slacon()

subroutine slacon	(	integer	n,
		real, dimension( * )	v,
		real, dimension( * )	x,
		integer, dimension( * )	isgn,
		real	est,
		integer	kase
	)

SLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

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Purpose:

 SLACON estimates the 1-norm of a square, real matrix A.
 Reverse communication is used for evaluating matrix-vector products.

Parameters

[in]	N	N is INTEGER The order of the matrix. N >= 1.
[out]	V	V is REAL array, dimension (N) On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned).
[in,out]	X	X is REAL array, dimension (N) On an intermediate return, X should be overwritten by A * X, if KASE=1, A**T * X, if KASE=2, and SLACON must be re-called with all the other parameters unchanged.
[out]	ISGN	ISGN is INTEGER array, dimension (N)
[in,out]	EST	EST is REAL On entry with KASE = 1 or 2 and JUMP = 3, EST should be unchanged from the previous call to SLACON. On exit, EST is an estimate (a lower bound) for norm(A).
[in,out]	KASE	KASE is INTEGER On the initial call to SLACON, 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**T * X. On the final return from SLACON, KASE will again be 0.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Nick Higham, University of Manchester.
Originally named SONEST, dated March 16, 1988.

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.

Definition at line 114 of file slacon.f.

*
*  -- LAPACK auxiliary 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            KASE, N
      REAL               EST
*     ..
*     .. Array Arguments ..
      INTEGER            ISGN( * )
      REAL               V( * ), X( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      parameter( itmax = 5 )
      REAL               ZERO, ONE, TWO
      parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ITER, J, JLAST, JUMP
      REAL               ALTSGN, ESTOLD, TEMP
*     ..
*     .. External Functions ..
      INTEGER            ISAMAX
      REAL               SASUM
      EXTERNAL           isamax, sasum
*     ..
*     .. External Subroutines ..
      EXTERNAL           scopy
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, nint, real, sign
*     ..
*     .. Save statement ..
      SAVE
*     ..
*     .. Executable Statements ..
*
      IF( kase.EQ.0 ) THEN
         DO 10 i = 1, n
            x( i ) = one / real( n )
   10    CONTINUE
         kase = 1
         jump = 1
         RETURN
      END IF
*
      GO TO ( 20, 40, 70, 110, 140 )jump
*
*     ................ ENTRY   (JUMP = 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 150
      END IF
      est = sasum( n, x, 1 )
*
      DO 30 i = 1, n
         x( i ) = sign( one, x( i ) )
         isgn( i ) = nint( x( i ) )
   30 CONTINUE
      kase = 2
      jump = 2
      RETURN
*
*     ................ ENTRY   (JUMP = 2)
*     FIRST ITERATION.  X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
   40 CONTINUE
      j = isamax( n, x, 1 )
      iter = 2
*
*     MAIN LOOP - ITERATIONS 2,3,...,ITMAX.
*
   50 CONTINUE
      DO 60 i = 1, n
         x( i ) = zero
   60 CONTINUE
      x( j ) = one
      kase = 1
      jump = 3
      RETURN
*
*     ................ ENTRY   (JUMP = 3)
*     X HAS BEEN OVERWRITTEN BY A*X.
*
   70 CONTINUE
      CALL scopy( n, x, 1, v, 1 )
      estold = est
      est = sasum( n, v, 1 )
      DO 80 i = 1, n
         IF( nint( sign( one, x( i ) ) ).NE.isgn( i ) )
     $      GO TO 90
   80 CONTINUE
*     REPEATED SIGN VECTOR DETECTED, HENCE ALGORITHM HAS CONVERGED.
      GO TO 120
*
   90 CONTINUE
*     TEST FOR CYCLING.
      IF( est.LE.estold )
     $   GO TO 120
*
      DO 100 i = 1, n
         x( i ) = sign( one, x( i ) )
         isgn( i ) = nint( x( i ) )
  100 CONTINUE
      kase = 2
      jump = 4
      RETURN
*
*     ................ ENTRY   (JUMP = 4)
*     X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
*
  110 CONTINUE
      jlast = j
      j = isamax( n, x, 1 )
      IF( ( x( jlast ).NE.abs( x( j ) ) ) .AND. ( iter.LT.itmax ) ) THEN
         iter = iter + 1
         GO TO 50
      END IF
*
*     ITERATION COMPLETE.  FINAL STAGE.
*
  120 CONTINUE
      altsgn = one
      DO 130 i = 1, n
         x( i ) = altsgn*( one+real( i-1 ) / real( n-1 ) )
         altsgn = -altsgn
  130 CONTINUE
      kase = 1
      jump = 5
      RETURN
*
*     ................ ENTRY   (JUMP = 5)
*     X HAS BEEN OVERWRITTEN BY A*X.
*
  140 CONTINUE
      temp = two*( sasum( n, x, 1 ) / real( 3*n ) )
      IF( temp.GT.est ) THEN
         CALL scopy( n, x, 1, v, 1 )
         est = temp
      END IF
*
  150 CONTINUE
      kase = 0
      RETURN
*
*     End of SLACON
*

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