LAPACK  3.9.1 LAPACK: Linear Algebra PACKage

## ◆ zlacon()

 subroutine zlacon ( integer N, complex*16, dimension( n ) V, complex*16, dimension( n ) X, double precision EST, integer KASE )

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

Download ZLACON + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` ZLACON estimates the 1-norm of a square, complex 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 COMPLEX*16 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 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 ZLACON must be re-called with all the other parameters unchanged.``` [in,out] EST ``` EST is DOUBLE PRECISION On entry with KASE = 1 or 2 and JUMP = 3, EST should be unchanged from the previous call to ZLACON. On exit, EST is an estimate (a lower bound) for norm(A).``` [in,out] KASE ``` KASE is INTEGER On the initial call to ZLACON, 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 ZLACON, KASE will again be 0.```
Further Details:
Originally named CONEST, dated March 16, 1988.
Contributors:
Nick Higham, University of Manchester
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 113 of file zlacon.f.

114 *
115 * -- LAPACK auxiliary routine --
116 * -- LAPACK is a software package provided by Univ. of Tennessee, --
117 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
118 *
119 * .. Scalar Arguments ..
120  INTEGER KASE, N
121  DOUBLE PRECISION EST
122 * ..
123 * .. Array Arguments ..
124  COMPLEX*16 V( N ), X( N )
125 * ..
126 *
127 * =====================================================================
128 *
129 * .. Parameters ..
130  INTEGER ITMAX
131  parameter( itmax = 5 )
132  DOUBLE PRECISION ONE, TWO
133  parameter( one = 1.0d0, two = 2.0d0 )
134  COMPLEX*16 CZERO, CONE
135  parameter( czero = ( 0.0d0, 0.0d0 ),
136  \$ cone = ( 1.0d0, 0.0d0 ) )
137 * ..
138 * .. Local Scalars ..
139  INTEGER I, ITER, J, JLAST, JUMP
140  DOUBLE PRECISION ABSXI, ALTSGN, ESTOLD, SAFMIN, TEMP
141 * ..
142 * .. External Functions ..
143  INTEGER IZMAX1
144  DOUBLE PRECISION DLAMCH, DZSUM1
145  EXTERNAL izmax1, dlamch, dzsum1
146 * ..
147 * .. External Subroutines ..
148  EXTERNAL zcopy
149 * ..
150 * .. Intrinsic Functions ..
151  INTRINSIC abs, dble, dcmplx, dimag
152 * ..
153 * .. Save statement ..
154  SAVE
155 * ..
156 * .. Executable Statements ..
157 *
158  safmin = dlamch( 'Safe minimum' )
159  IF( kase.EQ.0 ) THEN
160  DO 10 i = 1, n
161  x( i ) = dcmplx( one / dble( n ) )
162  10 CONTINUE
163  kase = 1
164  jump = 1
165  RETURN
166  END IF
167 *
168  GO TO ( 20, 40, 70, 90, 120 )jump
169 *
170 * ................ ENTRY (JUMP = 1)
171 * FIRST ITERATION. X HAS BEEN OVERWRITTEN BY A*X.
172 *
173  20 CONTINUE
174  IF( n.EQ.1 ) THEN
175  v( 1 ) = x( 1 )
176  est = abs( v( 1 ) )
177 * ... QUIT
178  GO TO 130
179  END IF
180  est = dzsum1( n, x, 1 )
181 *
182  DO 30 i = 1, n
183  absxi = abs( x( i ) )
184  IF( absxi.GT.safmin ) THEN
185  x( i ) = dcmplx( dble( x( i ) ) / absxi,
186  \$ dimag( x( i ) ) / absxi )
187  ELSE
188  x( i ) = cone
189  END IF
190  30 CONTINUE
191  kase = 2
192  jump = 2
193  RETURN
194 *
195 * ................ ENTRY (JUMP = 2)
196 * FIRST ITERATION. X HAS BEEN OVERWRITTEN BY CTRANS(A)*X.
197 *
198  40 CONTINUE
199  j = izmax1( n, x, 1 )
200  iter = 2
201 *
202 * MAIN LOOP - ITERATIONS 2,3,...,ITMAX.
203 *
204  50 CONTINUE
205  DO 60 i = 1, n
206  x( i ) = czero
207  60 CONTINUE
208  x( j ) = cone
209  kase = 1
210  jump = 3
211  RETURN
212 *
213 * ................ ENTRY (JUMP = 3)
214 * X HAS BEEN OVERWRITTEN BY A*X.
215 *
216  70 CONTINUE
217  CALL zcopy( n, x, 1, v, 1 )
218  estold = est
219  est = dzsum1( n, v, 1 )
220 *
221 * TEST FOR CYCLING.
222  IF( est.LE.estold )
223  \$ GO TO 100
224 *
225  DO 80 i = 1, n
226  absxi = abs( x( i ) )
227  IF( absxi.GT.safmin ) THEN
228  x( i ) = dcmplx( dble( x( i ) ) / absxi,
229  \$ dimag( x( i ) ) / absxi )
230  ELSE
231  x( i ) = cone
232  END IF
233  80 CONTINUE
234  kase = 2
235  jump = 4
236  RETURN
237 *
238 * ................ ENTRY (JUMP = 4)
239 * X HAS BEEN OVERWRITTEN BY CTRANS(A)*X.
240 *
241  90 CONTINUE
242  jlast = j
243  j = izmax1( n, x, 1 )
244  IF( ( abs( x( jlast ) ).NE.abs( x( j ) ) ) .AND.
245  \$ ( iter.LT.itmax ) ) THEN
246  iter = iter + 1
247  GO TO 50
248  END IF
249 *
250 * ITERATION COMPLETE. FINAL STAGE.
251 *
252  100 CONTINUE
253  altsgn = one
254  DO 110 i = 1, n
255  x( i ) = dcmplx( altsgn*( one+dble( i-1 ) / dble( n-1 ) ) )
256  altsgn = -altsgn
257  110 CONTINUE
258  kase = 1
259  jump = 5
260  RETURN
261 *
262 * ................ ENTRY (JUMP = 5)
263 * X HAS BEEN OVERWRITTEN BY A*X.
264 *
265  120 CONTINUE
266  temp = two*( dzsum1( n, x, 1 ) / dble( 3*n ) )
267  IF( temp.GT.est ) THEN
268  CALL zcopy( n, x, 1, v, 1 )
269  est = temp
270  END IF
271 *
272  130 CONTINUE
273  kase = 0
274  RETURN
275 *
276 * End of ZLACON
277 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
double precision function dzsum1(N, CX, INCX)
DZSUM1 forms the 1-norm of the complex vector using the true absolute value.
Definition: dzsum1.f:81
integer function izmax1(N, ZX, INCX)
IZMAX1 finds the index of the first vector element of maximum absolute value.
Definition: izmax1.f:81
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