 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dgbequb()

 subroutine dgbequb ( integer M, integer N, integer KL, integer KU, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, integer INFO )

DGBEQUB

Purpose:
``` DGBEQUB computes row and column scalings intended to equilibrate an
M-by-N matrix A and reduce its condition number.  R returns the row
scale factors and C the column scale factors, chosen to try to make
the largest element in each row and column of the matrix B with
elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most

R(i) and C(j) are restricted to be a power of the radix between
SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
of these scaling factors is not guaranteed to reduce the condition
number of A but works well in practice.

This routine differs from DGEEQU by restricting the scaling factors
to a power of the radix.  Barring over- and underflow, scaling by
these factors introduces no additional rounding errors.  However, the
scaled entries' magnitudes are no longer approximately 1 but lie
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in] KL ``` KL is INTEGER The number of subdiagonals within the band of A. KL >= 0.``` [in] KU ``` KU is INTEGER The number of superdiagonals within the band of A. KU >= 0.``` [in] AB ``` AB is DOUBLE PRECISION array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array A. LDAB >= max(1,M).``` [out] R ``` R is DOUBLE PRECISION array, dimension (M) If INFO = 0 or INFO > M, R contains the row scale factors for A.``` [out] C ``` C is DOUBLE PRECISION array, dimension (N) If INFO = 0, C contains the column scale factors for A.``` [out] ROWCND ``` ROWCND is DOUBLE PRECISION If INFO = 0 or INFO > M, ROWCND contains the ratio of the smallest R(i) to the largest R(i). If ROWCND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by R.``` [out] COLCND ``` COLCND is DOUBLE PRECISION If INFO = 0, COLCND contains the ratio of the smallest C(i) to the largest C(i). If COLCND >= 0.1, it is not worth scaling by C.``` [out] AMAX ``` AMAX is DOUBLE PRECISION Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is <= M: the i-th row of A is exactly zero > M: the (i-M)-th column of A is exactly zero```

Definition at line 158 of file dgbequb.f.

160*
161* -- LAPACK computational routine --
162* -- LAPACK is a software package provided by Univ. of Tennessee, --
163* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
164*
165* .. Scalar Arguments ..
166 INTEGER INFO, KL, KU, LDAB, M, N
167 DOUBLE PRECISION AMAX, COLCND, ROWCND
168* ..
169* .. Array Arguments ..
170 DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
171* ..
172*
173* =====================================================================
174*
175* .. Parameters ..
176 DOUBLE PRECISION ONE, ZERO
177 parameter( one = 1.0d+0, zero = 0.0d+0 )
178* ..
179* .. Local Scalars ..
180 INTEGER I, J, KD
181 DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
182* ..
183* .. External Functions ..
184 DOUBLE PRECISION DLAMCH
185 EXTERNAL dlamch
186* ..
187* .. External Subroutines ..
188 EXTERNAL xerbla
189* ..
190* .. Intrinsic Functions ..
191 INTRINSIC abs, max, min, log
192* ..
193* .. Executable Statements ..
194*
195* Test the input parameters.
196*
197 info = 0
198 IF( m.LT.0 ) THEN
199 info = -1
200 ELSE IF( n.LT.0 ) THEN
201 info = -2
202 ELSE IF( kl.LT.0 ) THEN
203 info = -3
204 ELSE IF( ku.LT.0 ) THEN
205 info = -4
206 ELSE IF( ldab.LT.kl+ku+1 ) THEN
207 info = -6
208 END IF
209 IF( info.NE.0 ) THEN
210 CALL xerbla( 'DGBEQUB', -info )
211 RETURN
212 END IF
213*
214* Quick return if possible.
215*
216 IF( m.EQ.0 .OR. n.EQ.0 ) THEN
217 rowcnd = one
218 colcnd = one
219 amax = zero
220 RETURN
221 END IF
222*
223* Get machine constants. Assume SMLNUM is a power of the radix.
224*
225 smlnum = dlamch( 'S' )
226 bignum = one / smlnum
227 radix = dlamch( 'B' )
229*
230* Compute row scale factors.
231*
232 DO 10 i = 1, m
233 r( i ) = zero
234 10 CONTINUE
235*
236* Find the maximum element in each row.
237*
238 kd = ku + 1
239 DO 30 j = 1, n
240 DO 20 i = max( j-ku, 1 ), min( j+kl, m )
241 r( i ) = max( r( i ), abs( ab( kd+i-j, j ) ) )
242 20 CONTINUE
243 30 CONTINUE
244 DO i = 1, m
245 IF( r( i ).GT.zero ) THEN
246 r( i ) = radix**int( log( r( i ) ) / logrdx )
247 END IF
248 END DO
249*
250* Find the maximum and minimum scale factors.
251*
252 rcmin = bignum
253 rcmax = zero
254 DO 40 i = 1, m
255 rcmax = max( rcmax, r( i ) )
256 rcmin = min( rcmin, r( i ) )
257 40 CONTINUE
258 amax = rcmax
259*
260 IF( rcmin.EQ.zero ) THEN
261*
262* Find the first zero scale factor and return an error code.
263*
264 DO 50 i = 1, m
265 IF( r( i ).EQ.zero ) THEN
266 info = i
267 RETURN
268 END IF
269 50 CONTINUE
270 ELSE
271*
272* Invert the scale factors.
273*
274 DO 60 i = 1, m
275 r( i ) = one / min( max( r( i ), smlnum ), bignum )
276 60 CONTINUE
277*
278* Compute ROWCND = min(R(I)) / max(R(I)).
279*
280 rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
281 END IF
282*
283* Compute column scale factors.
284*
285 DO 70 j = 1, n
286 c( j ) = zero
287 70 CONTINUE
288*
289* Find the maximum element in each column,
290* assuming the row scaling computed above.
291*
292 DO 90 j = 1, n
293 DO 80 i = max( j-ku, 1 ), min( j+kl, m )
294 c( j ) = max( c( j ), abs( ab( kd+i-j, j ) )*r( i ) )
295 80 CONTINUE
296 IF( c( j ).GT.zero ) THEN
297 c( j ) = radix**int( log( c( j ) ) / logrdx )
298 END IF
299 90 CONTINUE
300*
301* Find the maximum and minimum scale factors.
302*
303 rcmin = bignum
304 rcmax = zero
305 DO 100 j = 1, n
306 rcmin = min( rcmin, c( j ) )
307 rcmax = max( rcmax, c( j ) )
308 100 CONTINUE
309*
310 IF( rcmin.EQ.zero ) THEN
311*
312* Find the first zero scale factor and return an error code.
313*
314 DO 110 j = 1, n
315 IF( c( j ).EQ.zero ) THEN
316 info = m + j
317 RETURN
318 END IF
319 110 CONTINUE
320 ELSE
321*
322* Invert the scale factors.
323*
324 DO 120 j = 1, n
325 c( j ) = one / min( max( c( j ), smlnum ), bignum )
326 120 CONTINUE
327*
328* Compute COLCND = min(C(J)) / max(C(J)).
329*
330 colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
331 END IF
332*
333 RETURN
334*
335* End of DGBEQUB
336*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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