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zgebal.f
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1 *> \brief \b ZGEBAL
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER JOB
25 * INTEGER IHI, ILO, INFO, LDA, N
26 * ..
27 * .. Array Arguments ..
28 * DOUBLE PRECISION SCALE( * )
29 * COMPLEX*16 A( LDA, * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> ZGEBAL balances a general complex matrix A. This involves, first,
39 *> permuting A by a similarity transformation to isolate eigenvalues
40 *> in the first 1 to ILO-1 and last IHI+1 to N elements on the
41 *> diagonal; and second, applying a diagonal similarity transformation
42 *> to rows and columns ILO to IHI to make the rows and columns as
43 *> close in norm as possible. Both steps are optional.
44 *>
45 *> Balancing may reduce the 1-norm of the matrix, and improve the
46 *> accuracy of the computed eigenvalues and/or eigenvectors.
47 *> \endverbatim
48 *
49 * Arguments:
50 * ==========
51 *
52 *> \param[in] JOB
53 *> \verbatim
54 *> JOB is CHARACTER*1
55 *> Specifies the operations to be performed on A:
56 *> = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
57 *> for i = 1,...,N;
58 *> = 'P': permute only;
59 *> = 'S': scale only;
60 *> = 'B': both permute and scale.
61 *> \endverbatim
62 *>
63 *> \param[in] N
64 *> \verbatim
65 *> N is INTEGER
66 *> The order of the matrix A. N >= 0.
67 *> \endverbatim
68 *>
69 *> \param[in,out] A
70 *> \verbatim
71 *> A is COMPLEX*16 array, dimension (LDA,N)
72 *> On entry, the input matrix A.
73 *> On exit, A is overwritten by the balanced matrix.
74 *> If JOB = 'N', A is not referenced.
75 *> See Further Details.
76 *> \endverbatim
77 *>
78 *> \param[in] LDA
79 *> \verbatim
80 *> LDA is INTEGER
81 *> The leading dimension of the array A. LDA >= max(1,N).
82 *> \endverbatim
83 *>
84 *> \param[out] ILO
85 *> \verbatim
86 *> \endverbatim
87 *>
88 *> \param[out] IHI
89 *> \verbatim
90 *> ILO and IHI are set to INTEGER such that on exit
91 *> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
92 *> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
93 *> \endverbatim
94 *>
95 *> \param[out] SCALE
96 *> \verbatim
97 *> SCALE is DOUBLE PRECISION array, dimension (N)
98 *> Details of the permutations and scaling factors applied to
99 *> A. If P(j) is the index of the row and column interchanged
100 *> with row and column j and D(j) is the scaling factor
101 *> applied to row and column j, then
102 *> SCALE(j) = P(j) for j = 1,...,ILO-1
103 *> = D(j) for j = ILO,...,IHI
104 *> = P(j) for j = IHI+1,...,N.
105 *> The order in which the interchanges are made is N to IHI+1,
106 *> then 1 to ILO-1.
107 *> \endverbatim
108 *>
109 *> \param[out] INFO
110 *> \verbatim
111 *> INFO is INTEGER
112 *> = 0: successful exit.
113 *> < 0: if INFO = -i, the i-th argument had an illegal value.
114 *> \endverbatim
115 *
116 * Authors:
117 * ========
118 *
119 *> \author Univ. of Tennessee
120 *> \author Univ. of California Berkeley
121 *> \author Univ. of Colorado Denver
122 *> \author NAG Ltd.
123 *
124 *> \date November 2013
125 *
126 *> \ingroup complex16GEcomputational
127 *
128 *> \par Further Details:
129 * =====================
130 *>
131 *> \verbatim
132 *>
133 *> The permutations consist of row and column interchanges which put
134 *> the matrix in the form
135 *>
136 *> ( T1 X Y )
137 *> P A P = ( 0 B Z )
138 *> ( 0 0 T2 )
139 *>
140 *> where T1 and T2 are upper triangular matrices whose eigenvalues lie
141 *> along the diagonal. The column indices ILO and IHI mark the starting
142 *> and ending columns of the submatrix B. Balancing consists of applying
143 *> a diagonal similarity transformation inv(D) * B * D to make the
144 *> 1-norms of each row of B and its corresponding column nearly equal.
145 *> The output matrix is
146 *>
147 *> ( T1 X*D Y )
148 *> ( 0 inv(D)*B*D inv(D)*Z ).
149 *> ( 0 0 T2 )
150 *>
151 *> Information about the permutations P and the diagonal matrix D is
152 *> returned in the vector SCALE.
153 *>
154 *> This subroutine is based on the EISPACK routine CBAL.
155 *>
156 *> Modified by Tzu-Yi Chen, Computer Science Division, University of
157 *> California at Berkeley, USA
158 *> \endverbatim
159 *>
160 * =====================================================================
161  SUBROUTINE zgebal( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
162 *
163 * -- LAPACK computational routine (version 3.5.0) --
164 * -- LAPACK is a software package provided by Univ. of Tennessee, --
165 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166 * November 2013
167 *
168 * .. Scalar Arguments ..
169  CHARACTER job
170  INTEGER ihi, ilo, info, lda, n
171 * ..
172 * .. Array Arguments ..
173  DOUBLE PRECISION scale( * )
174  COMPLEX*16 a( lda, * )
175 * ..
176 *
177 * =====================================================================
178 *
179 * .. Parameters ..
180  DOUBLE PRECISION zero, one
181  parameter( zero = 0.0d+0, one = 1.0d+0 )
182  DOUBLE PRECISION sclfac
183  parameter( sclfac = 2.0d+0 )
184  DOUBLE PRECISION factor
185  parameter( factor = 0.95d+0 )
186 * ..
187 * .. Local Scalars ..
188  LOGICAL noconv
189  INTEGER i, ica, iexc, ira, j, k, l, m
190  DOUBLE PRECISION c, ca, f, g, r, ra, s, sfmax1, sfmax2, sfmin1,
191  $ sfmin2
192  COMPLEX*16 cdum
193 * ..
194 * .. External Functions ..
195  LOGICAL disnan, lsame
196  INTEGER izamax
197  DOUBLE PRECISION dlamch, dznrm2
198  EXTERNAL disnan, lsame, izamax, dlamch, dznrm2
199 * ..
200 * .. External Subroutines ..
201  EXTERNAL xerbla, zdscal, zswap
202 * ..
203 * .. Intrinsic Functions ..
204  INTRINSIC abs, dble, dimag, max, min
205 * ..
206 * .. Statement Functions ..
207  DOUBLE PRECISION cabs1
208 * ..
209 * .. Statement Function definitions ..
210  cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
211 * ..
212 * .. Executable Statements ..
213 *
214 * Test the input parameters
215 *
216  info = 0
217  IF( .NOT.lsame( job, 'N' ) .AND. .NOT.lsame( job, 'P' ) .AND.
218  $ .NOT.lsame( job, 'S' ) .AND. .NOT.lsame( job, 'B' ) ) THEN
219  info = -1
220  ELSE IF( n.LT.0 ) THEN
221  info = -2
222  ELSE IF( lda.LT.max( 1, n ) ) THEN
223  info = -4
224  END IF
225  IF( info.NE.0 ) THEN
226  CALL xerbla( 'ZGEBAL', -info )
227  RETURN
228  END IF
229 *
230  k = 1
231  l = n
232 *
233  IF( n.EQ.0 )
234  $ go to 210
235 *
236  IF( lsame( job, 'N' ) ) THEN
237  DO 10 i = 1, n
238  scale( i ) = one
239  10 CONTINUE
240  go to 210
241  END IF
242 *
243  IF( lsame( job, 'S' ) )
244  $ go to 120
245 *
246 * Permutation to isolate eigenvalues if possible
247 *
248  go to 50
249 *
250 * Row and column exchange.
251 *
252  20 CONTINUE
253  scale( m ) = j
254  IF( j.EQ.m )
255  $ go to 30
256 *
257  CALL zswap( l, a( 1, j ), 1, a( 1, m ), 1 )
258  CALL zswap( n-k+1, a( j, k ), lda, a( m, k ), lda )
259 *
260  30 CONTINUE
261  go to( 40, 80 )iexc
262 *
263 * Search for rows isolating an eigenvalue and push them down.
264 *
265  40 CONTINUE
266  IF( l.EQ.1 )
267  $ go to 210
268  l = l - 1
269 *
270  50 CONTINUE
271  DO 70 j = l, 1, -1
272 *
273  DO 60 i = 1, l
274  IF( i.EQ.j )
275  $ go to 60
276  IF( dble( a( j, i ) ).NE.zero .OR. dimag( a( j, i ) ).NE.
277  $ zero )go to 70
278  60 CONTINUE
279 *
280  m = l
281  iexc = 1
282  go to 20
283  70 CONTINUE
284 *
285  go to 90
286 *
287 * Search for columns isolating an eigenvalue and push them left.
288 *
289  80 CONTINUE
290  k = k + 1
291 *
292  90 CONTINUE
293  DO 110 j = k, l
294 *
295  DO 100 i = k, l
296  IF( i.EQ.j )
297  $ go to 100
298  IF( dble( a( i, j ) ).NE.zero .OR. dimag( a( i, j ) ).NE.
299  $ zero )go to 110
300  100 CONTINUE
301 *
302  m = k
303  iexc = 2
304  go to 20
305  110 CONTINUE
306 *
307  120 CONTINUE
308  DO 130 i = k, l
309  scale( i ) = one
310  130 CONTINUE
311 *
312  IF( lsame( job, 'P' ) )
313  $ go to 210
314 *
315 * Balance the submatrix in rows K to L.
316 *
317 * Iterative loop for norm reduction
318 *
319  sfmin1 = dlamch( 'S' ) / dlamch( 'P' )
320  sfmax1 = one / sfmin1
321  sfmin2 = sfmin1*sclfac
322  sfmax2 = one / sfmin2
323  140 CONTINUE
324  noconv = .false.
325 *
326  DO 200 i = k, l
327 *
328  c = dznrm2( l-k+1, a( k, i ), 1 )
329  r = dznrm2( l-k+1, a( i, k ), lda )
330  ica = izamax( l, a( 1, i ), 1 )
331  ca = abs( a( ica, i ) )
332  ira = izamax( n-k+1, a( i, k ), lda )
333  ra = abs( a( i, ira+k-1 ) )
334 *
335 * Guard against zero C or R due to underflow.
336 *
337  IF( c.EQ.zero .OR. r.EQ.zero )
338  $ go to 200
339  g = r / sclfac
340  f = one
341  s = c + r
342  160 CONTINUE
343  IF( c.GE.g .OR. max( f, c, ca ).GE.sfmax2 .OR.
344  $ min( r, g, ra ).LE.sfmin2 )go to 170
345  IF( disnan( c+f+ca+r+g+ra ) ) THEN
346 *
347 * Exit if NaN to avoid infinite loop
348 *
349  info = -3
350  CALL xerbla( 'ZGEBAL', -info )
351  RETURN
352  END IF
353  f = f*sclfac
354  c = c*sclfac
355  ca = ca*sclfac
356  r = r / sclfac
357  g = g / sclfac
358  ra = ra / sclfac
359  go to 160
360 *
361  170 CONTINUE
362  g = c / sclfac
363  180 CONTINUE
364  IF( g.LT.r .OR. max( r, ra ).GE.sfmax2 .OR.
365  $ min( f, c, g, ca ).LE.sfmin2 )go to 190
366  f = f / sclfac
367  c = c / sclfac
368  g = g / sclfac
369  ca = ca / sclfac
370  r = r*sclfac
371  ra = ra*sclfac
372  go to 180
373 *
374 * Now balance.
375 *
376  190 CONTINUE
377  IF( ( c+r ).GE.factor*s )
378  $ go to 200
379  IF( f.LT.one .AND. scale( i ).LT.one ) THEN
380  IF( f*scale( i ).LE.sfmin1 )
381  $ go to 200
382  END IF
383  IF( f.GT.one .AND. scale( i ).GT.one ) THEN
384  IF( scale( i ).GE.sfmax1 / f )
385  $ go to 200
386  END IF
387  g = one / f
388  scale( i ) = scale( i )*f
389  noconv = .true.
390 *
391  CALL zdscal( n-k+1, g, a( i, k ), lda )
392  CALL zdscal( l, f, a( 1, i ), 1 )
393 *
394  200 CONTINUE
395 *
396  IF( noconv )
397  $ go to 140
398 *
399  210 CONTINUE
400  ilo = k
401  ihi = l
402 *
403  RETURN
404 *
405 * End of ZGEBAL
406 *
407  END