LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zgbequ()

 subroutine zgbequ ( 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 )

ZGBEQU

Purpose:
``` ZGBEQU computes row and column scalings intended to equilibrate an
M-by-N band 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 absolute value 1.

R(i) and C(j) are restricted to be 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.```
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) The band matrix A, stored 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(m,j+kl).``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1.``` [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 152 of file zgbequ.f.

154 *
155 * -- LAPACK computational routine --
156 * -- LAPACK is a software package provided by Univ. of Tennessee, --
157 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
158 *
159 * .. Scalar Arguments ..
160  INTEGER INFO, KL, KU, LDAB, M, N
161  DOUBLE PRECISION AMAX, COLCND, ROWCND
162 * ..
163 * .. Array Arguments ..
164  DOUBLE PRECISION C( * ), R( * )
165  COMPLEX*16 AB( LDAB, * )
166 * ..
167 *
168 * =====================================================================
169 *
170 * .. Parameters ..
171  DOUBLE PRECISION ONE, ZERO
172  parameter( one = 1.0d+0, zero = 0.0d+0 )
173 * ..
174 * .. Local Scalars ..
175  INTEGER I, J, KD
176  DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM
177  COMPLEX*16 ZDUM
178 * ..
179 * .. External Functions ..
180  DOUBLE PRECISION DLAMCH
181  EXTERNAL dlamch
182 * ..
183 * .. External Subroutines ..
184  EXTERNAL xerbla
185 * ..
186 * .. Intrinsic Functions ..
187  INTRINSIC abs, dble, dimag, max, min
188 * ..
189 * .. Statement Functions ..
190  DOUBLE PRECISION CABS1
191 * ..
192 * .. Statement Function definitions ..
193  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
194 * ..
195 * .. Executable Statements ..
196 *
197 * Test the input parameters
198 *
199  info = 0
200  IF( m.LT.0 ) THEN
201  info = -1
202  ELSE IF( n.LT.0 ) THEN
203  info = -2
204  ELSE IF( kl.LT.0 ) THEN
205  info = -3
206  ELSE IF( ku.LT.0 ) THEN
207  info = -4
208  ELSE IF( ldab.LT.kl+ku+1 ) THEN
209  info = -6
210  END IF
211  IF( info.NE.0 ) THEN
212  CALL xerbla( 'ZGBEQU', -info )
213  RETURN
214  END IF
215 *
216 * Quick return if possible
217 *
218  IF( m.EQ.0 .OR. n.EQ.0 ) THEN
219  rowcnd = one
220  colcnd = one
221  amax = zero
222  RETURN
223  END IF
224 *
225 * Get machine constants.
226 *
227  smlnum = dlamch( 'S' )
228  bignum = one / smlnum
229 *
230 * Compute row scale factors.
231 *
232  DO 10 i = 1, m
233  r( i ) = zero
234  10 CONTINUE
235 *
236 * Find the maximum element in each row.
237 *
238  kd = ku + 1
239  DO 30 j = 1, n
240  DO 20 i = max( j-ku, 1 ), min( j+kl, m )
241  r( i ) = max( r( i ), cabs1( ab( kd+i-j, j ) ) )
242  20 CONTINUE
243  30 CONTINUE
244 *
245 * Find the maximum and minimum scale factors.
246 *
247  rcmin = bignum
248  rcmax = zero
249  DO 40 i = 1, m
250  rcmax = max( rcmax, r( i ) )
251  rcmin = min( rcmin, r( i ) )
252  40 CONTINUE
253  amax = rcmax
254 *
255  IF( rcmin.EQ.zero ) THEN
256 *
257 * Find the first zero scale factor and return an error code.
258 *
259  DO 50 i = 1, m
260  IF( r( i ).EQ.zero ) THEN
261  info = i
262  RETURN
263  END IF
264  50 CONTINUE
265  ELSE
266 *
267 * Invert the scale factors.
268 *
269  DO 60 i = 1, m
270  r( i ) = one / min( max( r( i ), smlnum ), bignum )
271  60 CONTINUE
272 *
273 * Compute ROWCND = min(R(I)) / max(R(I))
274 *
275  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
276  END IF
277 *
278 * Compute column scale factors
279 *
280  DO 70 j = 1, n
281  c( j ) = zero
282  70 CONTINUE
283 *
284 * Find the maximum element in each column,
285 * assuming the row scaling computed above.
286 *
287  kd = ku + 1
288  DO 90 j = 1, n
289  DO 80 i = max( j-ku, 1 ), min( j+kl, m )
290  c( j ) = max( c( j ), cabs1( ab( kd+i-j, j ) )*r( i ) )
291  80 CONTINUE
292  90 CONTINUE
293 *
294 * Find the maximum and minimum scale factors.
295 *
296  rcmin = bignum
297  rcmax = zero
298  DO 100 j = 1, n
299  rcmin = min( rcmin, c( j ) )
300  rcmax = max( rcmax, c( j ) )
301  100 CONTINUE
302 *
303  IF( rcmin.EQ.zero ) THEN
304 *
305 * Find the first zero scale factor and return an error code.
306 *
307  DO 110 j = 1, n
308  IF( c( j ).EQ.zero ) THEN
309  info = m + j
310  RETURN
311  END IF
312  110 CONTINUE
313  ELSE
314 *
315 * Invert the scale factors.
316 *
317  DO 120 j = 1, n
318  c( j ) = one / min( max( c( j ), smlnum ), bignum )
319  120 CONTINUE
320 *
321 * Compute COLCND = min(C(J)) / max(C(J))
322 *
323  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
324  END IF
325 *
326  RETURN
327 *
328 * End of ZGBEQU
329 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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