LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
cgebal.f
Go to the documentation of this file.
1 *> \brief \b CGEBAL
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CGEBAL + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgebal.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgebal.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgebal.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CGEBAL( 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 * REAL SCALE( * )
29 * COMPLEX A( LDA, * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> CGEBAL 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 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 *> ILO is INTEGER
87 *> \endverbatim
88 *> \param[out] IHI
89 *> \verbatim
90 *> IHI is INTEGER
91 *> ILO and IHI are set to integers such that on exit
92 *> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
93 *> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
94 *> \endverbatim
95 *>
96 *> \param[out] SCALE
97 *> \verbatim
98 *> SCALE is REAL array, dimension (N)
99 *> Details of the permutations and scaling factors applied to
100 *> A. If P(j) is the index of the row and column interchanged
101 *> with row and column j and D(j) is the scaling factor
102 *> applied to row and column j, then
103 *> SCALE(j) = P(j) for j = 1,...,ILO-1
104 *> = D(j) for j = ILO,...,IHI
105 *> = P(j) for j = IHI+1,...,N.
106 *> The order in which the interchanges are made is N to IHI+1,
107 *> then 1 to ILO-1.
108 *> \endverbatim
109 *>
110 *> \param[out] INFO
111 *> \verbatim
112 *> INFO is INTEGER
113 *> = 0: successful exit.
114 *> < 0: if INFO = -i, the i-th argument had an illegal value.
115 *> \endverbatim
116 *
117 * Authors:
118 * ========
119 *
120 *> \author Univ. of Tennessee
121 *> \author Univ. of California Berkeley
122 *> \author Univ. of Colorado Denver
123 *> \author NAG Ltd.
124 *
125 *> \date November 2015
126 *
127 *> \ingroup complexGEcomputational
128 *
129 *> \par Further Details:
130 * =====================
131 *>
132 *> \verbatim
133 *>
134 *> The permutations consist of row and column interchanges which put
135 *> the matrix in the form
136 *>
137 *> ( T1 X Y )
138 *> P A P = ( 0 B Z )
139 *> ( 0 0 T2 )
140 *>
141 *> where T1 and T2 are upper triangular matrices whose eigenvalues lie
142 *> along the diagonal. The column indices ILO and IHI mark the starting
143 *> and ending columns of the submatrix B. Balancing consists of applying
144 *> a diagonal similarity transformation inv(D) * B * D to make the
145 *> 1-norms of each row of B and its corresponding column nearly equal.
146 *> The output matrix is
147 *>
148 *> ( T1 X*D Y )
149 *> ( 0 inv(D)*B*D inv(D)*Z ).
150 *> ( 0 0 T2 )
151 *>
152 *> Information about the permutations P and the diagonal matrix D is
153 *> returned in the vector SCALE.
154 *>
155 *> This subroutine is based on the EISPACK routine CBAL.
156 *>
157 *> Modified by Tzu-Yi Chen, Computer Science Division, University of
158 *> California at Berkeley, USA
159 *> \endverbatim
160 *>
161 * =====================================================================
162  SUBROUTINE cgebal( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
163 *
164 * -- LAPACK computational routine (version 3.6.0) --
165 * -- LAPACK is a software package provided by Univ. of Tennessee, --
166 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
167 * November 2015
168 *
169 * .. Scalar Arguments ..
170  CHARACTER JOB
171  INTEGER IHI, ILO, INFO, LDA, N
172 * ..
173 * .. Array Arguments ..
174  REAL SCALE( * )
175  COMPLEX A( lda, * )
176 * ..
177 *
178 * =====================================================================
179 *
180 * .. Parameters ..
181  REAL ZERO, ONE
182  parameter( zero = 0.0e+0, one = 1.0e+0 )
183  REAL SCLFAC
184  parameter( sclfac = 2.0e+0 )
185  REAL FACTOR
186  parameter( factor = 0.95e+0 )
187 * ..
188 * .. Local Scalars ..
189  LOGICAL NOCONV
190  INTEGER I, ICA, IEXC, IRA, J, K, L, M
191  REAL C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
192  $ sfmin2
193  COMPLEX CDUM
194 * ..
195 * .. External Functions ..
196  LOGICAL SISNAN, LSAME
197  INTEGER ICAMAX
198  REAL SLAMCH, SCNRM2
199  EXTERNAL sisnan, lsame, icamax, slamch, scnrm2
200 * ..
201 * .. External Subroutines ..
202  EXTERNAL csscal, cswap, xerbla
203 * ..
204 * .. Intrinsic Functions ..
205  INTRINSIC abs, aimag, max, min, real
206 *
207 * Test the input parameters
208 *
209  info = 0
210  IF( .NOT.lsame( job, 'N' ) .AND. .NOT.lsame( job, 'P' ) .AND.
211  $ .NOT.lsame( job, 'S' ) .AND. .NOT.lsame( job, 'B' ) ) THEN
212  info = -1
213  ELSE IF( n.LT.0 ) THEN
214  info = -2
215  ELSE IF( lda.LT.max( 1, n ) ) THEN
216  info = -4
217  END IF
218  IF( info.NE.0 ) THEN
219  CALL xerbla( 'CGEBAL', -info )
220  RETURN
221  END IF
222 *
223  k = 1
224  l = n
225 *
226  IF( n.EQ.0 )
227  $ GO TO 210
228 *
229  IF( lsame( job, 'N' ) ) THEN
230  DO 10 i = 1, n
231  scale( i ) = one
232  10 CONTINUE
233  GO TO 210
234  END IF
235 *
236  IF( lsame( job, 'S' ) )
237  $ GO TO 120
238 *
239 * Permutation to isolate eigenvalues if possible
240 *
241  GO TO 50
242 *
243 * Row and column exchange.
244 *
245  20 CONTINUE
246  scale( m ) = j
247  IF( j.EQ.m )
248  $ GO TO 30
249 *
250  CALL cswap( l, a( 1, j ), 1, a( 1, m ), 1 )
251  CALL cswap( n-k+1, a( j, k ), lda, a( m, k ), lda )
252 *
253  30 CONTINUE
254  GO TO ( 40, 80 )iexc
255 *
256 * Search for rows isolating an eigenvalue and push them down.
257 *
258  40 CONTINUE
259  IF( l.EQ.1 )
260  $ GO TO 210
261  l = l - 1
262 *
263  50 CONTINUE
264  DO 70 j = l, 1, -1
265 *
266  DO 60 i = 1, l
267  IF( i.EQ.j )
268  $ GO TO 60
269  IF( REAL( A( J, I ) ).NE.zero .OR. aimag( A( j, i ) ).NE.
270  $ zero )GO TO 70
271  60 CONTINUE
272 *
273  m = l
274  iexc = 1
275  GO TO 20
276  70 CONTINUE
277 *
278  GO TO 90
279 *
280 * Search for columns isolating an eigenvalue and push them left.
281 *
282  80 CONTINUE
283  k = k + 1
284 *
285  90 CONTINUE
286  DO 110 j = k, l
287 *
288  DO 100 i = k, l
289  IF( i.EQ.j )
290  $ GO TO 100
291  IF( REAL( A( I, J ) ).NE.zero .OR. aimag( A( i, j ) ).NE.
292  $ zero )GO TO 110
293  100 CONTINUE
294 *
295  m = k
296  iexc = 2
297  GO TO 20
298  110 CONTINUE
299 *
300  120 CONTINUE
301  DO 130 i = k, l
302  scale( i ) = one
303  130 CONTINUE
304 *
305  IF( lsame( job, 'P' ) )
306  $ GO TO 210
307 *
308 * Balance the submatrix in rows K to L.
309 *
310 * Iterative loop for norm reduction
311 *
312  sfmin1 = slamch( 'S' ) / slamch( 'P' )
313  sfmax1 = one / sfmin1
314  sfmin2 = sfmin1*sclfac
315  sfmax2 = one / sfmin2
316  140 CONTINUE
317  noconv = .false.
318 *
319  DO 200 i = k, l
320 *
321  c = scnrm2( l-k+1, a( k, i ), 1 )
322  r = scnrm2( l-k+1, a( i , k ), lda )
323  ica = icamax( l, a( 1, i ), 1 )
324  ca = abs( a( ica, i ) )
325  ira = icamax( n-k+1, a( i, k ), lda )
326  ra = abs( a( i, ira+k-1 ) )
327 *
328 * Guard against zero C or R due to underflow.
329 *
330  IF( c.EQ.zero .OR. r.EQ.zero )
331  $ GO TO 200
332  g = r / sclfac
333  f = one
334  s = c + r
335  160 CONTINUE
336  IF( c.GE.g .OR. max( f, c, ca ).GE.sfmax2 .OR.
337  $ min( r, g, ra ).LE.sfmin2 )GO TO 170
338  IF( sisnan( c+f+ca+r+g+ra ) ) THEN
339 *
340 * Exit if NaN to avoid infinite loop
341 *
342  info = -3
343  CALL xerbla( 'CGEBAL', -info )
344  RETURN
345  END IF
346  f = f*sclfac
347  c = c*sclfac
348  ca = ca*sclfac
349  r = r / sclfac
350  g = g / sclfac
351  ra = ra / sclfac
352  GO TO 160
353 *
354  170 CONTINUE
355  g = c / sclfac
356  180 CONTINUE
357  IF( g.LT.r .OR. max( r, ra ).GE.sfmax2 .OR.
358  $ min( f, c, g, ca ).LE.sfmin2 )GO TO 190
359  f = f / sclfac
360  c = c / sclfac
361  g = g / sclfac
362  ca = ca / sclfac
363  r = r*sclfac
364  ra = ra*sclfac
365  GO TO 180
366 *
367 * Now balance.
368 *
369  190 CONTINUE
370  IF( ( c+r ).GE.factor*s )
371  $ GO TO 200
372  IF( f.LT.one .AND. scale( i ).LT.one ) THEN
373  IF( f*scale( i ).LE.sfmin1 )
374  $ GO TO 200
375  END IF
376  IF( f.GT.one .AND. scale( i ).GT.one ) THEN
377  IF( scale( i ).GE.sfmax1 / f )
378  $ GO TO 200
379  END IF
380  g = one / f
381  scale( i ) = scale( i )*f
382  noconv = .true.
383 *
384  CALL csscal( n-k+1, g, a( i, k ), lda )
385  CALL csscal( l, f, a( 1, i ), 1 )
386 *
387  200 CONTINUE
388 *
389  IF( noconv )
390  $ GO TO 140
391 *
392  210 CONTINUE
393  ilo = k
394  ihi = l
395 *
396  RETURN
397 *
398 * End of CGEBAL
399 *
400  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine cgebal(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
CGEBAL
Definition: cgebal.f:163
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:54
subroutine cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:52