LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine sgeequb ( integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO )

SGEEQUB

Purpose:
``` SGEEQUB 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 SGEEQU 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] A ``` A is REAL 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 REAL array, dimension (M) If INFO = 0 or INFO > M, R contains the row scale factors for A.``` [out] C ``` C is REAL array, dimension (N) If INFO = 0, C contains the column scale factors for A.``` [out] ROWCND ``` ROWCND is REAL 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 REAL 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 REAL 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 148 of file sgeequb.f.

148 *
149 * -- LAPACK computational routine (version 3.4.0) --
150 * -- LAPACK is a software package provided by Univ. of Tennessee, --
151 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
152 * November 2011
153 *
154 * .. Scalar Arguments ..
155  INTEGER info, lda, m, n
156  REAL amax, colcnd, rowcnd
157 * ..
158 * .. Array Arguments ..
159  REAL a( lda, * ), c( * ), r( * )
160 * ..
161 *
162 * =====================================================================
163 *
164 * .. Parameters ..
165  REAL one, zero
166  parameter ( one = 1.0e+0, zero = 0.0e+0 )
167 * ..
168 * .. Local Scalars ..
169  INTEGER i, j
170  REAL bignum, rcmax, rcmin, smlnum, radix, logrdx
171 * ..
172 * .. External Functions ..
173  REAL slamch
174  EXTERNAL slamch
175 * ..
176 * .. External Subroutines ..
177  EXTERNAL xerbla
178 * ..
179 * .. Intrinsic Functions ..
180  INTRINSIC abs, max, min, log
181 * ..
182 * .. Executable Statements ..
183 *
184 * Test the input parameters.
185 *
186  info = 0
187  IF( m.LT.0 ) THEN
188  info = -1
189  ELSE IF( n.LT.0 ) THEN
190  info = -2
191  ELSE IF( lda.LT.max( 1, m ) ) THEN
192  info = -4
193  END IF
194  IF( info.NE.0 ) THEN
195  CALL xerbla( 'SGEEQUB', -info )
196  RETURN
197  END IF
198 *
199 * Quick return if possible.
200 *
201  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
202  rowcnd = one
203  colcnd = one
204  amax = zero
205  RETURN
206  END IF
207 *
208 * Get machine constants. Assume SMLNUM is a power of the radix.
209 *
210  smlnum = slamch( 'S' )
211  bignum = one / smlnum
212  radix = slamch( 'B' )
213  logrdx = log( radix )
214 *
215 * Compute row scale factors.
216 *
217  DO 10 i = 1, m
218  r( i ) = zero
219  10 CONTINUE
220 *
221 * Find the maximum element in each row.
222 *
223  DO 30 j = 1, n
224  DO 20 i = 1, m
225  r( i ) = max( r( i ), abs( a( i, j ) ) )
226  20 CONTINUE
227  30 CONTINUE
228  DO i = 1, m
229  IF( r( i ).GT.zero ) THEN
230  r( i ) = radix**int( log( r( i ) ) / logrdx )
231  END IF
232  END DO
233 *
234 * Find the maximum and minimum scale factors.
235 *
236  rcmin = bignum
237  rcmax = zero
238  DO 40 i = 1, m
239  rcmax = max( rcmax, r( i ) )
240  rcmin = min( rcmin, r( i ) )
241  40 CONTINUE
242  amax = rcmax
243 *
244  IF( rcmin.EQ.zero ) THEN
245 *
246 * Find the first zero scale factor and return an error code.
247 *
248  DO 50 i = 1, m
249  IF( r( i ).EQ.zero ) THEN
250  info = i
251  RETURN
252  END IF
253  50 CONTINUE
254  ELSE
255 *
256 * Invert the scale factors.
257 *
258  DO 60 i = 1, m
259  r( i ) = one / min( max( r( i ), smlnum ), bignum )
260  60 CONTINUE
261 *
262 * Compute ROWCND = min(R(I)) / max(R(I)).
263 *
264  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
265  END IF
266 *
267 * Compute column scale factors
268 *
269  DO 70 j = 1, n
270  c( j ) = zero
271  70 CONTINUE
272 *
273 * Find the maximum element in each column,
274 * assuming the row scaling computed above.
275 *
276  DO 90 j = 1, n
277  DO 80 i = 1, m
278  c( j ) = max( c( j ), abs( a( i, j ) )*r( i ) )
279  80 CONTINUE
280  IF( c( j ).GT.zero ) THEN
281  c( j ) = radix**int( log( c( j ) ) / logrdx )
282  END IF
283  90 CONTINUE
284 *
285 * Find the maximum and minimum scale factors.
286 *
287  rcmin = bignum
288  rcmax = zero
289  DO 100 j = 1, n
290  rcmin = min( rcmin, c( j ) )
291  rcmax = max( rcmax, c( j ) )
292  100 CONTINUE
293 *
294  IF( rcmin.EQ.zero ) THEN
295 *
296 * Find the first zero scale factor and return an error code.
297 *
298  DO 110 j = 1, n
299  IF( c( j ).EQ.zero ) THEN
300  info = m + j
301  RETURN
302  END IF
303  110 CONTINUE
304  ELSE
305 *
306 * Invert the scale factors.
307 *
308  DO 120 j = 1, n
309  c( j ) = one / min( max( c( j ), smlnum ), bignum )
310  120 CONTINUE
311 *
312 * Compute COLCND = min(C(J)) / max(C(J)).
313 *
314  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
315  END IF
316 *
317  RETURN
318 *
319 * End of SGEEQUB
320 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69

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