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    1: *> \brief \b ZPTCON
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download ZPTCON + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zptcon.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zptcon.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zptcon.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZPTCON( N, D, E, ANORM, RCOND, RWORK, INFO )
   22: * 
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            INFO, N
   25: *       DOUBLE PRECISION   ANORM, RCOND
   26: *       ..
   27: *       .. Array Arguments ..
   28: *       DOUBLE PRECISION   D( * ), RWORK( * )
   29: *       COMPLEX*16         E( * )
   30: *       ..
   31: *  
   32: *
   33: *> \par Purpose:
   34: *  =============
   35: *>
   36: *> \verbatim
   37: *>
   38: *> ZPTCON computes the reciprocal of the condition number (in the
   39: *> 1-norm) of a complex Hermitian positive definite tridiagonal matrix
   40: *> using the factorization A = L*D*L**H or A = U**H*D*U computed by
   41: *> ZPTTRF.
   42: *>
   43: *> Norm(inv(A)) is computed by a direct method, and the reciprocal of
   44: *> the condition number is computed as
   45: *>                  RCOND = 1 / (ANORM * norm(inv(A))).
   46: *> \endverbatim
   47: *
   48: *  Arguments:
   49: *  ==========
   50: *
   51: *> \param[in] N
   52: *> \verbatim
   53: *>          N is INTEGER
   54: *>          The order of the matrix A.  N >= 0.
   55: *> \endverbatim
   56: *>
   57: *> \param[in] D
   58: *> \verbatim
   59: *>          D is DOUBLE PRECISION array, dimension (N)
   60: *>          The n diagonal elements of the diagonal matrix D from the
   61: *>          factorization of A, as computed by ZPTTRF.
   62: *> \endverbatim
   63: *>
   64: *> \param[in] E
   65: *> \verbatim
   66: *>          E is COMPLEX*16 array, dimension (N-1)
   67: *>          The (n-1) off-diagonal elements of the unit bidiagonal factor
   68: *>          U or L from the factorization of A, as computed by ZPTTRF.
   69: *> \endverbatim
   70: *>
   71: *> \param[in] ANORM
   72: *> \verbatim
   73: *>          ANORM is DOUBLE PRECISION
   74: *>          The 1-norm of the original matrix A.
   75: *> \endverbatim
   76: *>
   77: *> \param[out] RCOND
   78: *> \verbatim
   79: *>          RCOND is DOUBLE PRECISION
   80: *>          The reciprocal of the condition number of the matrix A,
   81: *>          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
   82: *>          1-norm of inv(A) computed in this routine.
   83: *> \endverbatim
   84: *>
   85: *> \param[out] RWORK
   86: *> \verbatim
   87: *>          RWORK is DOUBLE PRECISION array, dimension (N)
   88: *> \endverbatim
   89: *>
   90: *> \param[out] INFO
   91: *> \verbatim
   92: *>          INFO is INTEGER
   93: *>          = 0:  successful exit
   94: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
   95: *> \endverbatim
   96: *
   97: *  Authors:
   98: *  ========
   99: *
  100: *> \author Univ. of Tennessee 
  101: *> \author Univ. of California Berkeley 
  102: *> \author Univ. of Colorado Denver 
  103: *> \author NAG Ltd. 
  104: *
  105: *> \date November 2011
  106: *
  107: *> \ingroup complex16OTHERcomputational
  108: *
  109: *> \par Further Details:
  110: *  =====================
  111: *>
  112: *> \verbatim
  113: *>
  114: *>  The method used is described in Nicholas J. Higham, "Efficient
  115: *>  Algorithms for Computing the Condition Number of a Tridiagonal
  116: *>  Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
  117: *> \endverbatim
  118: *>
  119: *  =====================================================================
  120:       SUBROUTINE ZPTCON( N, D, E, ANORM, RCOND, RWORK, INFO )
  121: *
  122: *  -- LAPACK computational routine (version 3.4.0) --
  123: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  124: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  125: *     November 2011
  126: *
  127: *     .. Scalar Arguments ..
  128:       INTEGER            INFO, N
  129:       DOUBLE PRECISION   ANORM, RCOND
  130: *     ..
  131: *     .. Array Arguments ..
  132:       DOUBLE PRECISION   D( * ), RWORK( * )
  133:       COMPLEX*16         E( * )
  134: *     ..
  135: *
  136: *  =====================================================================
  137: *
  138: *     .. Parameters ..
  139:       DOUBLE PRECISION   ONE, ZERO
  140:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
  141: *     ..
  142: *     .. Local Scalars ..
  143:       INTEGER            I, IX
  144:       DOUBLE PRECISION   AINVNM
  145: *     ..
  146: *     .. External Functions ..
  147:       INTEGER            IDAMAX
  148:       EXTERNAL           IDAMAX
  149: *     ..
  150: *     .. External Subroutines ..
  151:       EXTERNAL           XERBLA
  152: *     ..
  153: *     .. Intrinsic Functions ..
  154:       INTRINSIC          ABS
  155: *     ..
  156: *     .. Executable Statements ..
  157: *
  158: *     Test the input arguments.
  159: *
  160:       INFO = 0
  161:       IF( N.LT.0 ) THEN
  162:          INFO = -1
  163:       ELSE IF( ANORM.LT.ZERO ) THEN
  164:          INFO = -4
  165:       END IF
  166:       IF( INFO.NE.0 ) THEN
  167:          CALL XERBLA( 'ZPTCON', -INFO )
  168:          RETURN
  169:       END IF
  170: *
  171: *     Quick return if possible
  172: *
  173:       RCOND = ZERO
  174:       IF( N.EQ.0 ) THEN
  175:          RCOND = ONE
  176:          RETURN
  177:       ELSE IF( ANORM.EQ.ZERO ) THEN
  178:          RETURN
  179:       END IF
  180: *
  181: *     Check that D(1:N) is positive.
  182: *
  183:       DO 10 I = 1, N
  184:          IF( D( I ).LE.ZERO )
  185:      $      RETURN
  186:    10 CONTINUE
  187: *
  188: *     Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
  189: *
  190: *        m(i,j) =  abs(A(i,j)), i = j,
  191: *        m(i,j) = -abs(A(i,j)), i .ne. j,
  192: *
  193: *     and e = [ 1, 1, ..., 1 ]**T.  Note M(A) = M(L)*D*M(L)**H.
  194: *
  195: *     Solve M(L) * x = e.
  196: *
  197:       RWORK( 1 ) = ONE
  198:       DO 20 I = 2, N
  199:          RWORK( I ) = ONE + RWORK( I-1 )*ABS( E( I-1 ) )
  200:    20 CONTINUE
  201: *
  202: *     Solve D * M(L)**H * x = b.
  203: *
  204:       RWORK( N ) = RWORK( N ) / D( N )
  205:       DO 30 I = N - 1, 1, -1
  206:          RWORK( I ) = RWORK( I ) / D( I ) + RWORK( I+1 )*ABS( E( I ) )
  207:    30 CONTINUE
  208: *
  209: *     Compute AINVNM = max(x(i)), 1<=i<=n.
  210: *
  211:       IX = IDAMAX( N, RWORK, 1 )
  212:       AINVNM = ABS( RWORK( IX ) )
  213: *
  214: *     Compute the reciprocal condition number.
  215: *
  216:       IF( AINVNM.NE.ZERO )
  217:      $   RCOND = ( ONE / AINVNM ) / ANORM
  218: *
  219:       RETURN
  220: *
  221: *     End of ZPTCON
  222: *
  223:       END