LAPACK  3.8.0 LAPACK: Linear Algebra PACKage
cgeequb.f
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1 *> \brief \b CGEEQUB
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
22 * INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER INFO, LDA, M, N
26 * REAL AMAX, COLCND, ROWCND
27 * ..
28 * .. Array Arguments ..
29 * REAL C( * ), R( * )
30 * COMPLEX A( LDA, * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> CGEEQUB computes row and column scalings intended to equilibrate an
40 *> M-by-N matrix A and reduce its condition number. R returns the row
41 *> scale factors and C the column scale factors, chosen to try to make
42 *> the largest element in each row and column of the matrix B with
43 *> elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
45 *>
46 *> R(i) and C(j) are restricted to be a power of the radix between
47 *> SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
48 *> of these scaling factors is not guaranteed to reduce the condition
49 *> number of A but works well in practice.
50 *>
51 *> This routine differs from CGEEQU by restricting the scaling factors
52 *> to a power of the radix. Barring over- and underflow, scaling by
53 *> these factors introduces no additional rounding errors. However, the
54 *> scaled entries' magnitudes are no longer approximately 1 but lie
56 *> \endverbatim
57 *
58 * Arguments:
59 * ==========
60 *
61 *> \param[in] M
62 *> \verbatim
63 *> M is INTEGER
64 *> The number of rows of the matrix A. M >= 0.
65 *> \endverbatim
66 *>
67 *> \param[in] N
68 *> \verbatim
69 *> N is INTEGER
70 *> The number of columns of the matrix A. N >= 0.
71 *> \endverbatim
72 *>
73 *> \param[in] A
74 *> \verbatim
75 *> A is COMPLEX array, dimension (LDA,N)
76 *> The M-by-N matrix whose equilibration factors are
77 *> to be computed.
78 *> \endverbatim
79 *>
80 *> \param[in] LDA
81 *> \verbatim
82 *> LDA is INTEGER
83 *> The leading dimension of the array A. LDA >= max(1,M).
84 *> \endverbatim
85 *>
86 *> \param[out] R
87 *> \verbatim
88 *> R is REAL array, dimension (M)
89 *> If INFO = 0 or INFO > M, R contains the row scale factors
90 *> for A.
91 *> \endverbatim
92 *>
93 *> \param[out] C
94 *> \verbatim
95 *> C is REAL array, dimension (N)
96 *> If INFO = 0, C contains the column scale factors for A.
97 *> \endverbatim
98 *>
99 *> \param[out] ROWCND
100 *> \verbatim
101 *> ROWCND is REAL
102 *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
103 *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
104 *> AMAX is neither too large nor too small, it is not worth
105 *> scaling by R.
106 *> \endverbatim
107 *>
108 *> \param[out] COLCND
109 *> \verbatim
110 *> COLCND is REAL
111 *> If INFO = 0, COLCND contains the ratio of the smallest
112 *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
113 *> worth scaling by C.
114 *> \endverbatim
115 *>
116 *> \param[out] AMAX
117 *> \verbatim
118 *> AMAX is REAL
119 *> Absolute value of largest matrix element. If AMAX is very
120 *> close to overflow or very close to underflow, the matrix
121 *> should be scaled.
122 *> \endverbatim
123 *>
124 *> \param[out] INFO
125 *> \verbatim
126 *> INFO is INTEGER
127 *> = 0: successful exit
128 *> < 0: if INFO = -i, the i-th argument had an illegal value
129 *> > 0: if INFO = i, and i is
130 *> <= M: the i-th row of A is exactly zero
131 *> > M: the (i-M)-th column of A is exactly zero
132 *> \endverbatim
133 *
134 * Authors:
135 * ========
136 *
137 *> \author Univ. of Tennessee
138 *> \author Univ. of California Berkeley
139 *> \author Univ. of Colorado Denver
140 *> \author NAG Ltd.
141 *
142 *> \date December 2016
143 *
144 *> \ingroup complexGEcomputational
145 *
146 * =====================================================================
147  SUBROUTINE cgeequb( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
148  \$ INFO )
149 *
150 * -- LAPACK computational routine (version 3.7.0) --
151 * -- LAPACK is a software package provided by Univ. of Tennessee, --
152 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
153 * December 2016
154 *
155 * .. Scalar Arguments ..
156  INTEGER INFO, LDA, M, N
157  REAL AMAX, COLCND, ROWCND
158 * ..
159 * .. Array Arguments ..
160  REAL C( * ), R( * )
161  COMPLEX A( lda, * )
162 * ..
163 *
164 * =====================================================================
165 *
166 * .. Parameters ..
167  REAL ONE, ZERO
168  parameter( one = 1.0e+0, zero = 0.0e+0 )
169 * ..
170 * .. Local Scalars ..
171  INTEGER I, J
172  REAL BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
173  COMPLEX ZDUM
174 * ..
175 * .. External Functions ..
176  REAL SLAMCH
177  EXTERNAL slamch
178 * ..
179 * .. External Subroutines ..
180  EXTERNAL xerbla
181 * ..
182 * .. Intrinsic Functions ..
183  INTRINSIC abs, max, min, log, REAL, AIMAG
184 * ..
185 * .. Statement Functions ..
186  REAL CABS1
187 * ..
188 * .. Statement Function definitions ..
189  cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( AIMAG( zdum ) )
190 * ..
191 * .. Executable Statements ..
192 *
193 * Test the input parameters.
194 *
195  info = 0
196  IF( m.LT.0 ) THEN
197  info = -1
198  ELSE IF( n.LT.0 ) THEN
199  info = -2
200  ELSE IF( lda.LT.max( 1, m ) ) THEN
201  info = -4
202  END IF
203  IF( info.NE.0 ) THEN
204  CALL xerbla( 'CGEEQUB', -info )
205  RETURN
206  END IF
207 *
208 * Quick return if possible.
209 *
210  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
211  rowcnd = one
212  colcnd = one
213  amax = zero
214  RETURN
215  END IF
216 *
217 * Get machine constants. Assume SMLNUM is a power of the radix.
218 *
219  smlnum = slamch( 'S' )
220  bignum = one / smlnum
221  radix = slamch( 'B' )
222  logrdx = log( radix )
223 *
224 * Compute row scale factors.
225 *
226  DO 10 i = 1, m
227  r( i ) = zero
228  10 CONTINUE
229 *
230 * Find the maximum element in each row.
231 *
232  DO 30 j = 1, n
233  DO 20 i = 1, m
234  r( i ) = max( r( i ), cabs1( a( i, j ) ) )
235  20 CONTINUE
236  30 CONTINUE
237  DO i = 1, m
238  IF( r( i ).GT.zero ) THEN
239  r( i ) = radix**int( log(r( i ) ) / logrdx )
240  END IF
241  END DO
242 *
243 * Find the maximum and minimum scale factors.
244 *
245  rcmin = bignum
246  rcmax = zero
247  DO 40 i = 1, m
248  rcmax = max( rcmax, r( i ) )
249  rcmin = min( rcmin, r( i ) )
250  40 CONTINUE
251  amax = rcmax
252 *
253  IF( rcmin.EQ.zero ) THEN
254 *
255 * Find the first zero scale factor and return an error code.
256 *
257  DO 50 i = 1, m
258  IF( r( i ).EQ.zero ) THEN
259  info = i
260  RETURN
261  END IF
262  50 CONTINUE
263  ELSE
264 *
265 * Invert the scale factors.
266 *
267  DO 60 i = 1, m
268  r( i ) = one / min( max( r( i ), smlnum ), bignum )
269  60 CONTINUE
270 *
271 * Compute ROWCND = min(R(I)) / max(R(I)).
272 *
273  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
274  END IF
275 *
276 * Compute column scale factors.
277 *
278  DO 70 j = 1, n
279  c( j ) = zero
280  70 CONTINUE
281 *
282 * Find the maximum element in each column,
283 * assuming the row scaling computed above.
284 *
285  DO 90 j = 1, n
286  DO 80 i = 1, m
287  c( j ) = max( c( j ), cabs1( a( i, j ) )*r( i ) )
288  80 CONTINUE
289  IF( c( j ).GT.zero ) THEN
290  c( j ) = radix**int( log( c( j ) ) / logrdx )
291  END IF
292  90 CONTINUE
293 *
294 * Find the maximum and minimum scale factors.
295 *
296  rcmin = bignum
297  rcmax = zero
298  DO 100 j = 1, n
299  rcmin = min( rcmin, c( j ) )
300  rcmax = max( rcmax, c( j ) )
301  100 CONTINUE
302 *
303  IF( rcmin.EQ.zero ) THEN
304 *
305 * Find the first zero scale factor and return an error code.
306 *
307  DO 110 j = 1, n
308  IF( c( j ).EQ.zero ) THEN
309  info = m + j
310  RETURN
311  END IF
312  110 CONTINUE
313  ELSE
314 *
315 * Invert the scale factors.
316 *
317  DO 120 j = 1, n
318  c( j ) = one / min( max( c( j ), smlnum ), bignum )
319  120 CONTINUE
320 *
321 * Compute COLCND = min(C(J)) / max(C(J)).
322 *
323  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
324  END IF
325 *
326  RETURN
327 *
328 * End of CGEEQUB
329 *
330  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine cgeequb(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
CGEEQUB
Definition: cgeequb.f:149