dtbrfs.f

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*> \brief \b DTBRFS
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
*                          LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, TRANS, UPLO
*       INTEGER            INFO, KD, LDAB, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), BERR( * ),
*      $                   FERR( * ), WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTBRFS provides error bounds and backward error estimates for the
*> solution to a system of linear equations with a triangular band
*> coefficient matrix.
*>
*> The solution matrix X must be computed by DTBTRS or some other
*> means before entering this routine.  DTBRFS does not do iterative
*> refinement because doing so cannot improve the backward error.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  A is upper triangular;
*>          = 'L':  A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations:
*>          = 'N':  A * X = B  (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          = 'N':  A is non-unit triangular;
*>          = 'U':  A is unit triangular.
*> \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 or subdiagonals of the
*>          triangular band matrix A.  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 DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangular 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).
*>          If DIAG = 'U', the diagonal elements of A are not referenced
*>          and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \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] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          The 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
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tbrfs
*
*  =====================================================================
      SUBROUTINE dtbrfs( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
     $                   LDB, X, LDX, FERR, BERR, WORK, IWORK, 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          DIAG, TRANS, UPLO
      INTEGER            INFO, KD, LDAB, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), BERR( * ),
     $                   ferr( * ), work( * ), x( ldx, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d+0 )
      DOUBLE PRECISION   ONE
      parameter( one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN, NOUNIT, UPPER
      CHARACTER          TRANST
      INTEGER            I, J, K, KASE, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           daxpy, dcopy, dlacn2, dtbmv, dtbsv,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           lsame, dlamch
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      upper = lsame( uplo, 'U' )
      notran = lsame( trans, 'N' )
      nounit = lsame( diag, 'N' )
*
      IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
         info = -1
      ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
     $         lsame( trans, 'C' ) ) THEN
         info = -2
      ELSE IF( .NOT.nounit .AND. .NOT.lsame( diag, 'U' ) ) THEN
         info = -3
      ELSE IF( n.LT.0 ) THEN
         info = -4
      ELSE IF( kd.LT.0 ) THEN
         info = -5
      ELSE IF( nrhs.LT.0 ) THEN
         info = -6
      ELSE IF( ldab.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( 'DTBRFS', -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
*
      IF( notran ) THEN
         transt = 'T'
      ELSE
         transt = 'N'
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      nz = kd + 2
      eps = dlamch( 'Epsilon' )
      safmin = dlamch( 'Safe minimum' )
      safe1 = nz*safmin
      safe2 = safe1 / eps
*
*     Do for each right hand side
*
      DO 250 j = 1, nrhs
*
*        Compute residual R = B - op(A) * X,
*        where op(A) = A or A**T, depending on TRANS.
*
         CALL dcopy( n, x( 1, j ), 1, work( n+1 ), 1 )
         CALL dtbmv( uplo, trans, diag, n, kd, ab, ldab, work( n+1 ),
     $               1 )
         CALL daxpy( n, -one, b( 1, j ), 1, work( n+1 ), 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(op(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 20 i = 1, n
            work( i ) = abs( b( i, j ) )
   20    CONTINUE
*
         IF( notran ) THEN
*
*           Compute abs(A)*abs(X) + abs(B).
*
            IF( upper ) THEN
               IF( nounit ) THEN
                  DO 40 k = 1, n
                     xk = abs( x( k, j ) )
                     DO 30 i = max( 1, k-kd ), k
                        work( i ) = work( i ) +
     $                              abs( ab( kd+1+i-k, k ) )*xk
   30                CONTINUE
   40             CONTINUE
               ELSE
                  DO 60 k = 1, n
                     xk = abs( x( k, j ) )
                     DO 50 i = max( 1, k-kd ), k - 1
                        work( i ) = work( i ) +
     $                              abs( ab( kd+1+i-k, k ) )*xk
   50                CONTINUE
                     work( k ) = work( k ) + xk
   60             CONTINUE
               END IF
            ELSE
               IF( nounit ) THEN
                  DO 80 k = 1, n
                     xk = abs( x( k, j ) )
                     DO 70 i = k, min( n, k+kd )
                        work( i ) = work( i ) + abs( ab( 1+i-k, k ) )*xk
   70                CONTINUE
   80             CONTINUE
               ELSE
                  DO 100 k = 1, n
                     xk = abs( x( k, j ) )
                     DO 90 i = k + 1, min( n, k+kd )
                        work( i ) = work( i ) + abs( ab( 1+i-k, k ) )*xk
   90                CONTINUE
                     work( k ) = work( k ) + xk
  100             CONTINUE
               END IF
            END IF
         ELSE
*
*           Compute abs(A**T)*abs(X) + abs(B).
*
            IF( upper ) THEN
               IF( nounit ) THEN
                  DO 120 k = 1, n
                     s = zero
                     DO 110 i = max( 1, k-kd ), k
                        s = s + abs( ab( kd+1+i-k, k ) )*
     $                      abs( x( i, j ) )
  110                CONTINUE
                     work( k ) = work( k ) + s
  120             CONTINUE
               ELSE
                  DO 140 k = 1, n
                     s = abs( x( k, j ) )
                     DO 130 i = max( 1, k-kd ), k - 1
                        s = s + abs( ab( kd+1+i-k, k ) )*
     $                      abs( x( i, j ) )
  130                CONTINUE
                     work( k ) = work( k ) + s
  140             CONTINUE
               END IF
            ELSE
               IF( nounit ) THEN
                  DO 160 k = 1, n
                     s = zero
                     DO 150 i = k, min( n, k+kd )
                        s = s + abs( ab( 1+i-k, k ) )*abs( x( i, j ) )
  150                CONTINUE
                     work( k ) = work( k ) + s
  160             CONTINUE
               ELSE
                  DO 180 k = 1, n
                     s = abs( x( k, j ) )
                     DO 170 i = k + 1, min( n, k+kd )
                        s = s + abs( ab( 1+i-k, k ) )*abs( x( i, j ) )
  170                CONTINUE
                     work( k ) = work( k ) + s
  180             CONTINUE
               END IF
            END IF
         END IF
         s = zero
         DO 190 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
  190    CONTINUE
         berr( j ) = s
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(op(A)))*
*           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(op(A)) is the inverse of op(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(op(A))*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
*        Use DLACN2 to estimate the infinity-norm of the matrix
*           inv(op(A)) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
         DO 200 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
  200    CONTINUE
*
         kase = 0
  210    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(op(A)**T).
*
               CALL dtbsv( uplo, transt, diag, n, kd, ab, ldab,
     $                     work( n+1 ), 1 )
               DO 220 i = 1, n
                  work( n+i ) = work( i )*work( n+i )
  220          CONTINUE
            ELSE
*
*              Multiply by inv(op(A))*diag(W).
*
               DO 230 i = 1, n
                  work( n+i ) = work( i )*work( n+i )
  230          CONTINUE
               CALL dtbsv( uplo, trans, diag, n, kd, ab, ldab,
     $                     work( n+1 ), 1 )
            END IF
            GO TO 210
         END IF
*
*        Normalize error.
*
         lstres = zero
         DO 240 i = 1, n
            lstres = max( lstres, abs( x( i, j ) ) )
  240    CONTINUE
         IF( lstres.NE.zero )
     $      ferr( j ) = ferr( j ) / lstres
*
  250 CONTINUE
*
      RETURN
*
*     End of DTBRFS
*
      END