LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
dgbequ.f
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1 *> \brief \b DGBEQU
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DGBEQU( 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 AB( LDAB, * ), C( * ), R( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> DGBEQU computes row and column scalings intended to equilibrate an
39 *> M-by-N band matrix A and reduce its condition number. R returns the
40 *> row scale factors and C the column scale factors, chosen to try to
41 *> make the largest element in each row and column of the matrix B with
42 *> elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
43 *>
44 *> R(i) and C(j) are restricted to be between SMLNUM = smallest safe
45 *> number and BIGNUM = largest safe number. Use of these scaling
46 *> factors is not guaranteed to reduce the condition number of A but
47 *> works well in practice.
48 *> \endverbatim
49 *
50 * Arguments:
51 * ==========
52 *
53 *> \param[in] M
54 *> \verbatim
55 *> M is INTEGER
56 *> The number of rows of the matrix A. M >= 0.
57 *> \endverbatim
58 *>
59 *> \param[in] N
60 *> \verbatim
61 *> N is INTEGER
62 *> The number of columns of the matrix A. N >= 0.
63 *> \endverbatim
64 *>
65 *> \param[in] KL
66 *> \verbatim
67 *> KL is INTEGER
68 *> The number of subdiagonals within the band of A. KL >= 0.
69 *> \endverbatim
70 *>
71 *> \param[in] KU
72 *> \verbatim
73 *> KU is INTEGER
74 *> The number of superdiagonals within the band of A. KU >= 0.
75 *> \endverbatim
76 *>
77 *> \param[in] AB
78 *> \verbatim
79 *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
80 *> The band matrix A, stored in rows 1 to KL+KU+1. The j-th
81 *> column of A is stored in the j-th column of the array AB as
82 *> follows:
83 *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
84 *> \endverbatim
85 *>
86 *> \param[in] LDAB
87 *> \verbatim
88 *> LDAB is INTEGER
89 *> The leading dimension of the array AB. LDAB >= KL+KU+1.
90 *> \endverbatim
91 *>
92 *> \param[out] R
93 *> \verbatim
94 *> R is DOUBLE PRECISION array, dimension (M)
95 *> If INFO = 0, or INFO > M, R contains the row scale factors
96 *> for A.
97 *> \endverbatim
98 *>
99 *> \param[out] C
100 *> \verbatim
101 *> C is DOUBLE PRECISION array, dimension (N)
102 *> If INFO = 0, C contains the column scale factors for A.
103 *> \endverbatim
104 *>
105 *> \param[out] ROWCND
106 *> \verbatim
107 *> ROWCND is DOUBLE PRECISION
108 *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
109 *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
110 *> AMAX is neither too large nor too small, it is not worth
111 *> scaling by R.
112 *> \endverbatim
113 *>
114 *> \param[out] COLCND
115 *> \verbatim
116 *> COLCND is DOUBLE PRECISION
117 *> If INFO = 0, COLCND contains the ratio of the smallest
118 *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
119 *> worth scaling by C.
120 *> \endverbatim
121 *>
122 *> \param[out] AMAX
123 *> \verbatim
124 *> AMAX is DOUBLE PRECISION
125 *> Absolute value of largest matrix element. If AMAX is very
126 *> close to overflow or very close to underflow, the matrix
127 *> should be scaled.
128 *> \endverbatim
129 *>
130 *> \param[out] INFO
131 *> \verbatim
132 *> INFO is INTEGER
133 *> = 0: successful exit
134 *> < 0: if INFO = -i, the i-th argument had an illegal value
135 *> > 0: if INFO = i, and i is
136 *> <= M: the i-th row of A is exactly zero
137 *> > M: the (i-M)-th column of A is exactly zero
138 *> \endverbatim
139 *
140 * Authors:
141 * ========
142 *
143 *> \author Univ. of Tennessee
144 *> \author Univ. of California Berkeley
145 *> \author Univ. of Colorado Denver
146 *> \author NAG Ltd.
147 *
148 *> \date November 2011
149 *
150 *> \ingroup doubleGBcomputational
151 *
152 * =====================================================================
153  SUBROUTINE dgbequ( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
154  \$ amax, info )
155 *
156 * -- LAPACK computational routine (version 3.4.0) --
157 * -- LAPACK is a software package provided by Univ. of Tennessee, --
158 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
159 * November 2011
160 *
161 * .. Scalar Arguments ..
162  INTEGER info, kl, ku, ldab, m, n
163  DOUBLE PRECISION amax, colcnd, rowcnd
164 * ..
165 * .. Array Arguments ..
166  DOUBLE PRECISION ab( ldab, * ), c( * ), r( * )
167 * ..
168 *
169 * =====================================================================
170 *
171 * .. Parameters ..
172  DOUBLE PRECISION one, zero
173  parameter( one = 1.0d+0, zero = 0.0d+0 )
174 * ..
175 * .. Local Scalars ..
176  INTEGER i, j, kd
177  DOUBLE PRECISION bignum, rcmax, rcmin, smlnum
178 * ..
179 * .. External Functions ..
180  DOUBLE PRECISION dlamch
181  EXTERNAL dlamch
182 * ..
183 * .. External Subroutines ..
184  EXTERNAL xerbla
185 * ..
186 * .. Intrinsic Functions ..
187  INTRINSIC abs, max, min
188 * ..
189 * .. Executable Statements ..
190 *
191 * Test the input parameters
192 *
193  info = 0
194  IF( m.LT.0 ) THEN
195  info = -1
196  ELSE IF( n.LT.0 ) THEN
197  info = -2
198  ELSE IF( kl.LT.0 ) THEN
199  info = -3
200  ELSE IF( ku.LT.0 ) THEN
201  info = -4
202  ELSE IF( ldab.LT.kl+ku+1 ) THEN
203  info = -6
204  END IF
205  IF( info.NE.0 ) THEN
206  CALL xerbla( 'DGBEQU', -info )
207  return
208  END IF
209 *
210 * Quick return if possible
211 *
212  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
213  rowcnd = one
214  colcnd = one
215  amax = zero
216  return
217  END IF
218 *
219 * Get machine constants.
220 *
221  smlnum = dlamch( 'S' )
222  bignum = one / smlnum
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  kd = ku + 1
233  DO 30 j = 1, n
234  DO 20 i = max( j-ku, 1 ), min( j+kl, m )
235  r( i ) = max( r( i ), abs( ab( kd+i-j, j ) ) )
236  20 continue
237  30 continue
238 *
239 * Find the maximum and minimum scale factors.
240 *
241  rcmin = bignum
242  rcmax = zero
243  DO 40 i = 1, m
244  rcmax = max( rcmax, r( i ) )
245  rcmin = min( rcmin, r( i ) )
246  40 continue
247  amax = rcmax
248 *
249  IF( rcmin.EQ.zero ) THEN
250 *
251 * Find the first zero scale factor and return an error code.
252 *
253  DO 50 i = 1, m
254  IF( r( i ).EQ.zero ) THEN
255  info = i
256  return
257  END IF
258  50 continue
259  ELSE
260 *
261 * Invert the scale factors.
262 *
263  DO 60 i = 1, m
264  r( i ) = one / min( max( r( i ), smlnum ), bignum )
265  60 continue
266 *
267 * Compute ROWCND = min(R(I)) / max(R(I))
268 *
269  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
270  END IF
271 *
272 * Compute column scale factors
273 *
274  DO 70 j = 1, n
275  c( j ) = zero
276  70 continue
277 *
278 * Find the maximum element in each column,
279 * assuming the row scaling computed above.
280 *
281  kd = ku + 1
282  DO 90 j = 1, n
283  DO 80 i = max( j-ku, 1 ), min( j+kl, m )
284  c( j ) = max( c( j ), abs( ab( kd+i-j, j ) )*r( i ) )
285  80 continue
286  90 continue
287 *
288 * Find the maximum and minimum scale factors.
289 *
290  rcmin = bignum
291  rcmax = zero
292  DO 100 j = 1, n
293  rcmin = min( rcmin, c( j ) )
294  rcmax = max( rcmax, c( j ) )
295  100 continue
296 *
297  IF( rcmin.EQ.zero ) THEN
298 *
299 * Find the first zero scale factor and return an error code.
300 *
301  DO 110 j = 1, n
302  IF( c( j ).EQ.zero ) THEN
303  info = m + j
304  return
305  END IF
306  110 continue
307  ELSE
308 *
309 * Invert the scale factors.
310 *
311  DO 120 j = 1, n
312  c( j ) = one / min( max( c( j ), smlnum ), bignum )
313  120 continue
314 *
315 * Compute COLCND = min(C(J)) / max(C(J))
316 *
317  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
318  END IF
319 *
320  return
321 *
322 * End of DGBEQU
323 *
324  END