*> \brief \b DPPRFS
*
* =========== DOCUMENTATION ===========
*
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* http://www.netlib.org/lapack/explore-html/
*
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*
* Definition:
* ===========
*
* SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
* BERR, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
* $ FERR( * ), WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPPRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is symmetric positive definite
*> and packed, and provides error bounds and backward error estimates
*> for the solution.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \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 matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The upper or lower triangle of the symmetric matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in] AFP
*> \verbatim
*> AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
*> packed columnwise in a linear array in the same format as A
*> (see AP).
*> \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 DPPTRS.
*> 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 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 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 (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.
*
*> \date December 2016
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
$ BERR, WORK, IWORK, 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 ..
CHARACTER UPLO
INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
$ 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 ..
LOGICAL UPPER
INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLACN2, DPPTRS, DSPMV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. 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 = -7
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPPRFS', -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 = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - A * X
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
$ 1 )
*
* 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.
*
DO 30 I = 1, N
WORK( I ) = ABS( B( I, J ) )
30 CONTINUE
*
* Compute abs(A)*abs(X) + abs(B).
*
KK = 1
IF( UPPER ) THEN
DO 50 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
IK = KK
DO 40 I = 1, K - 1
WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
IK = IK + 1
40 CONTINUE
WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
KK = KK + K
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
XK = ABS( X( K, J ) )
WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
IK = KK + 1
DO 60 I = K + 1, N
WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
IK = IK + 1
60 CONTINUE
WORK( K ) = WORK( K ) + S
KK = KK + ( N-K+1 )
70 CONTINUE
END IF
S = ZERO
DO 80 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
80 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 DPPTRS( UPLO, N, 1, AFP, 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.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(A) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
DO 90 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
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL DLACN2( 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(A**T).
*
CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
DO 110 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
110 CONTINUE
ELSE IF( KASE.EQ.2 ) THEN
*
* Multiply by inv(A)*diag(W).
*
DO 120 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
120 CONTINUE
CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of DPPRFS
*
END