*> \brief \b CPTRFS
*
* =========== DOCUMENTATION ===========
*
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* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*
* Definition:
* ===========
*
* SUBROUTINE CPTRFS( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
* FERR, BERR, WORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* REAL BERR( * ), D( * ), DF( * ), FERR( * ),
* $ RWORK( * )
* COMPLEX B( LDB, * ), E( * ), EF( * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CPTRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is Hermitian positive definite
*> and tridiagonal, and provides error bounds and backward error
*> estimates for the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the superdiagonal or the subdiagonal of the
*> tridiagonal matrix A is stored and the form of the
*> factorization:
*> = 'U': E is the superdiagonal of A, and A = U**H*D*U;
*> = 'L': E is the subdiagonal of A, and A = L*D*L**H.
*> (The two forms are equivalent if A is real.)
*> \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] D
*> \verbatim
*> D is REAL array, dimension (N)
*> The n real diagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is COMPLEX array, dimension (N-1)
*> The (n-1) off-diagonal elements of the tridiagonal matrix A
*> (see UPLO).
*> \endverbatim
*>
*> \param[in] DF
*> \verbatim
*> DF is REAL array, dimension (N)
*> The n diagonal elements of the diagonal matrix D from
*> the factorization computed by CPTTRF.
*> \endverbatim
*>
*> \param[in] EF
*> \verbatim
*> EF is COMPLEX array, dimension (N-1)
*> The (n-1) off-diagonal elements of the unit bidiagonal
*> factor U or L from the factorization computed by CPTTRF
*> (see UPLO).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)
*> On entry, the solution matrix X, as computed by CPTTRS.
*> 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 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 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 COMPLEX array, dimension (N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL 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.
*
*> \date September 2012
*
*> \ingroup complexPTcomputational
*
* =====================================================================
SUBROUTINE CPTRFS( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
$ FERR, BERR, WORK, RWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
REAL BERR( * ), D( * ), DF( * ), FERR( * ),
$ RWORK( * )
COMPLEX B( LDB, * ), E( * ), EF( * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
REAL ONE
PARAMETER ( ONE = 1.0E+0 )
REAL TWO
PARAMETER ( TWO = 2.0E+0 )
REAL THREE
PARAMETER ( THREE = 3.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER COUNT, I, IX, J, NZ
REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
COMPLEX BI, CX, DX, EX, ZDUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ISAMAX
REAL SLAMCH
EXTERNAL LSAME, ISAMAX, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CAXPY, CPTTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, REAL
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) 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 = -9
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CPTRFS', -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 = SLAMCH( 'Epsilon' )
SAFMIN = SLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 100 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( UPPER ) THEN
IF( N.EQ.1 ) THEN
BI = B( 1, J )
DX = D( 1 )*X( 1, J )
WORK( 1 ) = BI - DX
RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
ELSE
BI = B( 1, J )
DX = D( 1 )*X( 1, J )
EX = E( 1 )*X( 2, J )
WORK( 1 ) = BI - DX - EX
RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
$ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
DO 30 I = 2, N - 1
BI = B( I, J )
CX = CONJG( E( I-1 ) )*X( I-1, J )
DX = D( I )*X( I, J )
EX = E( I )*X( I+1, J )
WORK( I ) = BI - CX - DX - EX
RWORK( I ) = CABS1( BI ) +
$ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
$ CABS1( DX ) + CABS1( E( I ) )*
$ CABS1( X( I+1, J ) )
30 CONTINUE
BI = B( N, J )
CX = CONJG( E( N-1 ) )*X( N-1, J )
DX = D( N )*X( N, J )
WORK( N ) = BI - CX - DX
RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
$ CABS1( X( N-1, J ) ) + CABS1( DX )
END IF
ELSE
IF( N.EQ.1 ) THEN
BI = B( 1, J )
DX = D( 1 )*X( 1, J )
WORK( 1 ) = BI - DX
RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
ELSE
BI = B( 1, J )
DX = D( 1 )*X( 1, J )
EX = CONJG( E( 1 ) )*X( 2, J )
WORK( 1 ) = BI - DX - EX
RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
$ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
DO 40 I = 2, N - 1
BI = B( I, J )
CX = E( I-1 )*X( I-1, J )
DX = D( I )*X( I, J )
EX = CONJG( E( I ) )*X( I+1, J )
WORK( I ) = BI - CX - DX - EX
RWORK( I ) = CABS1( BI ) +
$ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
$ CABS1( DX ) + CABS1( E( I ) )*
$ CABS1( X( I+1, J ) )
40 CONTINUE
BI = B( N, J )
CX = E( N-1 )*X( N-1, J )
DX = D( N )*X( N, J )
WORK( N ) = BI - CX - DX
RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
$ CABS1( X( N-1, J ) ) + CABS1( DX )
END IF
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 50 I = 1, N
IF( RWORK( I ).GT.SAFE2 ) THEN
S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
ELSE
S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
$ ( RWORK( 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 CPTTRS( UPLO, N, 1, DF, EF, WORK, N, INFO )
CALL CAXPY( N, CMPLX( ONE ), WORK, 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 60 I = 1, N
IF( RWORK( I ).GT.SAFE2 ) THEN
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
ELSE
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
$ SAFE1
END IF
60 CONTINUE
IX = ISAMAX( N, RWORK, 1 )
FERR( J ) = RWORK( 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)**H.
*
* Solve M(L) * x = e.
*
RWORK( 1 ) = ONE
DO 70 I = 2, N
RWORK( I ) = ONE + RWORK( I-1 )*ABS( EF( I-1 ) )
70 CONTINUE
*
* Solve D * M(L)**H * x = b.
*
RWORK( N ) = RWORK( N ) / DF( N )
DO 80 I = N - 1, 1, -1
RWORK( I ) = RWORK( I ) / DF( I ) +
$ RWORK( I+1 )*ABS( EF( I ) )
80 CONTINUE
*
* Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
*
IX = ISAMAX( N, RWORK, 1 )
FERR( J ) = FERR( J )*ABS( RWORK( IX ) )
*
* Normalize error.
*
LSTRES = ZERO
DO 90 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
90 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
100 CONTINUE
*
RETURN
*
* End of CPTRFS
*
END