zptrfs.f

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*> \brief \b ZPTRFS
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZPTRFS( 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 ..
*       DOUBLE PRECISION   BERR( * ), D( * ), DF( * ), FERR( * ),
*      $                   RWORK( * )
*       COMPLEX*16         B( LDB, * ), E( * ), EF( * ), WORK( * ),
*      $                   X( LDX, * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZPTRFS 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 DOUBLE PRECISION array, dimension (N)
*>          The n real diagonal elements of the tridiagonal matrix A.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is COMPLEX*16 array, dimension (N-1)
*>          The (n-1) off-diagonal elements of the tridiagonal matrix A
*>          (see UPLO).
*> \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 ZPTTRF.
*> \endverbatim
*>
*> \param[in] EF
*> \verbatim
*>          EF is COMPLEX*16 array, dimension (N-1)
*>          The (n-1) off-diagonal elements of the unit bidiagonal
*>          factor U or L from the factorization computed by ZPTTRF
*>          (see UPLO).
*> \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 ZPTTRS.
*>          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 COMPLEX*16 array, dimension (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. 
*
*> \date September 2012
*
*> \ingroup complex16PTcomputational
*
*  =====================================================================
      SUBROUTINE zptrfs( 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 ..
      DOUBLE PRECISION   BERR( * ), D( * ), DF( * ), FERR( * ),
     $                   rwork( * )
      COMPLEX*16         B( ldb, * ), E( * ), EF( * ), 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, IX, J, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
      COMPLEX*16         BI, CX, DX, EX, ZDUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           lsame, idamax, dlamch
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zaxpy, zpttrs
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dcmplx, dconjg, dimag, max
*     ..
*     .. 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( 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( 'ZPTRFS', -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 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 = dconjg( 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 = dconjg( 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 = dconjg( 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 = dconjg( 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 zpttrs( uplo, n, 1, df, ef, work, n, info )
            CALL zaxpy( n, dcmplx( 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 = idamax( 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 = idamax( 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 ZPTRFS
*
      END