LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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zgbequb.f
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1*> \brief \b ZGBEQUB
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download ZGBEQUB + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbequb.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbequb.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbequb.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGBEQUB( 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*> ZGBEQUB 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
44*> the radix.
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 ZGEEQU 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
55*> between sqrt(radix) and 1/sqrt(radix).
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] KL
74*> \verbatim
75*> KL is INTEGER
76*> The number of subdiagonals within the band of A. KL >= 0.
77*> \endverbatim
78*>
79*> \param[in] KU
80*> \verbatim
81*> KU is INTEGER
82*> The number of superdiagonals within the band of A. KU >= 0.
83*> \endverbatim
84*>
85*> \param[in] AB
86*> \verbatim
87*> AB is COMPLEX*16 array, dimension (LDAB,N)
88*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
89*> The j-th column of A is stored in the j-th column of the
90*> array AB as follows:
91*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
92*> \endverbatim
93*>
94*> \param[in] LDAB
95*> \verbatim
96*> LDAB is INTEGER
97*> The leading dimension of the array A. LDAB >= max(1,M).
98*> \endverbatim
99*>
100*> \param[out] R
101*> \verbatim
102*> R is DOUBLE PRECISION array, dimension (M)
103*> If INFO = 0 or INFO > M, R contains the row scale factors
104*> for A.
105*> \endverbatim
106*>
107*> \param[out] C
108*> \verbatim
109*> C is DOUBLE PRECISION array, dimension (N)
110*> If INFO = 0, C contains the column scale factors for A.
111*> \endverbatim
112*>
113*> \param[out] ROWCND
114*> \verbatim
115*> ROWCND is DOUBLE PRECISION
116*> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
117*> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
118*> AMAX is neither too large nor too small, it is not worth
119*> scaling by R.
120*> \endverbatim
121*>
122*> \param[out] COLCND
123*> \verbatim
124*> COLCND is DOUBLE PRECISION
125*> If INFO = 0, COLCND contains the ratio of the smallest
126*> C(i) to the largest C(i). If COLCND >= 0.1, it is not
127*> worth scaling by C.
128*> \endverbatim
129*>
130*> \param[out] AMAX
131*> \verbatim
132*> AMAX is DOUBLE PRECISION
133*> Absolute value of largest matrix element. If AMAX is very
134*> close to overflow or very close to underflow, the matrix
135*> should be scaled.
136*> \endverbatim
137*>
138*> \param[out] INFO
139*> \verbatim
140*> INFO is INTEGER
141*> = 0: successful exit
142*> < 0: if INFO = -i, the i-th argument had an illegal value
143*> > 0: if INFO = i, and i is
144*> <= M: the i-th row of A is exactly zero
145*> > M: the (i-M)-th column of A is exactly zero
146*> \endverbatim
147*
148* Authors:
149* ========
150*
151*> \author Univ. of Tennessee
152*> \author Univ. of California Berkeley
153*> \author Univ. of Colorado Denver
154*> \author NAG Ltd.
155*
156*> \ingroup complex16GBcomputational
157*
158* =====================================================================
159 SUBROUTINE zgbequb( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
160 $ AMAX, INFO )
161*
162* -- LAPACK computational routine --
163* -- LAPACK is a software package provided by Univ. of Tennessee, --
164* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
165*
166* .. Scalar Arguments ..
167 INTEGER INFO, KL, KU, LDAB, M, N
168 DOUBLE PRECISION AMAX, COLCND, ROWCND
169* ..
170* .. Array Arguments ..
171 DOUBLE PRECISION C( * ), R( * )
172 COMPLEX*16 AB( LDAB, * )
173* ..
174*
175* =====================================================================
176*
177* .. Parameters ..
178 DOUBLE PRECISION ONE, ZERO
179 parameter( one = 1.0d+0, zero = 0.0d+0 )
180* ..
181* .. Local Scalars ..
182 INTEGER I, J, KD
183 DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX,
184 $ logrdx
185 COMPLEX*16 ZDUM
186* ..
187* .. External Functions ..
188 DOUBLE PRECISION DLAMCH
189 EXTERNAL dlamch
190* ..
191* .. External Subroutines ..
192 EXTERNAL xerbla
193* ..
194* .. Intrinsic Functions ..
195 INTRINSIC abs, max, min, log, real, dimag
196* ..
197* .. Statement Functions ..
198 DOUBLE PRECISION CABS1
199* ..
200* .. Statement Function definitions ..
201 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
202* ..
203* .. Executable Statements ..
204*
205* Test the input parameters.
206*
207 info = 0
208 IF( m.LT.0 ) THEN
209 info = -1
210 ELSE IF( n.LT.0 ) THEN
211 info = -2
212 ELSE IF( kl.LT.0 ) THEN
213 info = -3
214 ELSE IF( ku.LT.0 ) THEN
215 info = -4
216 ELSE IF( ldab.LT.kl+ku+1 ) THEN
217 info = -6
218 END IF
219 IF( info.NE.0 ) THEN
220 CALL xerbla( 'ZGBEQUB', -info )
221 RETURN
222 END IF
223*
224* Quick return if possible.
225*
226 IF( m.EQ.0 .OR. n.EQ.0 ) THEN
227 rowcnd = one
228 colcnd = one
229 amax = zero
230 RETURN
231 END IF
232*
233* Get machine constants. Assume SMLNUM is a power of the radix.
234*
235 smlnum = dlamch( 'S' )
236 bignum = one / smlnum
237 radix = dlamch( 'B' )
238 logrdx = log(radix)
239*
240* Compute row scale factors.
241*
242 DO 10 i = 1, m
243 r( i ) = zero
244 10 CONTINUE
245*
246* Find the maximum element in each row.
247*
248 kd = ku + 1
249 DO 30 j = 1, n
250 DO 20 i = max( j-ku, 1 ), min( j+kl, m )
251 r( i ) = max( r( i ), cabs1( ab( kd+i-j, j ) ) )
252 20 CONTINUE
253 30 CONTINUE
254 DO i = 1, m
255 IF( r( i ).GT.zero ) THEN
256 r( i ) = radix**int( log( r( i ) ) / logrdx )
257 END IF
258 END DO
259*
260* Find the maximum and minimum scale factors.
261*
262 rcmin = bignum
263 rcmax = zero
264 DO 40 i = 1, m
265 rcmax = max( rcmax, r( i ) )
266 rcmin = min( rcmin, r( i ) )
267 40 CONTINUE
268 amax = rcmax
269*
270 IF( rcmin.EQ.zero ) THEN
271*
272* Find the first zero scale factor and return an error code.
273*
274 DO 50 i = 1, m
275 IF( r( i ).EQ.zero ) THEN
276 info = i
277 RETURN
278 END IF
279 50 CONTINUE
280 ELSE
281*
282* Invert the scale factors.
283*
284 DO 60 i = 1, m
285 r( i ) = one / min( max( r( i ), smlnum ), bignum )
286 60 CONTINUE
287*
288* Compute ROWCND = min(R(I)) / max(R(I)).
289*
290 rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
291 END IF
292*
293* Compute column scale factors.
294*
295 DO 70 j = 1, n
296 c( j ) = zero
297 70 CONTINUE
298*
299* Find the maximum element in each column,
300* assuming the row scaling computed above.
301*
302 DO 90 j = 1, n
303 DO 80 i = max( j-ku, 1 ), min( j+kl, m )
304 c( j ) = max( c( j ), cabs1( ab( kd+i-j, j ) )*r( i ) )
305 80 CONTINUE
306 IF( c( j ).GT.zero ) THEN
307 c( j ) = radix**int( log( c( j ) ) / logrdx )
308 END IF
309 90 CONTINUE
310*
311* Find the maximum and minimum scale factors.
312*
313 rcmin = bignum
314 rcmax = zero
315 DO 100 j = 1, n
316 rcmin = min( rcmin, c( j ) )
317 rcmax = max( rcmax, c( j ) )
318 100 CONTINUE
319*
320 IF( rcmin.EQ.zero ) THEN
321*
322* Find the first zero scale factor and return an error code.
323*
324 DO 110 j = 1, n
325 IF( c( j ).EQ.zero ) THEN
326 info = m + j
327 RETURN
328 END IF
329 110 CONTINUE
330 ELSE
331*
332* Invert the scale factors.
333*
334 DO 120 j = 1, n
335 c( j ) = one / min( max( c( j ), smlnum ), bignum )
336 120 CONTINUE
337*
338* Compute COLCND = min(C(J)) / max(C(J)).
339*
340 colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
341 END IF
342*
343 RETURN
344*
345* End of ZGBEQUB
346*
347 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zgbequb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
ZGBEQUB
Definition: zgbequb.f:161