zlacn2.f

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*> \brief \b ZLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZLACN2( N, V, X, EST, KASE, ISAVE )
* 
*       .. Scalar Arguments ..
*       INTEGER            KASE, N
*       DOUBLE PRECISION   EST
*       ..
*       .. Array Arguments ..
*       INTEGER            ISAVE( 3 )
*       COMPLEX*16         V( * ), X( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZLACN2 estimates the 1-norm of a square, complex matrix A.
*> Reverse communication is used for evaluating matrix-vector products.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The order of the matrix.  N >= 1.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is COMPLEX*16 array, dimension (N)
*>         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
*>         (W is not returned).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is COMPLEX*16 array, dimension (N)
*>         On an intermediate return, X should be overwritten by
*>               A * X,   if KASE=1,
*>               A**H * X,  if KASE=2,
*>         where A**H is the conjugate transpose of A, and ZLACN2 must be
*>         re-called with all the other parameters unchanged.
*> \endverbatim
*>
*> \param[in,out] EST
*> \verbatim
*>          EST is DOUBLE PRECISION
*>         On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
*>         unchanged from the previous call to ZLACN2.
*>         On exit, EST is an estimate (a lower bound) for norm(A). 
*> \endverbatim
*>
*> \param[in,out] KASE
*> \verbatim
*>          KASE is INTEGER
*>         On the initial call to ZLACN2, KASE should be 0.
*>         On an intermediate return, KASE will be 1 or 2, indicating
*>         whether X should be overwritten by A * X  or A**H * X.
*>         On the final return from ZLACN2, KASE will again be 0.
*> \endverbatim
*>
*> \param[in,out] ISAVE
*> \verbatim
*>          ISAVE is INTEGER array, dimension (3)
*>         ISAVE is used to save variables between calls to ZLACN2
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date September 2012
*
*> \ingroup complex16OTHERauxiliary
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Originally named CONEST, dated March 16, 1988.
*>
*>  Last modified:  April, 1999
*>
*>  This is a thread safe version of ZLACON, which uses the array ISAVE
*>  in place of a SAVE statement, as follows:
*>
*>     ZLACON     ZLACN2
*>      JUMP     ISAVE(1)
*>      J        ISAVE(2)
*>      ITER     ISAVE(3)
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Nick Higham, University of Manchester
*
*> \par References:
*  ================
*>
*>  N.J. Higham, "FORTRAN codes for estimating the one-norm of
*>  a real or complex matrix, with applications to condition estimation",
*>  ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
*>
*  =====================================================================
      SUBROUTINE zlacn2( N, V, X, EST, KASE, ISAVE )
*
*  -- LAPACK auxiliary routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      INTEGER            KASE, N
      DOUBLE PRECISION   EST
*     ..
*     .. Array Arguments ..
      INTEGER            ISAVE( 3 )
      COMPLEX*16         V( * ), X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER              ITMAX
      parameter                ( itmax = 5 )
      DOUBLE PRECISION     ONE,         TWO
      parameter                ( one = 1.0d0, two = 2.0d0 )
      COMPLEX*16           CZERO, CONE
      parameter                ( czero = ( 0.0d0, 0.0d0 ),
     $                            cone = ( 1.0d0, 0.0d0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER            I, JLAST
      DOUBLE PRECISION   ABSXI, ALTSGN, ESTOLD, SAFMIN, TEMP
*     ..
*     .. External Functions ..
      INTEGER            IZMAX1
      DOUBLE PRECISION   DLAMCH, DZSUM1
      EXTERNAL           izmax1, dlamch, dzsum1
*     ..
*     .. External Subroutines ..
      EXTERNAL           zcopy
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dcmplx, dimag
*     ..
*     .. Executable Statements ..
*
      safmin = dlamch( 'Safe minimum' )
      IF( kase.EQ.0 ) THEN
         DO 10 i = 1, n
            x( i ) = dcmplx( one / dble( n ) )
   10    CONTINUE
         kase = 1
         isave( 1 ) = 1
         RETURN
      END IF
*
      GO TO ( 20, 40, 70, 90, 120 )isave( 1 )
*
*     ................ ENTRY   (ISAVE( 1 ) = 1)
*     FIRST ITERATION.  X HAS BEEN OVERWRITTEN BY A*X.
*
   20 CONTINUE
      IF( n.EQ.1 ) THEN
         v( 1 ) = x( 1 )
         est = abs( v( 1 ) )
*        ... QUIT
         GO TO 130
      END IF
      est = dzsum1( n, x, 1 )
*
      DO 30 i = 1, n
         absxi = abs( x( i ) )
         IF( absxi.GT.safmin ) THEN
            x( i ) = dcmplx( dble( x( i ) ) / absxi,
     $               dimag( x( i ) ) / absxi )
         ELSE
            x( i ) = cone
         END IF
   30 CONTINUE
      kase = 2
      isave( 1 ) = 2
      RETURN
*
*     ................ ENTRY   (ISAVE( 1 ) = 2)
*     FIRST ITERATION.  X HAS BEEN OVERWRITTEN BY CTRANS(A)*X.
*
   40 CONTINUE
      isave( 2 ) = izmax1( n, x, 1 )
      isave( 3 ) = 2
*
*     MAIN LOOP - ITERATIONS 2,3,...,ITMAX.
*
   50 CONTINUE
      DO 60 i = 1, n
         x( i ) = czero
   60 CONTINUE
      x( isave( 2 ) ) = cone
      kase = 1
      isave( 1 ) = 3
      RETURN
*
*     ................ ENTRY   (ISAVE( 1 ) = 3)
*     X HAS BEEN OVERWRITTEN BY A*X.
*
   70 CONTINUE
      CALL zcopy( n, x, 1, v, 1 )
      estold = est
      est = dzsum1( n, v, 1 )
*
*     TEST FOR CYCLING.
      IF( est.LE.estold )
     $   GO TO 100
*
      DO 80 i = 1, n
         absxi = abs( x( i ) )
         IF( absxi.GT.safmin ) THEN
            x( i ) = dcmplx( dble( x( i ) ) / absxi,
     $               dimag( x( i ) ) / absxi )
         ELSE
            x( i ) = cone
         END IF
   80 CONTINUE
      kase = 2
      isave( 1 ) = 4
      RETURN
*
*     ................ ENTRY   (ISAVE( 1 ) = 4)
*     X HAS BEEN OVERWRITTEN BY CTRANS(A)*X.
*
   90 CONTINUE
      jlast = isave( 2 )
      isave( 2 ) = izmax1( n, x, 1 )
      IF( ( abs( x( jlast ) ).NE.abs( x( isave( 2 ) ) ) ) .AND.
     $    ( isave( 3 ).LT.itmax ) ) THEN
         isave( 3 ) = isave( 3 ) + 1
         GO TO 50
      END IF
*
*     ITERATION COMPLETE.  FINAL STAGE.
*
  100 CONTINUE
      altsgn = one
      DO 110 i = 1, n
         x( i ) = dcmplx( altsgn*( one+dble( i-1 ) / dble( n-1 ) ) )
         altsgn = -altsgn
  110 CONTINUE
      kase = 1
      isave( 1 ) = 5
      RETURN
*
*     ................ ENTRY   (ISAVE( 1 ) = 5)
*     X HAS BEEN OVERWRITTEN BY A*X.
*
  120 CONTINUE
      temp = two*( dzsum1( n, x, 1 ) / dble( 3*n ) )
      IF( temp.GT.est ) THEN
         CALL zcopy( n, x, 1, v, 1 )
         est = temp
      END IF
*
  130 CONTINUE
      kase = 0
      RETURN
*
*     End of ZLACN2
*
      END