LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
sgbequb.f
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1 *> \brief \b SGBEQUB
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGBEQUB( 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 *> SGBEQUB computes row and column scalings intended to equilibrate an
39 *> M-by-N matrix A and reduce its condition number. R returns the row
40 *> scale factors and C the column scale factors, chosen to try to make
41 *> 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 an absolute value of at most
43 *> the radix.
44 *>
45 *> R(i) and C(j) are restricted to be a power of the radix between
46 *> SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
47 *> of these scaling factors is not guaranteed to reduce the condition
48 *> number of A but works well in practice.
49 *>
50 *> This routine differs from SGEEQU by restricting the scaling factors
51 *> to a power of the radix. Baring over- and underflow, scaling by
52 *> these factors introduces no additional rounding errors. However, the
53 *> scaled entries' magnitured are no longer approximately 1 but lie
55 *> \endverbatim
56 *
57 * Arguments:
58 * ==========
59 *
60 *> \param[in] M
61 *> \verbatim
62 *> M is INTEGER
63 *> The number of rows of the matrix A. M >= 0.
64 *> \endverbatim
65 *>
66 *> \param[in] N
67 *> \verbatim
68 *> N is INTEGER
69 *> The number of columns of the matrix A. N >= 0.
70 *> \endverbatim
71 *>
72 *> \param[in] KL
73 *> \verbatim
74 *> KL is INTEGER
75 *> The number of subdiagonals within the band of A. KL >= 0.
76 *> \endverbatim
77 *>
78 *> \param[in] KU
79 *> \verbatim
80 *> KU is INTEGER
81 *> The number of superdiagonals within the band of A. KU >= 0.
82 *> \endverbatim
83 *>
84 *> \param[in] AB
85 *> \verbatim
86 *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
87 *> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
88 *> The j-th column of A is stored in the j-th column of the
89 *> array AB as follows:
90 *> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
91 *> \endverbatim
92 *>
93 *> \param[in] LDAB
94 *> \verbatim
95 *> LDAB is INTEGER
96 *> The leading dimension of the array A. LDAB >= max(1,M).
97 *> \endverbatim
98 *>
99 *> \param[out] R
100 *> \verbatim
101 *> R is REAL array, dimension (M)
102 *> If INFO = 0 or INFO > M, R contains the row scale factors
103 *> for A.
104 *> \endverbatim
105 *>
106 *> \param[out] C
107 *> \verbatim
108 *> C is REAL array, dimension (N)
109 *> If INFO = 0, C contains the column scale factors for A.
110 *> \endverbatim
111 *>
112 *> \param[out] ROWCND
113 *> \verbatim
114 *> ROWCND is REAL
115 *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
116 *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
117 *> AMAX is neither too large nor too small, it is not worth
118 *> scaling by R.
119 *> \endverbatim
120 *>
121 *> \param[out] COLCND
122 *> \verbatim
123 *> COLCND is REAL
124 *> If INFO = 0, COLCND contains the ratio of the smallest
125 *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
126 *> worth scaling by C.
127 *> \endverbatim
128 *>
129 *> \param[out] AMAX
130 *> \verbatim
131 *> AMAX is REAL
132 *> Absolute value of largest matrix element. If AMAX is very
133 *> close to overflow or very close to underflow, the matrix
134 *> should be scaled.
135 *> \endverbatim
136 *>
137 *> \param[out] INFO
138 *> \verbatim
139 *> INFO is INTEGER
140 *> = 0: successful exit
141 *> < 0: if INFO = -i, the i-th argument had an illegal value
142 *> > 0: if INFO = i, and i is
143 *> <= M: the i-th row of A is exactly zero
144 *> > M: the (i-M)-th column of A is exactly zero
145 *> \endverbatim
146 *
147 * Authors:
148 * ========
149 *
150 *> \author Univ. of Tennessee
151 *> \author Univ. of California Berkeley
152 *> \author Univ. of Colorado Denver
153 *> \author NAG Ltd.
154 *
155 *> \date November 2011
156 *
157 *> \ingroup realGBcomputational
158 *
159 * =====================================================================
160  SUBROUTINE sgbequb( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
161  \$ amax, info )
162 *
163 * -- LAPACK computational routine (version 3.4.0) --
164 * -- LAPACK is a software package provided by Univ. of Tennessee, --
165 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166 * November 2011
167 *
168 * .. Scalar Arguments ..
169  INTEGER info, kl, ku, ldab, m, n
170  REAL amax, colcnd, rowcnd
171 * ..
172 * .. Array Arguments ..
173  REAL ab( ldab, * ), c( * ), r( * )
174 * ..
175 *
176 * =====================================================================
177 *
178 * .. Parameters ..
179  REAL one, zero
180  parameter( one = 1.0e+0, zero = 0.0e+0 )
181 * ..
182 * .. Local Scalars ..
183  INTEGER i, j, kd
184  REAL bignum, rcmax, rcmin, smlnum, radix, logrdx
185 * ..
186 * .. External Functions ..
187  REAL slamch
188  EXTERNAL slamch
189 * ..
190 * .. External Subroutines ..
191  EXTERNAL xerbla
192 * ..
193 * .. Intrinsic Functions ..
194  INTRINSIC abs, max, min, log
195 * ..
196 * .. Executable Statements ..
197 *
198 * Test the input parameters.
199 *
200  info = 0
201  IF( m.LT.0 ) THEN
202  info = -1
203  ELSE IF( n.LT.0 ) THEN
204  info = -2
205  ELSE IF( kl.LT.0 ) THEN
206  info = -3
207  ELSE IF( ku.LT.0 ) THEN
208  info = -4
209  ELSE IF( ldab.LT.kl+ku+1 ) THEN
210  info = -6
211  END IF
212  IF( info.NE.0 ) THEN
213  CALL xerbla( 'SGBEQUB', -info )
214  return
215  END IF
216 *
217 * Quick return if possible.
218 *
219  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
220  rowcnd = one
221  colcnd = one
222  amax = zero
223  return
224  END IF
225 *
226 * Get machine constants. Assume SMLNUM is a power of the radix.
227 *
228  smlnum = slamch( 'S' )
229  bignum = one / smlnum
230  radix = slamch( 'B' )
231  logrdx = log(radix)
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 ), abs( ab( kd+i-j, j ) ) )
245  20 continue
246  30 continue
247  DO i = 1, m
248  IF( r( i ).GT.zero ) THEN
249  r( i ) = radix**int( log( r( i ) ) / logrdx )
250  END IF
251  END DO
252 *
253 * Find the maximum and minimum scale factors.
254 *
255  rcmin = bignum
256  rcmax = zero
257  DO 40 i = 1, m
258  rcmax = max( rcmax, r( i ) )
259  rcmin = min( rcmin, r( i ) )
260  40 continue
261  amax = rcmax
262 *
263  IF( rcmin.EQ.zero ) THEN
264 *
265 * Find the first zero scale factor and return an error code.
266 *
267  DO 50 i = 1, m
268  IF( r( i ).EQ.zero ) THEN
269  info = i
270  return
271  END IF
272  50 continue
273  ELSE
274 *
275 * Invert the scale factors.
276 *
277  DO 60 i = 1, m
278  r( i ) = one / min( max( r( i ), smlnum ), bignum )
279  60 continue
280 *
281 * Compute ROWCND = min(R(I)) / max(R(I)).
282 *
283  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
284  END IF
285 *
286 * Compute column scale factors.
287 *
288  DO 70 j = 1, n
289  c( j ) = zero
290  70 continue
291 *
292 * Find the maximum element in each column,
293 * assuming the row scaling computed above.
294 *
295  DO 90 j = 1, n
296  DO 80 i = max( j-ku, 1 ), min( j+kl, m )
297  c( j ) = max( c( j ), abs( ab( kd+i-j, j ) )*r( i ) )
298  80 continue
299  IF( c( j ).GT.zero ) THEN
300  c( j ) = radix**int( log( c( j ) ) / logrdx )
301  END IF
302  90 continue
303 *
304 * Find the maximum and minimum scale factors.
305 *
306  rcmin = bignum
307  rcmax = zero
308  DO 100 j = 1, n
309  rcmin = min( rcmin, c( j ) )
310  rcmax = max( rcmax, c( j ) )
311  100 continue
312 *
313  IF( rcmin.EQ.zero ) THEN
314 *
315 * Find the first zero scale factor and return an error code.
316 *
317  DO 110 j = 1, n
318  IF( c( j ).EQ.zero ) THEN
319  info = m + j
320  return
321  END IF
322  110 continue
323  ELSE
324 *
325 * Invert the scale factors.
326 *
327  DO 120 j = 1, n
328  c( j ) = one / min( max( c( j ), smlnum ), bignum )
329  120 continue
330 *
331 * Compute COLCND = min(C(J)) / max(C(J)).
332 *
333  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
334  END IF
335 *
336  return
337 *
338 * End of SGBEQUB
339 *
340  END