Julie
bc6a755be4
This is really old school, but a lot of times we have users sending us copy pasting of codes, and that is the only way to know the version of the code.
396 lines
12 KiB
FortranFixed
396 lines
12 KiB
FortranFixed
*> \brief \b DPTRFS
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download DPTRFS + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptrfs.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptrfs.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptrfs.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
|
|
* BERR, WORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* INTEGER INFO, LDB, LDX, N, NRHS
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
|
|
* $ E( * ), EF( * ), FERR( * ), WORK( * ),
|
|
* $ X( LDX, * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> DPTRFS improves the computed solution to a system of linear
|
|
*> equations when the coefficient matrix is symmetric positive definite
|
|
*> and tridiagonal, and provides error bounds and backward error
|
|
*> estimates for the solution.
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \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] D
|
|
*> \verbatim
|
|
*> D is DOUBLE PRECISION array, dimension (N)
|
|
*> The n diagonal elements of the tridiagonal matrix A.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] E
|
|
*> \verbatim
|
|
*> E is DOUBLE PRECISION array, dimension (N-1)
|
|
*> The (n-1) subdiagonal elements of the tridiagonal matrix A.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] DF
|
|
*> \verbatim
|
|
*> DF is DOUBLE PRECISION array, dimension (N)
|
|
*> The n diagonal elements of the diagonal matrix D from the
|
|
*> factorization computed by DPTTRF.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] EF
|
|
*> \verbatim
|
|
*> EF is DOUBLE PRECISION array, dimension (N-1)
|
|
*> The (n-1) subdiagonal elements of the unit bidiagonal factor
|
|
*> L from the factorization computed by DPTTRF.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] B
|
|
*> \verbatim
|
|
*> B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)
|
|
*> On entry, the solution matrix X, as computed by DPTTRS.
|
|
*> 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 DOUBLE PRECISION array, dimension (NRHS)
|
|
*> The 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).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] BERR
|
|
*> \verbatim
|
|
*> 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).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] WORK
|
|
*> \verbatim
|
|
*> WORK is DOUBLE PRECISION array, dimension (2*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.
|
|
*
|
|
*> \date December 2016
|
|
*
|
|
*> \ingroup doublePTcomputational
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
|
|
$ BERR, WORK, INFO )
|
|
*
|
|
* -- LAPACK computational routine (version 3.7.0) --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
* December 2016
|
|
*
|
|
* .. Scalar Arguments ..
|
|
INTEGER INFO, LDB, LDX, N, NRHS
|
|
* ..
|
|
* .. Array Arguments ..
|
|
DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
|
|
$ E( * ), EF( * ), FERR( * ), WORK( * ),
|
|
$ X( LDX, * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
INTEGER ITMAX
|
|
PARAMETER ( ITMAX = 5 )
|
|
DOUBLE PRECISION ZERO
|
|
PARAMETER ( ZERO = 0.0D+0 )
|
|
DOUBLE PRECISION ONE
|
|
PARAMETER ( ONE = 1.0D+0 )
|
|
DOUBLE PRECISION TWO
|
|
PARAMETER ( TWO = 2.0D+0 )
|
|
DOUBLE PRECISION THREE
|
|
PARAMETER ( THREE = 3.0D+0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
INTEGER COUNT, I, IX, J, NZ
|
|
DOUBLE PRECISION BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
|
|
$ SAFMIN
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL DAXPY, DPTTRS, XERBLA
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC ABS, MAX
|
|
* ..
|
|
* .. External Functions ..
|
|
INTEGER IDAMAX
|
|
DOUBLE PRECISION DLAMCH
|
|
EXTERNAL IDAMAX, DLAMCH
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input parameters.
|
|
*
|
|
INFO = 0
|
|
IF( N.LT.0 ) THEN
|
|
INFO = -1
|
|
ELSE IF( NRHS.LT.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -8
|
|
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
|
INFO = -10
|
|
END IF
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'DPTRFS', -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
|
|
*
|
|
* NZ = maximum number of nonzero elements in each row of A, plus 1
|
|
*
|
|
NZ = 4
|
|
EPS = DLAMCH( 'Epsilon' )
|
|
SAFMIN = DLAMCH( 'Safe minimum' )
|
|
SAFE1 = NZ*SAFMIN
|
|
SAFE2 = SAFE1 / EPS
|
|
*
|
|
* Do for each right hand side
|
|
*
|
|
DO 90 J = 1, NRHS
|
|
*
|
|
COUNT = 1
|
|
LSTRES = THREE
|
|
20 CONTINUE
|
|
*
|
|
* Loop until stopping criterion is satisfied.
|
|
*
|
|
* Compute residual R = B - A * X. Also compute
|
|
* abs(A)*abs(x) + abs(b) for use in the backward error bound.
|
|
*
|
|
IF( N.EQ.1 ) THEN
|
|
BI = B( 1, J )
|
|
DX = D( 1 )*X( 1, J )
|
|
WORK( N+1 ) = BI - DX
|
|
WORK( 1 ) = ABS( BI ) + ABS( DX )
|
|
ELSE
|
|
BI = B( 1, J )
|
|
DX = D( 1 )*X( 1, J )
|
|
EX = E( 1 )*X( 2, J )
|
|
WORK( N+1 ) = BI - DX - EX
|
|
WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
|
|
DO 30 I = 2, N - 1
|
|
BI = B( I, J )
|
|
CX = E( I-1 )*X( I-1, J )
|
|
DX = D( I )*X( I, J )
|
|
EX = E( I )*X( I+1, J )
|
|
WORK( N+I ) = BI - CX - DX - EX
|
|
WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
|
|
30 CONTINUE
|
|
BI = B( N, J )
|
|
CX = E( N-1 )*X( N-1, J )
|
|
DX = D( N )*X( N, J )
|
|
WORK( N+N ) = BI - CX - DX
|
|
WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
|
|
END IF
|
|
*
|
|
* Compute componentwise relative backward error from formula
|
|
*
|
|
* max(i) ( abs(R(i)) / ( abs(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 40 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
|
|
40 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 DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO )
|
|
CALL DAXPY( 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(A))*
|
|
* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
|
|
*
|
|
* where
|
|
* norm(Z) is the magnitude of the largest component of Z
|
|
* inv(A) is the inverse of 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(A)*abs(X)+abs(B))
|
|
* is incremented by SAFE1 if the i-th component of
|
|
* abs(A)*abs(X) + abs(B) is less than SAFE2.
|
|
*
|
|
DO 50 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
|
|
50 CONTINUE
|
|
IX = IDAMAX( N, WORK, 1 )
|
|
FERR( J ) = WORK( IX )
|
|
*
|
|
* Estimate the norm of inv(A).
|
|
*
|
|
* Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
|
|
*
|
|
* m(i,j) = abs(A(i,j)), i = j,
|
|
* m(i,j) = -abs(A(i,j)), i .ne. j,
|
|
*
|
|
* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
|
|
*
|
|
* Solve M(L) * x = e.
|
|
*
|
|
WORK( 1 ) = ONE
|
|
DO 60 I = 2, N
|
|
WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
|
|
60 CONTINUE
|
|
*
|
|
* Solve D * M(L)**T * x = b.
|
|
*
|
|
WORK( N ) = WORK( N ) / DF( N )
|
|
DO 70 I = N - 1, 1, -1
|
|
WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
|
|
70 CONTINUE
|
|
*
|
|
* Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
|
|
*
|
|
IX = IDAMAX( N, WORK, 1 )
|
|
FERR( J ) = FERR( J )*ABS( WORK( IX ) )
|
|
*
|
|
* Normalize error.
|
|
*
|
|
LSTRES = ZERO
|
|
DO 80 I = 1, N
|
|
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
|
|
80 CONTINUE
|
|
IF( LSTRES.NE.ZERO )
|
|
$ FERR( J ) = FERR( J ) / LSTRES
|
|
*
|
|
90 CONTINUE
|
|
*
|
|
RETURN
|
|
*
|
|
* End of DPTRFS
|
|
*
|
|
END
|