LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ sgbequb()

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

SGBEQUB

Download SGBEQUB + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SGBEQUB 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 SGEEQU 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]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 REAL 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 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
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
June 2016

Definition at line 162 of file sgbequb.f.

162 *
163 * -- LAPACK computational routine (version 3.7.0) --
164 * -- LAPACK is a software package provided by Univ. of Tennessee, --
165 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166 * June 2016
167 *
168 * .. Scalar Arguments ..
169  INTEGER info, kl, ku, ldab, m, n
170  REAL amax, colcnd, rowcnd
171 * ..
172 * .. Array Arguments ..
173  REAL ab( ldab, * ), c( * ), r( * )
174 * ..
175 *
176 * =====================================================================
177 *
178 * .. Parameters ..
179  REAL one, zero
180  parameter( one = 1.0e+0, zero = 0.0e+0 )
181 * ..
182 * .. Local Scalars ..
183  INTEGER i, j, kd
184  REAL bignum, rcmax, rcmin, smlnum, radix, logrdx
185 * ..
186 * .. External Functions ..
187  REAL slamch
188  EXTERNAL slamch
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( 'SGBEQUB', -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 = slamch( 'S' )
229  bignum = one / smlnum
230  radix = slamch( 'B' )
231  logrdx = log(radix)
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 SGBEQUB
339 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
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