LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
zgbequ.f
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1 *> \brief \b ZGBEQU
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
22 * AMAX, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER INFO, KL, KU, LDAB, M, N
26 * DOUBLE PRECISION AMAX, COLCND, ROWCND
27 * ..
28 * .. Array Arguments ..
29 * DOUBLE PRECISION C( * ), R( * )
30 * COMPLEX*16 AB( LDAB, * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> ZGBEQU computes row and column scalings intended to equilibrate an
40 *> M-by-N band matrix A and reduce its condition number. R returns the
41 *> row scale factors and C the column scale factors, chosen to try to
42 *> make 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 absolute value 1.
44 *>
45 *> R(i) and C(j) are restricted to be between SMLNUM = smallest safe
46 *> number and BIGNUM = largest safe number. Use of these scaling
47 *> factors is not guaranteed to reduce the condition number of A but
48 *> works well in practice.
49 *> \endverbatim
50 *
51 * Arguments:
52 * ==========
53 *
54 *> \param[in] M
55 *> \verbatim
56 *> M is INTEGER
57 *> The number of rows of the matrix A. M >= 0.
58 *> \endverbatim
59 *>
60 *> \param[in] N
61 *> \verbatim
62 *> N is INTEGER
63 *> The number of columns of the matrix A. N >= 0.
64 *> \endverbatim
65 *>
66 *> \param[in] KL
67 *> \verbatim
68 *> KL is INTEGER
69 *> The number of subdiagonals within the band of A. KL >= 0.
70 *> \endverbatim
71 *>
72 *> \param[in] KU
73 *> \verbatim
74 *> KU is INTEGER
75 *> The number of superdiagonals within the band of A. KU >= 0.
76 *> \endverbatim
77 *>
78 *> \param[in] AB
79 *> \verbatim
80 *> AB is COMPLEX*16 array, dimension (LDAB,N)
81 *> The band matrix A, stored in rows 1 to KL+KU+1. The j-th
82 *> column of A is stored in the j-th column of the array AB as
83 *> follows:
84 *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
85 *> \endverbatim
86 *>
87 *> \param[in] LDAB
88 *> \verbatim
89 *> LDAB is INTEGER
90 *> The leading dimension of the array AB. LDAB >= KL+KU+1.
91 *> \endverbatim
92 *>
93 *> \param[out] R
94 *> \verbatim
95 *> R is DOUBLE PRECISION array, dimension (M)
96 *> If INFO = 0, or INFO > M, R contains the row scale factors
97 *> for A.
98 *> \endverbatim
99 *>
100 *> \param[out] C
101 *> \verbatim
102 *> C is DOUBLE PRECISION array, dimension (N)
103 *> If INFO = 0, C contains the column scale factors for A.
104 *> \endverbatim
105 *>
106 *> \param[out] ROWCND
107 *> \verbatim
108 *> ROWCND is DOUBLE PRECISION
109 *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
110 *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
111 *> AMAX is neither too large nor too small, it is not worth
112 *> scaling by R.
113 *> \endverbatim
114 *>
115 *> \param[out] COLCND
116 *> \verbatim
117 *> COLCND is DOUBLE PRECISION
118 *> If INFO = 0, COLCND contains the ratio of the smallest
119 *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
120 *> worth scaling by C.
121 *> \endverbatim
122 *>
123 *> \param[out] AMAX
124 *> \verbatim
125 *> AMAX is DOUBLE PRECISION
126 *> Absolute value of largest matrix element. If AMAX is very
127 *> close to overflow or very close to underflow, the matrix
128 *> should be scaled.
129 *> \endverbatim
130 *>
131 *> \param[out] INFO
132 *> \verbatim
133 *> INFO is INTEGER
134 *> = 0: successful exit
135 *> < 0: if INFO = -i, the i-th argument had an illegal value
136 *> > 0: if INFO = i, and i is
137 *> <= M: the i-th row of A is exactly zero
138 *> > M: the (i-M)-th column of A is exactly zero
139 *> \endverbatim
140 *
141 * Authors:
142 * ========
143 *
144 *> \author Univ. of Tennessee
145 *> \author Univ. of California Berkeley
146 *> \author Univ. of Colorado Denver
147 *> \author NAG Ltd.
148 *
149 *> \date November 2011
150 *
151 *> \ingroup complex16GBcomputational
152 *
153 * =====================================================================
154  SUBROUTINE zgbequ( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
155  \$ amax, info )
156 *
157 * -- LAPACK computational routine (version 3.4.0) --
158 * -- LAPACK is a software package provided by Univ. of Tennessee, --
159 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
160 * November 2011
161 *
162 * .. Scalar Arguments ..
163  INTEGER INFO, KL, KU, LDAB, M, N
164  DOUBLE PRECISION AMAX, COLCND, ROWCND
165 * ..
166 * .. Array Arguments ..
167  DOUBLE PRECISION C( * ), R( * )
168  COMPLEX*16 AB( ldab, * )
169 * ..
170 *
171 * =====================================================================
172 *
173 * .. Parameters ..
174  DOUBLE PRECISION ONE, ZERO
175  parameter ( one = 1.0d+0, zero = 0.0d+0 )
176 * ..
177 * .. Local Scalars ..
178  INTEGER I, J, KD
179  DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
180  COMPLEX*16 ZDUM
181 * ..
182 * .. External Functions ..
183  DOUBLE PRECISION DLAMCH
184  EXTERNAL dlamch
185 * ..
186 * .. External Subroutines ..
187  EXTERNAL xerbla
188 * ..
189 * .. Intrinsic Functions ..
190  INTRINSIC abs, dble, dimag, max, min
191 * ..
192 * .. Statement Functions ..
193  DOUBLE PRECISION CABS1
194 * ..
195 * .. Statement Function definitions ..
196  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
197 * ..
198 * .. Executable Statements ..
199 *
200 * Test the input parameters
201 *
202  info = 0
203  IF( m.LT.0 ) THEN
204  info = -1
205  ELSE IF( n.LT.0 ) THEN
206  info = -2
207  ELSE IF( kl.LT.0 ) THEN
208  info = -3
209  ELSE IF( ku.LT.0 ) THEN
210  info = -4
211  ELSE IF( ldab.LT.kl+ku+1 ) THEN
212  info = -6
213  END IF
214  IF( info.NE.0 ) THEN
215  CALL xerbla( 'ZGBEQU', -info )
216  RETURN
217  END IF
218 *
219 * Quick return if possible
220 *
221  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
222  rowcnd = one
223  colcnd = one
224  amax = zero
225  RETURN
226  END IF
227 *
228 * Get machine constants.
229 *
230  smlnum = dlamch( 'S' )
231  bignum = one / smlnum
232 *
233 * Compute row scale factors.
234 *
235  DO 10 i = 1, m
236  r( i ) = zero
237  10 CONTINUE
238 *
239 * Find the maximum element in each row.
240 *
241  kd = ku + 1
242  DO 30 j = 1, n
243  DO 20 i = max( j-ku, 1 ), min( j+kl, m )
244  r( i ) = max( r( i ), cabs1( ab( kd+i-j, j ) ) )
245  20 CONTINUE
246  30 CONTINUE
247 *
248 * Find the maximum and minimum scale factors.
249 *
250  rcmin = bignum
251  rcmax = zero
252  DO 40 i = 1, m
253  rcmax = max( rcmax, r( i ) )
254  rcmin = min( rcmin, r( i ) )
255  40 CONTINUE
256  amax = rcmax
257 *
258  IF( rcmin.EQ.zero ) THEN
259 *
260 * Find the first zero scale factor and return an error code.
261 *
262  DO 50 i = 1, m
263  IF( r( i ).EQ.zero ) THEN
264  info = i
265  RETURN
266  END IF
267  50 CONTINUE
268  ELSE
269 *
270 * Invert the scale factors.
271 *
272  DO 60 i = 1, m
273  r( i ) = one / min( max( r( i ), smlnum ), bignum )
274  60 CONTINUE
275 *
276 * Compute ROWCND = min(R(I)) / max(R(I))
277 *
278  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
279  END IF
280 *
281 * Compute column scale factors
282 *
283  DO 70 j = 1, n
284  c( j ) = zero
285  70 CONTINUE
286 *
287 * Find the maximum element in each column,
288 * assuming the row scaling computed above.
289 *
290  kd = ku + 1
291  DO 90 j = 1, n
292  DO 80 i = max( j-ku, 1 ), min( j+kl, m )
293  c( j ) = max( c( j ), cabs1( ab( kd+i-j, j ) )*r( i ) )
294  80 CONTINUE
295  90 CONTINUE
296 *
297 * Find the maximum and minimum scale factors.
298 *
299  rcmin = bignum
300  rcmax = zero
301  DO 100 j = 1, n
302  rcmin = min( rcmin, c( j ) )
303  rcmax = max( rcmax, c( j ) )
304  100 CONTINUE
305 *
306  IF( rcmin.EQ.zero ) THEN
307 *
308 * Find the first zero scale factor and return an error code.
309 *
310  DO 110 j = 1, n
311  IF( c( j ).EQ.zero ) THEN
312  info = m + j
313  RETURN
314  END IF
315  110 CONTINUE
316  ELSE
317 *
318 * Invert the scale factors.
319 *
320  DO 120 j = 1, n
321  c( j ) = one / min( max( c( j ), smlnum ), bignum )
322  120 CONTINUE
323 *
324 * Compute COLCND = min(C(J)) / max(C(J))
325 *
326  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
327  END IF
328 *
329  RETURN
330 *
331 * End of ZGBEQU
332 *
333  END
subroutine zgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
ZGBEQU
Definition: zgbequ.f:156
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62