sgbcon.f

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*> \brief \b SGBCON
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE SGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          NORM
*       INTEGER            INFO, KL, KU, LDAB, N
*       REAL               ANORM, RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       REAL               AB( LDAB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SGBCON estimates the reciprocal of the condition number of a real
*> general band matrix A, in either the 1-norm or the infinity-norm,
*> using the LU factorization computed by SGBTRF.
*>
*> An estimate is obtained for norm(inv(A)), and the reciprocal of the
*> condition number is computed as
*>    RCOND = 1 / ( norm(A) * norm(inv(A)) ).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NORM
*> \verbatim
*>          NORM is CHARACTER*1
*>          Specifies whether the 1-norm condition number or the
*>          infinity-norm condition number is required:
*>          = '1' or 'O':  1-norm;
*>          = 'I':         Infinity-norm.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          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)
*>          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] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          The pivot indices; for 1 <= i <= N, row i of the matrix was
*>          interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*>          ANORM is REAL
*>          If NORM = '1' or 'O', the 1-norm of the original matrix A.
*>          If NORM = 'I', the infinity-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is REAL
*>          The reciprocal of the condition number of the matrix A,
*>          computed as RCOND = 1/(norm(A) * norm(inv(A))).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL 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.
*
*> \date December 2016
*
*> \ingroup realGBcomputational
*
*  =====================================================================
      SUBROUTINE sgbcon( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND,
     $                   WORK, IWORK, INFO )
*
*  -- 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          NORM
      INTEGER            INFO, KL, KU, LDAB, N
      REAL               ANORM, RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      REAL               AB( ldab, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE, ZERO
      parameter( one = 1.0e+0, zero = 0.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LNOTI, ONENRM
      CHARACTER          NORMIN
      INTEGER            IX, J, JP, KASE, KASE1, KD, LM
      REAL               AINVNM, SCALE, SMLNUM, T
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ISAMAX
      REAL               SDOT, SLAMCH
      EXTERNAL           lsame, isamax, sdot, slamch
*     ..
*     .. External Subroutines ..
      EXTERNAL           saxpy, slacn2, slatbs, srscl, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, min
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      onenrm = norm.EQ.'1' .OR. lsame( norm, 'O' )
      IF( .NOT.onenrm .AND. .NOT.lsame( norm, 'I' ) ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( kl.LT.0 ) THEN
         info = -3
      ELSE IF( ku.LT.0 ) THEN
         info = -4
      ELSE IF( ldab.LT.2*kl+ku+1 ) THEN
         info = -6
      ELSE IF( anorm.LT.zero ) THEN
         info = -8
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SGBCON', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      rcond = zero
      IF( n.EQ.0 ) THEN
         rcond = one
         RETURN
      ELSE IF( anorm.EQ.zero ) THEN
         RETURN
      END IF
*
      smlnum = slamch( 'Safe minimum' )
*
*     Estimate the norm of inv(A).
*
      ainvnm = zero
      normin = 'N'
      IF( onenrm ) THEN
         kase1 = 1
      ELSE
         kase1 = 2
      END IF
      kd = kl + ku + 1
      lnoti = kl.GT.0
      kase = 0
   10 CONTINUE
      CALL slacn2( n, work( n+1 ), work, iwork, ainvnm, kase, isave )
      IF( kase.NE.0 ) THEN
         IF( kase.EQ.kase1 ) THEN
*
*           Multiply by inv(L).
*
            IF( lnoti ) THEN
               DO 20 j = 1, n - 1
                  lm = min( kl, n-j )
                  jp = ipiv( j )
                  t = work( jp )
                  IF( jp.NE.j ) THEN
                     work( jp ) = work( j )
                     work( j ) = t
                  END IF
                  CALL saxpy( lm, -t, ab( kd+1, j ), 1, work( j+1 ), 1 )
   20          CONTINUE
            END IF
*
*           Multiply by inv(U).
*
            CALL slatbs( 'Upper', 'No transpose', 'Non-unit', normin, n,
     $                   kl+ku, ab, ldab, work, scale, work( 2*n+1 ),
     $                   info )
         ELSE
*
*           Multiply by inv(U**T).
*
            CALL slatbs( 'Upper', 'Transpose', 'Non-unit', normin, n,
     $                   kl+ku, ab, ldab, work, scale, work( 2*n+1 ),
     $                   info )
*
*           Multiply by inv(L**T).
*
            IF( lnoti ) THEN
               DO 30 j = n - 1, 1, -1
                  lm = min( kl, n-j )
                  work( j ) = work( j ) - sdot( lm, ab( kd+1, j ), 1,
     $                        work( j+1 ), 1 )
                  jp = ipiv( j )
                  IF( jp.NE.j ) THEN
                     t = work( jp )
                     work( jp ) = work( j )
                     work( j ) = t
                  END IF
   30          CONTINUE
            END IF
         END IF
*
*        Divide X by 1/SCALE if doing so will not cause overflow.
*
         normin = 'Y'
         IF( scale.NE.one ) THEN
            ix = isamax( n, work, 1 )
            IF( scale.LT.abs( work( ix ) )*smlnum .OR. scale.EQ.zero )
     $         GO TO 40
            CALL srscl( n, scale, work, 1 )
         END IF
         GO TO 10
      END IF
*
*     Compute the estimate of the reciprocal condition number.
*
      IF( ainvnm.NE.zero )
     $   rcond = ( one / ainvnm ) / anorm
*
   40 CONTINUE
      RETURN
*
*     End of SGBCON
*
      END