zpbrfs.f

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*> \brief \b ZPBRFS
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
*                          LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
*       COMPLEX*16         AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
*      $                   WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZPBRFS improves the computed solution to a system of linear
*> equations when the coefficient matrix is Hermitian positive definite
*> and banded, 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] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 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] AB
*> \verbatim
*>          AB is COMPLEX*16 array, dimension (LDAB,N)
*>          The upper or lower triangle of the Hermitian band matrix A,
*>          stored in the first KD+1 rows of the array.  The j-th column
*>          of A is stored in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] AFB
*> \verbatim
*>          AFB is COMPLEX*16 array, dimension (LDAFB,N)
*>          The triangular factor U or L from the Cholesky factorization
*>          A = U**H*U or A = L*L**H of the band matrix A as computed by
*>          ZPBTRF, in the same storage format as A (see AB).
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*>          LDAFB is INTEGER
*>          The leading dimension of the array AFB.  LDAFB >= KD+1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is COMPLEX*16 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*16 array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by ZPBTRS.
*>          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 COMPLEX*16 array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION 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 pbrfs
*
*  =====================================================================
      SUBROUTINE zpbrfs( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
     $                   LDB, X, LDX, FERR, BERR, WORK, RWORK, 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          UPLO
      INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
      COMPLEX*16         AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
     $                   work( * ), x( ldx, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      parameter( itmax = 5 )
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d+0 )
      COMPLEX*16         ONE
      parameter( one = ( 1.0d+0, 0.0d+0 ) )
      DOUBLE PRECISION   TWO
      parameter( two = 2.0d+0 )
      DOUBLE PRECISION   THREE
      parameter( three = 3.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            COUNT, I, J, K, KASE, L, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
      COMPLEX*16         ZDUM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zaxpy, zcopy, zhbmv, zlacn2, zpbtrs
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dimag, max, min
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           lsame, dlamch
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( 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( kd.LT.0 ) THEN
         info = -3
      ELSE IF( nrhs.LT.0 ) THEN
         info = -4
      ELSE IF( ldab.LT.kd+1 ) THEN
         info = -6
      ELSE IF( ldafb.LT.kd+1 ) THEN
         info = -8
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
         info = -10
      ELSE IF( ldx.LT.max( 1, n ) ) THEN
         info = -12
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZPBRFS', -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 = min( n+1, 2*kd+2 )
      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 zcopy( n, b( 1, j ), 1, work, 1 )
         CALL zhbmv( uplo, n, kd, -one, ab, ldab, x( 1, j ), 1, one,
     $               work, 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
            rwork( i ) = cabs1( b( i, j ) )
   30    CONTINUE
*
*        Compute abs(A)*abs(X) + abs(B).
*
         IF( upper ) THEN
            DO 50 k = 1, n
               s = zero
               xk = cabs1( x( k, j ) )
               l = kd + 1 - k
               DO 40 i = max( 1, k-kd ), k - 1
                  rwork( i ) = rwork( i ) + cabs1( ab( l+i, k ) )*xk
                  s = s + cabs1( ab( l+i, k ) )*cabs1( x( i, j ) )
   40          CONTINUE
               rwork( k ) = rwork( k ) + abs( dble( ab( kd+1, k ) ) )*
     $                      xk + s
   50       CONTINUE
         ELSE
            DO 70 k = 1, n
               s = zero
               xk = cabs1( x( k, j ) )
               rwork( k ) = rwork( k ) + abs( dble( ab( 1, k ) ) )*xk
               l = 1 - k
               DO 60 i = k + 1, min( n, k+kd )
                  rwork( i ) = rwork( i ) + cabs1( ab( l+i, k ) )*xk
                  s = s + cabs1( ab( l+i, k ) )*cabs1( x( i, j ) )
   60          CONTINUE
               rwork( k ) = rwork( k ) + s
   70       CONTINUE
         END IF
         s = zero
         DO 80 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
   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 zpbtrs( uplo, n, kd, 1, afb, ldafb, work, n, info )
            CALL zaxpy( n, 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.
*
*        Use ZLACN2 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( 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
   90    CONTINUE
*
         kase = 0
  100    CONTINUE
         CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
         IF( kase.NE.0 ) THEN
            IF( kase.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(A**H).
*
               CALL zpbtrs( uplo, n, kd, 1, afb, ldafb, work, n, info )
               DO 110 i = 1, n
                  work( i ) = rwork( i )*work( i )
  110          CONTINUE
            ELSE IF( kase.EQ.2 ) THEN
*
*              Multiply by inv(A)*diag(W).
*
               DO 120 i = 1, n
                  work( i ) = rwork( i )*work( i )
  120          CONTINUE
               CALL zpbtrs( uplo, n, kd, 1, afb, ldafb, work, n, info )
            END IF
            GO TO 100
         END IF
*
*        Normalize error.
*
         lstres = zero
         DO 130 i = 1, n
            lstres = max( lstres, cabs1( x( i, j ) ) )
  130    CONTINUE
         IF( lstres.NE.zero )
     $      ferr( j ) = ferr( j ) / lstres
*
  140 CONTINUE
*
      RETURN
*
*     End of ZPBRFS
*
      END