 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zgbequb()

 subroutine zgbequb ( integer M, integer N, integer KL, integer KU, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, integer INFO )

ZGBEQUB

Purpose:
``` ZGBEQUB 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 ZGEEQU 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 COMPLEX*16 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 159 of file zgbequb.f.

161 *
162 * -- LAPACK computational routine --
163 * -- LAPACK is a software package provided by Univ. of Tennessee, --
164 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
165 *
166 * .. Scalar Arguments ..
167  INTEGER INFO, KL, KU, LDAB, M, N
168  DOUBLE PRECISION AMAX, COLCND, ROWCND
169 * ..
170 * .. Array Arguments ..
171  DOUBLE PRECISION C( * ), R( * )
172  COMPLEX*16 AB( LDAB, * )
173 * ..
174 *
175 * =====================================================================
176 *
177 * .. Parameters ..
178  DOUBLE PRECISION ONE, ZERO
179  parameter( one = 1.0d+0, zero = 0.0d+0 )
180 * ..
181 * .. Local Scalars ..
182  INTEGER I, J, KD
183  DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX,
184  \$ LOGRDX
185  COMPLEX*16 ZDUM
186 * ..
187 * .. External Functions ..
188  DOUBLE PRECISION DLAMCH
189  EXTERNAL dlamch
190 * ..
191 * .. External Subroutines ..
192  EXTERNAL xerbla
193 * ..
194 * .. Intrinsic Functions ..
195  INTRINSIC abs, max, min, log, real, dimag
196 * ..
197 * .. Statement Functions ..
198  DOUBLE PRECISION CABS1
199 * ..
200 * .. Statement Function definitions ..
201  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
202 * ..
203 * .. Executable Statements ..
204 *
205 * Test the input parameters.
206 *
207  info = 0
208  IF( m.LT.0 ) THEN
209  info = -1
210  ELSE IF( n.LT.0 ) THEN
211  info = -2
212  ELSE IF( kl.LT.0 ) THEN
213  info = -3
214  ELSE IF( ku.LT.0 ) THEN
215  info = -4
216  ELSE IF( ldab.LT.kl+ku+1 ) THEN
217  info = -6
218  END IF
219  IF( info.NE.0 ) THEN
220  CALL xerbla( 'ZGBEQUB', -info )
221  RETURN
222  END IF
223 *
224 * Quick return if possible.
225 *
226  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
227  rowcnd = one
228  colcnd = one
229  amax = zero
230  RETURN
231  END IF
232 *
233 * Get machine constants. Assume SMLNUM is a power of the radix.
234 *
235  smlnum = dlamch( 'S' )
236  bignum = one / smlnum
237  radix = dlamch( 'B' )
239 *
240 * Compute row scale factors.
241 *
242  DO 10 i = 1, m
243  r( i ) = zero
244  10 CONTINUE
245 *
246 * Find the maximum element in each row.
247 *
248  kd = ku + 1
249  DO 30 j = 1, n
250  DO 20 i = max( j-ku, 1 ), min( j+kl, m )
251  r( i ) = max( r( i ), cabs1( ab( kd+i-j, j ) ) )
252  20 CONTINUE
253  30 CONTINUE
254  DO i = 1, m
255  IF( r( i ).GT.zero ) THEN
256  r( i ) = radix**int( log( r( i ) ) / logrdx )
257  END IF
258  END DO
259 *
260 * Find the maximum and minimum scale factors.
261 *
262  rcmin = bignum
263  rcmax = zero
264  DO 40 i = 1, m
265  rcmax = max( rcmax, r( i ) )
266  rcmin = min( rcmin, r( i ) )
267  40 CONTINUE
268  amax = rcmax
269 *
270  IF( rcmin.EQ.zero ) THEN
271 *
272 * Find the first zero scale factor and return an error code.
273 *
274  DO 50 i = 1, m
275  IF( r( i ).EQ.zero ) THEN
276  info = i
277  RETURN
278  END IF
279  50 CONTINUE
280  ELSE
281 *
282 * Invert the scale factors.
283 *
284  DO 60 i = 1, m
285  r( i ) = one / min( max( r( i ), smlnum ), bignum )
286  60 CONTINUE
287 *
288 * Compute ROWCND = min(R(I)) / max(R(I)).
289 *
290  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
291  END IF
292 *
293 * Compute column scale factors.
294 *
295  DO 70 j = 1, n
296  c( j ) = zero
297  70 CONTINUE
298 *
299 * Find the maximum element in each column,
300 * assuming the row scaling computed above.
301 *
302  DO 90 j = 1, n
303  DO 80 i = max( j-ku, 1 ), min( j+kl, m )
304  c( j ) = max( c( j ), cabs1( ab( kd+i-j, j ) )*r( i ) )
305  80 CONTINUE
306  IF( c( j ).GT.zero ) THEN
307  c( j ) = radix**int( log( c( j ) ) / logrdx )
308  END IF
309  90 CONTINUE
310 *
311 * Find the maximum and minimum scale factors.
312 *
313  rcmin = bignum
314  rcmax = zero
315  DO 100 j = 1, n
316  rcmin = min( rcmin, c( j ) )
317  rcmax = max( rcmax, c( j ) )
318  100 CONTINUE
319 *
320  IF( rcmin.EQ.zero ) THEN
321 *
322 * Find the first zero scale factor and return an error code.
323 *
324  DO 110 j = 1, n
325  IF( c( j ).EQ.zero ) THEN
326  info = m + j
327  RETURN
328  END IF
329  110 CONTINUE
330  ELSE
331 *
332 * Invert the scale factors.
333 *
334  DO 120 j = 1, n
335  c( j ) = one / min( max( c( j ), smlnum ), bignum )
336  120 CONTINUE
337 *
338 * Compute COLCND = min(C(J)) / max(C(J)).
339 *
340  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
341  END IF
342 *
343  RETURN
344 *
345 * End of ZGBEQUB
346 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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