cla_gbrcond_x.f

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*> \brief \b CLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       REAL FUNCTION CLA_GBRCOND_X( TRANS, N, KL, KU, AB, LDAB, AFB,
*                                    LDAFB, IPIV, X, INFO, WORK, RWORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            N, KL, KU, KD, KE, LDAB, LDAFB, INFO
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       COMPLEX            AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ),
*      $                   X( * )
*       REAL               RWORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    CLA_GBRCOND_X Computes the infinity norm condition number of
*>    op(A) * diag(X) where X is a COMPLEX vector.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \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] N
*> \verbatim
*>          N is INTEGER
*>     The number of linear equations, i.e., the order of the
*>     matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>     The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>     The number of superdiagonals within the band of A.  KU >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is COMPLEX array, dimension (LDAB,N)
*>     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
*>     The j-th column of A is stored in the j-th column of the
*>     array AB as follows:
*>     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>     The leading dimension of the array AB.  LDAB >= KL+KU+1.
*> \endverbatim
*>
*> \param[in] AFB
*> \verbatim
*>          AFB is COMPLEX array, dimension (LDAFB,N)
*>     Details of the LU factorization of the band matrix A, as
*>     computed by CGBTRF.  U is stored as an upper triangular
*>     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
*>     and the multipliers used during the factorization are stored
*>     in rows KL+KU+2 to 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*>          LDAFB is INTEGER
*>     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>     The pivot indices from the factorization A = P*L*U
*>     as computed by CGBTRF; row i of the matrix was interchanged
*>     with row IPIV(i).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*>          X is COMPLEX array, dimension (N)
*>     The vector X in the formula op(A) * diag(X).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>       = 0:  Successful exit.
*>     i > 0:  The ith argument is invalid.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX array, dimension (2*N).
*>     Workspace.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (N).
*>     Workspace.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup la_gbrcond
*
*  =====================================================================
      REAL function cla_gbrcond_x( trans, n, kl, ku, ab, ldab, afb,
     $                             ldafb, ipiv, x, info, work, rwork )
*
*  -- 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          trans
      INTEGER            n, kl, ku, kd, ke, ldab, ldafb, info
*     ..
*     .. Array Arguments ..
      INTEGER            ipiv( * )
      COMPLEX            ab( ldab, * ), afb( ldafb, * ), work( * ),
     $                   x( * )
      REAL               rwork( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            notrans
      INTEGER            kase, i, j
      REAL               ainvnm, anorm, tmp
      COMPLEX            zdum
*     ..
*     .. Local Arrays ..
      INTEGER            isave( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            lsame
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           clacn2, cgbtrs, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max
*     ..
*     .. Statement Functions ..
      REAL               cabs1
*     ..
*     .. Statement Function Definitions ..
      cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
*     ..
*     .. Executable Statements ..
*
      cla_gbrcond_x = 0.0e+0
*
      info = 0
      notrans = lsame( trans, 'N' )
      IF ( .NOT. notrans .AND. .NOT. lsame(trans, 'T') .AND. .NOT.
     $     lsame( trans, 'C' ) ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( kl.LT.0 .OR. kl.GT.n-1 ) THEN
         info = -3
      ELSE IF( ku.LT.0 .OR. ku.GT.n-1 ) THEN
         info = -4
      ELSE IF( ldab.LT.kl+ku+1 ) THEN
         info = -6
      ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
         info = -8
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'CLA_GBRCOND_X', -info )
         RETURN
      END IF
*
*     Compute norm of op(A)*op2(C).
*
      kd = ku + 1
      ke = kl + 1
      anorm = 0.0
      IF ( notrans ) THEN
         DO i = 1, n
            tmp = 0.0e+0
            DO j = max( i-kl, 1 ), min( i+ku, n )
               tmp = tmp + cabs1( ab( kd+i-j, j) * x( j ) )
            END DO
            rwork( i ) = tmp
            anorm = max( anorm, tmp )
         END DO
      ELSE
         DO i = 1, n
            tmp = 0.0e+0
            DO j = max( i-kl, 1 ), min( i+ku, n )
               tmp = tmp + cabs1( ab( ke-i+j, i ) * x( j ) )
            END DO
            rwork( i ) = tmp
            anorm = max( anorm, tmp )
         END DO
      END IF
*
*     Quick return if possible.
*
      IF( n.EQ.0 ) THEN
         cla_gbrcond_x = 1.0e+0
         RETURN
      ELSE IF( anorm .EQ. 0.0e+0 ) THEN
         RETURN
      END IF
*
*     Estimate the norm of inv(op(A)).
*
      ainvnm = 0.0e+0
*
      kase = 0
   10 CONTINUE
      CALL clacn2( n, work( n+1 ), work, ainvnm, kase, isave )
      IF( kase.NE.0 ) THEN
         IF( kase.EQ.2 ) THEN
*
*           Multiply by R.
*
            DO i = 1, n
               work( i ) = work( i ) * rwork( i )
            END DO
*
            IF ( notrans ) THEN
               CALL cgbtrs( 'No transpose', n, kl, ku, 1, afb, ldafb,
     $              ipiv, work, n, info )
            ELSE
               CALL cgbtrs( 'Conjugate transpose', n, kl, ku, 1, afb,
     $              ldafb, ipiv, work, n, info )
            ENDIF
*
*           Multiply by inv(X).
*
            DO i = 1, n
               work( i ) = work( i ) / x( i )
            END DO
         ELSE
*
*           Multiply by inv(X**H).
*
            DO i = 1, n
               work( i ) = work( i ) / x( i )
            END DO
*
            IF ( notrans ) THEN
               CALL cgbtrs( 'Conjugate transpose', n, kl, ku, 1, afb,
     $              ldafb, ipiv, work, n, info )
            ELSE
               CALL cgbtrs( 'No transpose', n, kl, ku, 1, afb, ldafb,
     $              ipiv, work, n, info )
            END IF
*
*           Multiply by R.
*
            DO i = 1, n
               work( i ) = work( i ) * rwork( i )
            END DO
         END IF
         GO TO 10
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( ainvnm .NE. 0.0e+0 )
     $   cla_gbrcond_x = 1.0e+0 / ainvnm
*
      RETURN
*
*     End of CLA_GBRCOND_X
*
      END