LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
sgbequ.f
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1 *> \brief \b SGBEQU
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGBEQU( 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 * REAL AMAX, COLCND, ROWCND
27 * ..
28 * .. Array Arguments ..
29 * REAL AB( LDAB, * ), C( * ), R( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> SGBEQU 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 REAL 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 REAL 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 REAL 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 REAL
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 REAL
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 REAL
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 *> \ingroup realGBcomputational
149 *
150 * =====================================================================
151  SUBROUTINE sgbequ( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
152  \$ AMAX, INFO )
153 *
154 * -- LAPACK computational routine --
155 * -- LAPACK is a software package provided by Univ. of Tennessee, --
156 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
157 *
158 * .. Scalar Arguments ..
159  INTEGER INFO, KL, KU, LDAB, M, N
160  REAL AMAX, COLCND, ROWCND
161 * ..
162 * .. Array Arguments ..
163  REAL AB( LDAB, * ), C( * ), R( * )
164 * ..
165 *
166 * =====================================================================
167 *
168 * .. Parameters ..
169  REAL ONE, ZERO
170  parameter( one = 1.0e+0, zero = 0.0e+0 )
171 * ..
172 * .. Local Scalars ..
173  INTEGER I, J, KD
174  REAL BIGNUM, RCMAX, RCMIN, SMLNUM
175 * ..
176 * .. External Functions ..
177  REAL SLAMCH
178  EXTERNAL slamch
179 * ..
180 * .. External Subroutines ..
181  EXTERNAL xerbla
182 * ..
183 * .. Intrinsic Functions ..
184  INTRINSIC abs, max, min
185 * ..
186 * .. Executable Statements ..
187 *
188 * Test the input parameters
189 *
190  info = 0
191  IF( m.LT.0 ) THEN
192  info = -1
193  ELSE IF( n.LT.0 ) THEN
194  info = -2
195  ELSE IF( kl.LT.0 ) THEN
196  info = -3
197  ELSE IF( ku.LT.0 ) THEN
198  info = -4
199  ELSE IF( ldab.LT.kl+ku+1 ) THEN
200  info = -6
201  END IF
202  IF( info.NE.0 ) THEN
203  CALL xerbla( 'SGBEQU', -info )
204  RETURN
205  END IF
206 *
207 * Quick return if possible
208 *
209  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
210  rowcnd = one
211  colcnd = one
212  amax = zero
213  RETURN
214  END IF
215 *
216 * Get machine constants.
217 *
218  smlnum = slamch( 'S' )
219  bignum = one / smlnum
220 *
221 * Compute row scale factors.
222 *
223  DO 10 i = 1, m
224  r( i ) = zero
225  10 CONTINUE
226 *
227 * Find the maximum element in each row.
228 *
229  kd = ku + 1
230  DO 30 j = 1, n
231  DO 20 i = max( j-ku, 1 ), min( j+kl, m )
232  r( i ) = max( r( i ), abs( ab( kd+i-j, j ) ) )
233  20 CONTINUE
234  30 CONTINUE
235 *
236 * Find the maximum and minimum scale factors.
237 *
238  rcmin = bignum
239  rcmax = zero
240  DO 40 i = 1, m
241  rcmax = max( rcmax, r( i ) )
242  rcmin = min( rcmin, r( i ) )
243  40 CONTINUE
244  amax = rcmax
245 *
246  IF( rcmin.EQ.zero ) THEN
247 *
248 * Find the first zero scale factor and return an error code.
249 *
250  DO 50 i = 1, m
251  IF( r( i ).EQ.zero ) THEN
252  info = i
253  RETURN
254  END IF
255  50 CONTINUE
256  ELSE
257 *
258 * Invert the scale factors.
259 *
260  DO 60 i = 1, m
261  r( i ) = one / min( max( r( i ), smlnum ), bignum )
262  60 CONTINUE
263 *
264 * Compute ROWCND = min(R(I)) / max(R(I))
265 *
266  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
267  END IF
268 *
269 * Compute column scale factors
270 *
271  DO 70 j = 1, n
272  c( j ) = zero
273  70 CONTINUE
274 *
275 * Find the maximum element in each column,
276 * assuming the row scaling computed above.
277 *
278  kd = ku + 1
279  DO 90 j = 1, n
280  DO 80 i = max( j-ku, 1 ), min( j+kl, m )
281  c( j ) = max( c( j ), abs( ab( kd+i-j, j ) )*r( i ) )
282  80 CONTINUE
283  90 CONTINUE
284 *
285 * Find the maximum and minimum scale factors.
286 *
287  rcmin = bignum
288  rcmax = zero
289  DO 100 j = 1, n
290  rcmin = min( rcmin, c( j ) )
291  rcmax = max( rcmax, c( j ) )
292  100 CONTINUE
293 *
294  IF( rcmin.EQ.zero ) THEN
295 *
296 * Find the first zero scale factor and return an error code.
297 *
298  DO 110 j = 1, n
299  IF( c( j ).EQ.zero ) THEN
300  info = m + j
301  RETURN
302  END IF
303  110 CONTINUE
304  ELSE
305 *
306 * Invert the scale factors.
307 *
308  DO 120 j = 1, n
309  c( j ) = one / min( max( c( j ), smlnum ), bignum )
310  120 CONTINUE
311 *
312 * Compute COLCND = min(C(J)) / max(C(J))
313 *
314  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
315  END IF
316 *
317  RETURN
318 *
319 * End of SGBEQU
320 *
321  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
SGBEQU
Definition: sgbequ.f:153