LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ 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*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
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