sla_gbrcond.f

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*> \brief \b SLA_GBRCOND estimates the Skeel condition number for a general banded matrix.
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       REAL FUNCTION SLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB, AFB, LDAFB,
*                                  IPIV, CMODE, C, INFO, WORK, IWORK )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            N, LDAB, LDAFB, INFO, KL, KU, CMODE
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * ), IPIV( * )
*       REAL               AB( LDAB, * ), AFB( LDAFB, * ), WORK( * ),
*      $                   C( * )
*      ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    SLA_GBRCOND Estimates the Skeel condition number of  op(A) * op2(C)
*>    where op2 is determined by CMODE as follows
*>    CMODE =  1    op2(C) = C
*>    CMODE =  0    op2(C) = I
*>    CMODE = -1    op2(C) = inv(C)
*>    The Skeel condition number  cond(A) = norminf( |inv(A)||A| )
*>    is computed by computing scaling factors R such that
*>    diag(R)*A*op2(C) is row equilibrated and computing the standard
*>    infinity-norm condition number.
*> \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 REAL 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 REAL array, dimension (LDAFB,N)
*>     Details of the LU factorization of the band matrix A, as
*>     computed by SGBTRF.  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 SGBTRF; row i of the matrix was interchanged
*>     with row IPIV(i).
*> \endverbatim
*>
*> \param[in] CMODE
*> \verbatim
*>          CMODE is INTEGER
*>     Determines op2(C) in the formula op(A) * op2(C) as follows:
*>     CMODE =  1    op2(C) = C
*>     CMODE =  0    op2(C) = I
*>     CMODE = -1    op2(C) = inv(C)
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is REAL array, dimension (N)
*>     The vector C in the formula op(A) * op2(C).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>       = 0:  Successful exit.
*>     i > 0:  The ith argument is invalid.
*> \endverbatim
*>
*> \param[in] WORK
*> \verbatim
*>          WORK is REAL array, dimension (5*N).
*>     Workspace.
*> \endverbatim
*>
*> \param[in] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N).
*>     Workspace.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup realGBcomputational
*
*  =====================================================================
      REAL FUNCTION sla_gbrcond( TRANS, N, KL, KU, AB, LDAB, AFB, LDAFB,
     $                           IPIV, CMODE, C, INFO, WORK, IWORK )
*
*  -- LAPACK computational routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            N, LDAB, LDAFB, INFO, KL, KU, CMODE
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * ), IPIV( * )
      REAL               AB( ldab, * ), AFB( ldafb, * ), WORK( * ),
     $                   c( * )
*    ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            NOTRANS
      INTEGER            KASE, I, J, KD, KE
      REAL               AINVNM, TMP
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           slacn2, sgbtrs, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max
*     ..
*     .. Executable Statements ..
*
      sla_gbrcond = 0.0
*
      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( 'SLA_GBRCOND', -info )
         RETURN
      END IF
      IF( n.EQ.0 ) THEN
         sla_gbrcond = 1.0
         RETURN
      END IF
*
*     Compute the equilibration matrix R such that
*     inv(R)*A*C has unit 1-norm.
*
      kd = ku + 1
      ke = kl + 1
      IF ( notrans ) THEN
         DO i = 1, n
            tmp = 0.0
               IF ( cmode .EQ. 1 ) THEN
               DO j = max( i-kl, 1 ), min( i+ku, n )
                  tmp = tmp + abs( ab( kd+i-j, j ) * c( j ) )
               END DO
               ELSE IF ( cmode .EQ. 0 ) THEN
                  DO j = max( i-kl, 1 ), min( i+ku, n )
                     tmp = tmp + abs( ab( kd+i-j, j ) )
                  END DO
               ELSE
                  DO j = max( i-kl, 1 ), min( i+ku, n )
                     tmp = tmp + abs( ab( kd+i-j, j ) / c( j ) )
                  END DO
               END IF
            work( 2*n+i ) = tmp
         END DO
      ELSE
         DO i = 1, n
            tmp = 0.0
            IF ( cmode .EQ. 1 ) THEN
               DO j = max( i-kl, 1 ), min( i+ku, n )
                  tmp = tmp + abs( ab( ke-i+j, i ) * c( j ) )
               END DO
            ELSE IF ( cmode .EQ. 0 ) THEN
               DO j = max( i-kl, 1 ), min( i+ku, n )
                  tmp = tmp + abs( ab( ke-i+j, i ) )
               END DO
            ELSE
               DO j = max( i-kl, 1 ), min( i+ku, n )
                  tmp = tmp + abs( ab( ke-i+j, i ) / c( j ) )
               END DO
            END IF
            work( 2*n+i ) = tmp
         END DO
      END IF
*
*     Estimate the norm of inv(op(A)).
*
      ainvnm = 0.0

      kase = 0
   10 CONTINUE
      CALL slacn2( n, work( n+1 ), work, iwork, ainvnm, kase, isave )
      IF( kase.NE.0 ) THEN
         IF( kase.EQ.2 ) THEN
*
*           Multiply by R.
*
            DO i = 1, n
               work( i ) = work( i ) * work( 2*n+i )
            END DO

            IF ( notrans ) THEN
               CALL sgbtrs( 'No transpose', n, kl, ku, 1, afb, ldafb,
     $              ipiv, work, n, info )
            ELSE
               CALL sgbtrs( 'Transpose', n, kl, ku, 1, afb, ldafb, ipiv,
     $              work, n, info )
            END IF
*
*           Multiply by inv(C).
*
            IF ( cmode .EQ. 1 ) THEN
               DO i = 1, n
                  work( i ) = work( i ) / c( i )
               END DO
            ELSE IF ( cmode .EQ. -1 ) THEN
               DO i = 1, n
                  work( i ) = work( i ) * c( i )
               END DO
            END IF
         ELSE
*
*           Multiply by inv(C**T).
*
            IF ( cmode .EQ. 1 ) THEN
               DO i = 1, n
                  work( i ) = work( i ) / c( i )
               END DO
            ELSE IF ( cmode .EQ. -1 ) THEN
               DO i = 1, n
                  work( i ) = work( i ) * c( i )
               END DO
            END IF

            IF ( notrans ) THEN
               CALL sgbtrs( 'Transpose', n, kl, ku, 1, afb, ldafb, ipiv,
     $              work, n, info )
            ELSE
               CALL sgbtrs( '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 ) * work( 2*n+i )
            END DO
         END IF
         GO TO 10
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( ainvnm .NE. 0.0 )
     $   sla_gbrcond = ( 1.0 / ainvnm )
*
      RETURN
*
      END