LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
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/
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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 *> ILO is INTEGER
87 *> \endverbatim
88 *>
89 *> \param[out] IHI
90 *> \verbatim
91 *> IHI is INTEGER
92 *> ILO and IHI are set to INTEGER such that on exit
93 *> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
94 *> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
95 *> \endverbatim
96 *>
97 *> \param[out] SCALE
98 *> \verbatim
99 *> SCALE is DOUBLE PRECISION array, dimension (N)
100 *> Details of the permutations and scaling factors applied to
101 *> A. If P(j) is the index of the row and column interchanged
102 *> with row and column j and D(j) is the scaling factor
103 *> applied to row and column j, then
104 *> SCALE(j) = P(j) for j = 1,...,ILO-1
105 *> = D(j) for j = ILO,...,IHI
106 *> = P(j) for j = IHI+1,...,N.
107 *> The order in which the interchanges are made is N to IHI+1,
108 *> then 1 to ILO-1.
109 *> \endverbatim
110 *>
111 *> \param[out] INFO
112 *> \verbatim
113 *> INFO is INTEGER
114 *> = 0: successful exit.
115 *> < 0: if INFO = -i, the i-th argument had an illegal value.
116 *> \endverbatim
117 *
118 * Authors:
119 * ========
120 *
121 *> \author Univ. of Tennessee
122 *> \author Univ. of California Berkeley
123 *> \author Univ. of Colorado Denver
124 *> \author NAG Ltd.
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 --
164 * -- LAPACK is a software package provided by Univ. of Tennessee, --
165 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166 *
167 * .. Scalar Arguments ..
168  CHARACTER JOB
169  INTEGER IHI, ILO, INFO, LDA, N
170 * ..
171 * .. Array Arguments ..
172  DOUBLE PRECISION SCALE( * )
173  COMPLEX*16 A( LDA, * )
174 * ..
175 *
176 * =====================================================================
177 *
178 * .. Parameters ..
179  DOUBLE PRECISION ZERO, ONE
180  parameter( zero = 0.0d+0, one = 1.0d+0 )
181  DOUBLE PRECISION SCLFAC
182  parameter( sclfac = 2.0d+0 )
183  DOUBLE PRECISION FACTOR
184  parameter( factor = 0.95d+0 )
185 * ..
186 * .. Local Scalars ..
187  LOGICAL NOCONV
188  INTEGER I, ICA, IEXC, IRA, J, K, L, M
189  DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
190  $ SFMIN2
191 * ..
192 * .. External Functions ..
193  LOGICAL DISNAN, LSAME
194  INTEGER IZAMAX
195  DOUBLE PRECISION DLAMCH, DZNRM2
196  EXTERNAL disnan, lsame, izamax, dlamch, dznrm2
197 * ..
198 * .. External Subroutines ..
199  EXTERNAL xerbla, zdscal, zswap
200 * ..
201 * .. Intrinsic Functions ..
202  INTRINSIC abs, dble, dimag, max, min
203 *
204 * Test the input parameters
205 *
206  info = 0
207  IF( .NOT.lsame( job, 'N' ) .AND. .NOT.lsame( job, 'P' ) .AND.
208  $ .NOT.lsame( job, 'S' ) .AND. .NOT.lsame( job, 'B' ) ) THEN
209  info = -1
210  ELSE IF( n.LT.0 ) THEN
211  info = -2
212  ELSE IF( lda.LT.max( 1, n ) ) THEN
213  info = -4
214  END IF
215  IF( info.NE.0 ) THEN
216  CALL xerbla( 'ZGEBAL', -info )
217  RETURN
218  END IF
219 *
220  k = 1
221  l = n
222 *
223  IF( n.EQ.0 )
224  $ GO TO 210
225 *
226  IF( lsame( job, 'N' ) ) THEN
227  DO 10 i = 1, n
228  scale( i ) = one
229  10 CONTINUE
230  GO TO 210
231  END IF
232 *
233  IF( lsame( job, 'S' ) )
234  $ GO TO 120
235 *
236 * Permutation to isolate eigenvalues if possible
237 *
238  GO TO 50
239 *
240 * Row and column exchange.
241 *
242  20 CONTINUE
243  scale( m ) = j
244  IF( j.EQ.m )
245  $ GO TO 30
246 *
247  CALL zswap( l, a( 1, j ), 1, a( 1, m ), 1 )
248  CALL zswap( n-k+1, a( j, k ), lda, a( m, k ), lda )
249 *
250  30 CONTINUE
251  GO TO ( 40, 80 )iexc
252 *
253 * Search for rows isolating an eigenvalue and push them down.
254 *
255  40 CONTINUE
256  IF( l.EQ.1 )
257  $ GO TO 210
258  l = l - 1
259 *
260  50 CONTINUE
261  DO 70 j = l, 1, -1
262 *
263  DO 60 i = 1, l
264  IF( i.EQ.j )
265  $ GO TO 60
266  IF( dble( a( j, i ) ).NE.zero .OR. dimag( a( j, i ) ).NE.
267  $ zero )GO TO 70
268  60 CONTINUE
269 *
270  m = l
271  iexc = 1
272  GO TO 20
273  70 CONTINUE
274 *
275  GO TO 90
276 *
277 * Search for columns isolating an eigenvalue and push them left.
278 *
279  80 CONTINUE
280  k = k + 1
281 *
282  90 CONTINUE
283  DO 110 j = k, l
284 *
285  DO 100 i = k, l
286  IF( i.EQ.j )
287  $ GO TO 100
288  IF( dble( a( i, j ) ).NE.zero .OR. dimag( a( i, j ) ).NE.
289  $ zero )GO TO 110
290  100 CONTINUE
291 *
292  m = k
293  iexc = 2
294  GO TO 20
295  110 CONTINUE
296 *
297  120 CONTINUE
298  DO 130 i = k, l
299  scale( i ) = one
300  130 CONTINUE
301 *
302  IF( lsame( job, 'P' ) )
303  $ GO TO 210
304 *
305 * Balance the submatrix in rows K to L.
306 *
307 * Iterative loop for norm reduction
308 *
309  sfmin1 = dlamch( 'S' ) / dlamch( 'P' )
310  sfmax1 = one / sfmin1
311  sfmin2 = sfmin1*sclfac
312  sfmax2 = one / sfmin2
313  140 CONTINUE
314  noconv = .false.
315 *
316  DO 200 i = k, l
317 *
318  c = dznrm2( l-k+1, a( k, i ), 1 )
319  r = dznrm2( l-k+1, a( i, k ), lda )
320  ica = izamax( l, a( 1, i ), 1 )
321  ca = abs( a( ica, i ) )
322  ira = izamax( n-k+1, a( i, k ), lda )
323  ra = abs( a( i, ira+k-1 ) )
324 *
325 * Guard against zero C or R due to underflow.
326 *
327  IF( c.EQ.zero .OR. r.EQ.zero )
328  $ GO TO 200
329  g = r / sclfac
330  f = one
331  s = c + r
332  160 CONTINUE
333  IF( c.GE.g .OR. max( f, c, ca ).GE.sfmax2 .OR.
334  $ min( r, g, ra ).LE.sfmin2 )GO TO 170
335  IF( disnan( c+f+ca+r+g+ra ) ) THEN
336 *
337 * Exit if NaN to avoid infinite loop
338 *
339  info = -3
340  CALL xerbla( 'ZGEBAL', -info )
341  RETURN
342  END IF
343  f = f*sclfac
344  c = c*sclfac
345  ca = ca*sclfac
346  r = r / sclfac
347  g = g / sclfac
348  ra = ra / sclfac
349  GO TO 160
350 *
351  170 CONTINUE
352  g = c / sclfac
353  180 CONTINUE
354  IF( g.LT.r .OR. max( r, ra ).GE.sfmax2 .OR.
355  $ min( f, c, g, ca ).LE.sfmin2 )GO TO 190
356  f = f / sclfac
357  c = c / sclfac
358  g = g / sclfac
359  ca = ca / sclfac
360  r = r*sclfac
361  ra = ra*sclfac
362  GO TO 180
363 *
364 * Now balance.
365 *
366  190 CONTINUE
367  IF( ( c+r ).GE.factor*s )
368  $ GO TO 200
369  IF( f.LT.one .AND. scale( i ).LT.one ) THEN
370  IF( f*scale( i ).LE.sfmin1 )
371  $ GO TO 200
372  END IF
373  IF( f.GT.one .AND. scale( i ).GT.one ) THEN
374  IF( scale( i ).GE.sfmax1 / f )
375  $ GO TO 200
376  END IF
377  g = one / f
378  scale( i ) = scale( i )*f
379  noconv = .true.
380 *
381  CALL zdscal( n-k+1, g, a( i, k ), lda )
382  CALL zdscal( l, f, a( 1, i ), 1 )
383 *
384  200 CONTINUE
385 *
386  IF( noconv )
387  $ GO TO 140
388 *
389  210 CONTINUE
390  ilo = k
391  ihi = l
392 *
393  RETURN
394 *
395 * End of ZGEBAL
396 *
397  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine zgebal(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
ZGEBAL
Definition: zgebal.f:162