cla_gbrpvgrw.f

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*> \brief \b CLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
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*
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*
*  Definition:
*  ===========
*
*       REAL FUNCTION CLA_GBRPVGRW( N, KL, KU, NCOLS, AB, LDAB, AFB,
*                                   LDAFB )
*
*       .. Scalar Arguments ..
*       INTEGER            N, KL, KU, NCOLS, LDAB, LDAFB
*       ..
*       .. Array Arguments ..
*       COMPLEX            AB( LDAB, * ), AFB( LDAFB, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CLA_GBRPVGRW computes the reciprocal pivot growth factor
*> norm(A)/norm(U). The "max absolute element" norm is used. If this is
*> much less than 1, the stability of the LU factorization of the
*> (equilibrated) matrix A could be poor. This also means that the
*> solution X, estimated condition numbers, and error bounds could be
*> unreliable.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \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] NCOLS
*> \verbatim
*>          NCOLS is INTEGER
*>     The number of columns of the matrix A.  NCOLS >= 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
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexGBcomputational
*
*  =====================================================================
      REAL function cla_gbrpvgrw( n, kl, ku, ncols, ab, ldab, afb,
     $                            ldafb )
*
*  -- 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 ..
      INTEGER            n, kl, ku, ncols, ldab, ldafb
*     ..
*     .. Array Arguments ..
      COMPLEX            ab( ldab, * ), afb( ldafb, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            i, j, kd
      REAL               amax, umax, rpvgrw
      COMPLEX            zdum
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, real, aimag
*     ..
*     .. Statement Functions ..
      REAL               cabs1
*     ..
*     .. Statement Function Definitions ..
      cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
*     ..
*     .. Executable Statements ..
*
      rpvgrw = 1.0
 
      kd = ku + 1
      DO j = 1, ncols
         amax = 0.0
         umax = 0.0
         DO i = max( j-ku, 1 ), min( j+kl, n )
            amax = max( cabs1( ab( kd+i-j, j ) ), amax )
         END DO
         DO i = max( j-ku, 1 ), j
            umax = max( cabs1( afb( kd+i-j, j ) ), umax )
         END DO
         IF ( umax /= 0.0 ) THEN
            rpvgrw = min( amax / umax, rpvgrw )
         END IF
      END DO
      cla_gbrpvgrw = rpvgrw
*
*     End of CLA_GBRPVGRW
*
      END