472 lines
14 KiB
Fortran
472 lines
14 KiB
Fortran
*> \brief \b SGTRFS
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download SGTRFS + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgtrfs.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgtrfs.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgtrfs.f">
|
|
*> [TXT]</a>
|
|
*> \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
|