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SUBROUTINE <a name="SLACN2.1"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a>( N, V, X, ISGN, EST, KASE, ISAVE )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> INTEGER KASE, N
REAL EST
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER ISGN( * ), ISAVE( 3 )
REAL V( * ), X( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="SLACN2.19"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a> estimates the 1-norm of a square, real matrix A.
</span><span class="comment">*</span><span class="comment"> Reverse communication is used for evaluating matrix-vector products.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrix. N >= 1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> V (workspace) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On the final return, V = A*W, where EST = norm(V)/norm(W)
</span><span class="comment">*</span><span class="comment"> (W is not returned).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> X (input/output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On an intermediate return, X should be overwritten by
</span><span class="comment">*</span><span class="comment"> A * X, if KASE=1,
</span><span class="comment">*</span><span class="comment"> A' * X, if KASE=2,
</span><span class="comment">*</span><span class="comment"> and <a name="SLACN2.36"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a> must be re-called with all the other parameters
</span><span class="comment">*</span><span class="comment"> unchanged.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ISGN (workspace) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> EST (input/output) REAL
</span><span class="comment">*</span><span class="comment"> On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
</span><span class="comment">*</span><span class="comment"> unchanged from the previous call to <a name="SLACN2.43"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a>.
</span><span class="comment">*</span><span class="comment"> On exit, EST is an estimate (a lower bound) for norm(A).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> KASE (input/output) INTEGER
</span><span class="comment">*</span><span class="comment"> On the initial call to <a name="SLACN2.47"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a>, KASE should be 0.
</span><span class="comment">*</span><span class="comment"> On an intermediate return, KASE will be 1 or 2, indicating
</span><span class="comment">*</span><span class="comment"> whether X should be overwritten by A * X or A' * X.
</span><span class="comment">*</span><span class="comment"> On the final return from <a name="SLACN2.50"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a>, KASE will again be 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ISAVE (input/output) INTEGER array, dimension (3)
</span><span class="comment">*</span><span class="comment"> ISAVE is used to save variables between calls to <a name="SLACN2.53"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a>
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ======= =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Contributed by Nick Higham, University of Manchester.
</span><span class="comment">*</span><span class="comment"> Originally named SONEST, dated March 16, 1988.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of
</span><span class="comment">*</span><span class="comment"> a real or complex matrix, with applications to condition estimation",
</span><span class="comment">*</span><span class="comment"> ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This is a thread safe version of <a name="SLACON.65"></a><a href="slacon.f.html#SLACON.1">SLACON</a>, which uses the array ISAVE
</span><span class="comment">*</span><span class="comment"> in place of a SAVE statement, as follows:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="SLACON.68"></a><a href="slacon.f.html#SLACON.1">SLACON</a> <a name="SLACN2.68"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a>
</span><span class="comment">*</span><span class="comment"> JUMP ISAVE(1)
</span><span class="comment">*</span><span class="comment"> J ISAVE(2)
</span><span class="comment">*</span><span class="comment"> ITER ISAVE(3)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
REAL ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> INTEGER I, JLAST
REAL ALTSGN, ESTOLD, TEMP
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Functions ..
</span> INTEGER ISAMAX
REAL SASUM
EXTERNAL ISAMAX, SASUM
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Subroutines ..
</span> EXTERNAL SCOPY
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
</span> INTRINSIC ABS, NINT, REAL, SIGN
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span> IF( KASE.EQ.0 ) THEN
DO 10 I = 1, N
X( I ) = ONE / REAL( N )
10 CONTINUE
KASE = 1
ISAVE( 1 ) = 1
RETURN
END IF
<span class="comment">*</span><span class="comment">
</span> GO TO ( 20, 40, 70, 110, 140 )ISAVE( 1 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ................ ENTRY (ISAVE( 1 ) = 1)
</span><span class="comment">*</span><span class="comment"> FIRST ITERATION. X HAS BEEN OVERWRITTEN BY A*X.
</span><span class="comment">*</span><span class="comment">
</span> 20 CONTINUE
IF( N.EQ.1 ) THEN
V( 1 ) = X( 1 )
EST = ABS( V( 1 ) )
<span class="comment">*</span><span class="comment"> ... QUIT
</span> GO TO 150
END IF
EST = SASUM( N, X, 1 )
<span class="comment">*</span><span class="comment">
</span> DO 30 I = 1, N
X( I ) = SIGN( ONE, X( I ) )
ISGN( I ) = NINT( X( I ) )
30 CONTINUE
KASE = 2
ISAVE( 1 ) = 2
RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ................ ENTRY (ISAVE( 1 ) = 2)
</span><span class="comment">*</span><span class="comment"> FIRST ITERATION. X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
</span><span class="comment">*</span><span class="comment">
</span> 40 CONTINUE
ISAVE( 2 ) = ISAMAX( N, X, 1 )
ISAVE( 3 ) = 2
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> MAIN LOOP - ITERATIONS 2,3,...,ITMAX.
</span><span class="comment">*</span><span class="comment">
</span> 50 CONTINUE
DO 60 I = 1, N
X( I ) = ZERO
60 CONTINUE
X( ISAVE( 2 ) ) = ONE
KASE = 1
ISAVE( 1 ) = 3
RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ................ ENTRY (ISAVE( 1 ) = 3)
</span><span class="comment">*</span><span class="comment"> X HAS BEEN OVERWRITTEN BY A*X.
</span><span class="comment">*</span><span class="comment">
</span> 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
<span class="comment">*</span><span class="comment"> REPEATED SIGN VECTOR DETECTED, HENCE ALGORITHM HAS CONVERGED.
</span> GO TO 120
<span class="comment">*</span><span class="comment">
</span> 90 CONTINUE
<span class="comment">*</span><span class="comment"> TEST FOR CYCLING.
</span> IF( EST.LE.ESTOLD )
$ GO TO 120
<span class="comment">*</span><span class="comment">
</span> DO 100 I = 1, N
X( I ) = SIGN( ONE, X( I ) )
ISGN( I ) = NINT( X( I ) )
100 CONTINUE
KASE = 2
ISAVE( 1 ) = 4
RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ................ ENTRY (ISAVE( 1 ) = 4)
</span><span class="comment">*</span><span class="comment"> X HAS BEEN OVERWRITTEN BY TRANSPOSE(A)*X.
</span><span class="comment">*</span><span class="comment">
</span> 110 CONTINUE
JLAST = ISAVE( 2 )
ISAVE( 2 ) = ISAMAX( N, X, 1 )
IF( ( X( JLAST ).NE.ABS( X( ISAVE( 2 ) ) ) ) .AND.
$ ( ISAVE( 3 ).LT.ITMAX ) ) THEN
ISAVE( 3 ) = ISAVE( 3 ) + 1
GO TO 50
END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ITERATION COMPLETE. FINAL STAGE.
</span><span class="comment">*</span><span class="comment">
</span> 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
ISAVE( 1 ) = 5
RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ................ ENTRY (ISAVE( 1 ) = 5)
</span><span class="comment">*</span><span class="comment"> X HAS BEEN OVERWRITTEN BY A*X.
</span><span class="comment">*</span><span class="comment">
</span> 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
<span class="comment">*</span><span class="comment">
</span> 150 CONTINUE
KASE = 0
RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> End of <a name="SLACN2.212"></a><a href="slacn2.f.html#SLACN2.1">SLACN2</a>
</span><span class="comment">*</span><span class="comment">
</span> END
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