS, 'C' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -13 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -15 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGTRFS', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF * IF( NOTRAN ) THEN TRANSN = 'N' TRANST = 'T' ELSE TRANSN = 'T' TRANST = 'N' END IF * * NZ = maximum number of nonzero elements in each row of A, plus 1 * NZ = 4 EPS = SLAMCH( 'Epsilon' ) SAFMIN = SLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS * * Do for each right hand side * DO 110 J = 1, NRHS * COUNT = 1 LSTRES = THREE 20 CONTINUE * * Loop until stopping criterion is satisfied. * * Compute residual R = B - op(A) * X, * where op(A) = A, A**T, or A**H, depending on TRANS. * CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL SLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE, \$ WORK( N+1 ), N ) * * Compute abs(op(A))*abs(x) + abs(b) for use in the backward * error bound. * IF( NOTRAN ) THEN IF( N.EQ.1 ) THEN WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) ELSE WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + \$ ABS( DU( 1 )*X( 2, J ) ) DO 30 I = 2, N - 1 WORK( I ) = ABS( B( I, J ) ) + \$ ABS( DL( I-1 )*X( I-1, J ) ) + \$ ABS( D( I )*X( I, J ) ) + \$ ABS( DU( I )*X( I+1, J ) ) 30 CONTINUE WORK( N ) = ABS( B( N, J ) ) + \$ ABS( DL( N-1 )*X( N-1, J ) ) + \$ ABS( D( N )*X( N, J ) ) END IF ELSE IF( N.EQ.1 ) THEN WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) ELSE WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + \$ ABS( DL( 1 )*X( 2, J ) ) DO 40 I = 2, N - 1 WORK( I ) = ABS( B( I, J ) ) + \$ ABS( DU( I-1 )*X( I-1, J ) ) + \$ ABS( D( I )*X( I, J ) ) + \$ ABS( DL( I )*X( I+1, J ) ) 40 CONTINUE WORK( N ) = ABS( B( N, J ) ) + \$ ABS( DU( N-1 )*X( N-1, J ) ) + \$ ABS( D( N )*X( N, J ) ) END IF END IF * * Compute componentwise relative backward error from formula * * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) * * where abs(Z) is the componentwise absolute value of the matrix * or vector Z. If the i-th component of the denominator is less * than SAFE2, then SAFE1 is added to the i-th components of the * numerator and denominator before dividing. * S = ZERO DO 50 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / \$ ( WORK( I )+SAFE1 ) ) END IF 50 CONTINUE BERR( J ) = S * * Test stopping criterion. Continue iterating if * 1) The residual BERR(J) is larger than machine epsilon, and * 2) BERR(J) decreased by at least a factor of 2 during the * last iteration, and * 3) At most ITMAX iterations tried. * IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. \$ COUNT.LE.ITMAX ) THEN * * Update solution and try again. * CALL SGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, \$ WORK( N+1 ), N, INFO ) CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF * * Bound error from formula * * norm(X - XTRUE) / norm(X) .le. FERR = * norm( abs(inv(op(A)))* * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) * * where * norm(Z) is the magnitude of the largest component of Z * inv(op(A)) is the inverse of op(A) * abs(Z) is the componentwise absolute value of the matrix or * vector Z * NZ is the maximum number of nonzeros in any row of A, plus 1 * EPS is machine epsilon * * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) * is incremented by SAFE1 if the i-th component of * abs(op(A))*abs(X) + abs(B) is less than SAFE2. * * Use SLACN2 to estimate the infinity-norm of the matrix * inv(op(A)) * diag(W), * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) * DO 60 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 60 CONTINUE * KASE = 0 70 CONTINUE CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), \$ KASE, ISAVE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * * Multiply by diag(W)*inv(op(A)**T). * CALL SGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, \$ WORK( N+1 ), N, INFO ) DO 80 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 80 CONTINUE ELSE * * Multiply by inv(op(A))*diag(W). * DO 90 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 90 CONTINUE CALL SGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, \$ WORK( N+1 ), N, INFO ) END IF GO TO 70 END IF * * Normalize error. * LSTRES = ZERO DO 100 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 100 CONTINUE IF( LSTRES.NE.ZERO ) \$ FERR( J ) = FERR( J ) / LSTRES * 110 CONTINUE * RETURN * * End of SGTRFS * END