LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dgeequb()

 subroutine dgeequb ( integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, integer INFO )

DGEEQUB

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Purpose:
``` DGEEQUB 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
the radix.

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
between sqrt(radix) and 1/sqrt(radix).```
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] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) The M-by-N matrix whose equilibration factors are to be computed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= 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 144 of file dgeequb.f.

146 *
147 * -- LAPACK computational routine --
148 * -- LAPACK is a software package provided by Univ. of Tennessee, --
149 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
150 *
151 * .. Scalar Arguments ..
152  INTEGER INFO, LDA, M, N
153  DOUBLE PRECISION AMAX, COLCND, ROWCND
154 * ..
155 * .. Array Arguments ..
156  DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
157 * ..
158 *
159 * =====================================================================
160 *
161 * .. Parameters ..
162  DOUBLE PRECISION ONE, ZERO
163  parameter( one = 1.0d+0, zero = 0.0d+0 )
164 * ..
165 * .. Local Scalars ..
166  INTEGER I, J
167  DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
168 * ..
169 * .. External Functions ..
170  DOUBLE PRECISION DLAMCH
171  EXTERNAL dlamch
172 * ..
173 * .. External Subroutines ..
174  EXTERNAL xerbla
175 * ..
176 * .. Intrinsic Functions ..
177  INTRINSIC abs, max, min, log
178 * ..
179 * .. Executable Statements ..
180 *
181 * Test the input parameters.
182 *
183  info = 0
184  IF( m.LT.0 ) THEN
185  info = -1
186  ELSE IF( n.LT.0 ) THEN
187  info = -2
188  ELSE IF( lda.LT.max( 1, m ) ) THEN
189  info = -4
190  END IF
191  IF( info.NE.0 ) THEN
192  CALL xerbla( 'DGEEQUB', -info )
193  RETURN
194  END IF
195 *
196 * Quick return if possible.
197 *
198  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
199  rowcnd = one
200  colcnd = one
201  amax = zero
202  RETURN
203  END IF
204 *
205 * Get machine constants. Assume SMLNUM is a power of the radix.
206 *
207  smlnum = dlamch( 'S' )
208  bignum = one / smlnum
209  radix = dlamch( 'B' )
210  logrdx = log( radix )
211 *
212 * Compute row scale factors.
213 *
214  DO 10 i = 1, m
215  r( i ) = zero
216  10 CONTINUE
217 *
218 * Find the maximum element in each row.
219 *
220  DO 30 j = 1, n
221  DO 20 i = 1, m
222  r( i ) = max( r( i ), abs( a( i, j ) ) )
223  20 CONTINUE
224  30 CONTINUE
225  DO i = 1, m
226  IF( r( i ).GT.zero ) THEN
227  r( i ) = radix**int( log( r( i ) ) / logrdx )
228  END IF
229  END DO
230 *
231 * Find the maximum and minimum scale factors.
232 *
233  rcmin = bignum
234  rcmax = zero
235  DO 40 i = 1, m
236  rcmax = max( rcmax, r( i ) )
237  rcmin = min( rcmin, r( i ) )
238  40 CONTINUE
239  amax = rcmax
240 *
241  IF( rcmin.EQ.zero ) THEN
242 *
243 * Find the first zero scale factor and return an error code.
244 *
245  DO 50 i = 1, m
246  IF( r( i ).EQ.zero ) THEN
247  info = i
248  RETURN
249  END IF
250  50 CONTINUE
251  ELSE
252 *
253 * Invert the scale factors.
254 *
255  DO 60 i = 1, m
256  r( i ) = one / min( max( r( i ), smlnum ), bignum )
257  60 CONTINUE
258 *
259 * Compute ROWCND = min(R(I)) / max(R(I)).
260 *
261  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
262  END IF
263 *
264 * Compute column scale factors
265 *
266  DO 70 j = 1, n
267  c( j ) = zero
268  70 CONTINUE
269 *
270 * Find the maximum element in each column,
271 * assuming the row scaling computed above.
272 *
273  DO 90 j = 1, n
274  DO 80 i = 1, m
275  c( j ) = max( c( j ), abs( a( i, j ) )*r( i ) )
276  80 CONTINUE
277  IF( c( j ).GT.zero ) THEN
278  c( j ) = radix**int( log( c( j ) ) / logrdx )
279  END IF
280  90 CONTINUE
281 *
282 * Find the maximum and minimum scale factors.
283 *
284  rcmin = bignum
285  rcmax = zero
286  DO 100 j = 1, n
287  rcmin = min( rcmin, c( j ) )
288  rcmax = max( rcmax, c( j ) )
289  100 CONTINUE
290 *
291  IF( rcmin.EQ.zero ) THEN
292 *
293 * Find the first zero scale factor and return an error code.
294 *
295  DO 110 j = 1, n
296  IF( c( j ).EQ.zero ) THEN
297  info = m + j
298  RETURN
299  END IF
300  110 CONTINUE
301  ELSE
302 *
303 * Invert the scale factors.
304 *
305  DO 120 j = 1, n
306  c( j ) = one / min( max( c( j ), smlnum ), bignum )
307  120 CONTINUE
308 *
309 * Compute COLCND = min(C(J)) / max(C(J)).
310 *
311  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
312  END IF
313 *
314  RETURN
315 *
316 * End of DGEEQUB
317 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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