LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 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.  Baring over- and underflow, scaling by
these factors introduces no additional rounding errors.  However, the
scaled entries' magnitured 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```
Date
November 2011

Definition at line 162 of file dgbequb.f.

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  DOUBLE PRECISION amax, colcnd, rowcnd
171 * ..
172 * .. Array Arguments ..
173  DOUBLE PRECISION ab( ldab, * ), c( * ), r( * )
174 * ..
175 *
176 * =====================================================================
177 *
178 * .. Parameters ..
179  DOUBLE PRECISION one, zero
180  parameter ( one = 1.0d+0, zero = 0.0d+0 )
181 * ..
182 * .. Local Scalars ..
183  INTEGER i, j, kd
184  DOUBLE PRECISION bignum, rcmax, rcmin, smlnum, radix, logrdx
185 * ..
186 * .. External Functions ..
187  DOUBLE PRECISION dlamch
188  EXTERNAL dlamch
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( 'DGBEQUB', -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 = dlamch( 'S' )
229  bignum = one / smlnum
230  radix = dlamch( 'B' )
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 DGBEQUB
339 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62

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