 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dgbcon()

 subroutine dgbcon ( character NORM, integer N, integer KL, integer KU, double precision, dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

DGBCON

Purpose:
``` DGBCON estimates the reciprocal of the condition number of a real
general band matrix A, in either the 1-norm or the infinity-norm,
using the LU factorization computed by DGBTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).```
Parameters
 [in] NORM ``` NORM is CHARACTER*1 Specifies whether the 1-norm condition number or the infinity-norm condition number is required: = '1' or 'O': 1-norm; = 'I': Infinity-norm.``` [in] N ``` N is INTEGER The order 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 DOUBLE PRECISION array, dimension (LDAB,N) Details of the LU factorization of the band matrix A, as computed by DGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1.``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= 2*KL+KU+1.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= N, row i of the matrix was interchanged with row IPIV(i).``` [in] ANORM ``` ANORM is DOUBLE PRECISION If NORM = '1' or 'O', the 1-norm of the original matrix A. If NORM = 'I', the infinity-norm of the original matrix A.``` [out] RCOND ``` RCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(norm(A) * norm(inv(A))).``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (3*N)` [out] IWORK ` IWORK is INTEGER array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```

Definition at line 144 of file dgbcon.f.

146 *
147 * -- LAPACK computational routine --
148 * -- LAPACK is a software package provided by Univ. of Tennessee, --
149 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
150 *
151 * .. Scalar Arguments ..
152  CHARACTER NORM
153  INTEGER INFO, KL, KU, LDAB, N
154  DOUBLE PRECISION ANORM, RCOND
155 * ..
156 * .. Array Arguments ..
157  INTEGER IPIV( * ), IWORK( * )
158  DOUBLE PRECISION AB( LDAB, * ), WORK( * )
159 * ..
160 *
161 * =====================================================================
162 *
163 * .. Parameters ..
164  DOUBLE PRECISION ONE, ZERO
165  parameter( one = 1.0d+0, zero = 0.0d+0 )
166 * ..
167 * .. Local Scalars ..
168  LOGICAL LNOTI, ONENRM
169  CHARACTER NORMIN
170  INTEGER IX, J, JP, KASE, KASE1, KD, LM
171  DOUBLE PRECISION AINVNM, SCALE, SMLNUM, T
172 * ..
173 * .. Local Arrays ..
174  INTEGER ISAVE( 3 )
175 * ..
176 * .. External Functions ..
177  LOGICAL LSAME
178  INTEGER IDAMAX
179  DOUBLE PRECISION DDOT, DLAMCH
180  EXTERNAL lsame, idamax, ddot, dlamch
181 * ..
182 * .. External Subroutines ..
183  EXTERNAL daxpy, dlacn2, dlatbs, drscl, xerbla
184 * ..
185 * .. Intrinsic Functions ..
186  INTRINSIC abs, min
187 * ..
188 * .. Executable Statements ..
189 *
190 * Test the input parameters.
191 *
192  info = 0
193  onenrm = norm.EQ.'1' .OR. lsame( norm, 'O' )
194  IF( .NOT.onenrm .AND. .NOT.lsame( norm, 'I' ) ) THEN
195  info = -1
196  ELSE IF( n.LT.0 ) THEN
197  info = -2
198  ELSE IF( kl.LT.0 ) THEN
199  info = -3
200  ELSE IF( ku.LT.0 ) THEN
201  info = -4
202  ELSE IF( ldab.LT.2*kl+ku+1 ) THEN
203  info = -6
204  ELSE IF( anorm.LT.zero ) THEN
205  info = -8
206  END IF
207  IF( info.NE.0 ) THEN
208  CALL xerbla( 'DGBCON', -info )
209  RETURN
210  END IF
211 *
212 * Quick return if possible
213 *
214  rcond = zero
215  IF( n.EQ.0 ) THEN
216  rcond = one
217  RETURN
218  ELSE IF( anorm.EQ.zero ) THEN
219  RETURN
220  END IF
221 *
222  smlnum = dlamch( 'Safe minimum' )
223 *
224 * Estimate the norm of inv(A).
225 *
226  ainvnm = zero
227  normin = 'N'
228  IF( onenrm ) THEN
229  kase1 = 1
230  ELSE
231  kase1 = 2
232  END IF
233  kd = kl + ku + 1
234  lnoti = kl.GT.0
235  kase = 0
236  10 CONTINUE
237  CALL dlacn2( n, work( n+1 ), work, iwork, ainvnm, kase, isave )
238  IF( kase.NE.0 ) THEN
239  IF( kase.EQ.kase1 ) THEN
240 *
241 * Multiply by inv(L).
242 *
243  IF( lnoti ) THEN
244  DO 20 j = 1, n - 1
245  lm = min( kl, n-j )
246  jp = ipiv( j )
247  t = work( jp )
248  IF( jp.NE.j ) THEN
249  work( jp ) = work( j )
250  work( j ) = t
251  END IF
252  CALL daxpy( lm, -t, ab( kd+1, j ), 1, work( j+1 ), 1 )
253  20 CONTINUE
254  END IF
255 *
256 * Multiply by inv(U).
257 *
258  CALL dlatbs( 'Upper', 'No transpose', 'Non-unit', normin, n,
259  \$ kl+ku, ab, ldab, work, scale, work( 2*n+1 ),
260  \$ info )
261  ELSE
262 *
263 * Multiply by inv(U**T).
264 *
265  CALL dlatbs( 'Upper', 'Transpose', 'Non-unit', normin, n,
266  \$ kl+ku, ab, ldab, work, scale, work( 2*n+1 ),
267  \$ info )
268 *
269 * Multiply by inv(L**T).
270 *
271  IF( lnoti ) THEN
272  DO 30 j = n - 1, 1, -1
273  lm = min( kl, n-j )
274  work( j ) = work( j ) - ddot( lm, ab( kd+1, j ), 1,
275  \$ work( j+1 ), 1 )
276  jp = ipiv( j )
277  IF( jp.NE.j ) THEN
278  t = work( jp )
279  work( jp ) = work( j )
280  work( j ) = t
281  END IF
282  30 CONTINUE
283  END IF
284  END IF
285 *
286 * Divide X by 1/SCALE if doing so will not cause overflow.
287 *
288  normin = 'Y'
289  IF( scale.NE.one ) THEN
290  ix = idamax( n, work, 1 )
291  IF( scale.LT.abs( work( ix ) )*smlnum .OR. scale.EQ.zero )
292  \$ GO TO 40
293  CALL drscl( n, scale, work, 1 )
294  END IF
295  GO TO 10
296  END IF
297 *
298 * Compute the estimate of the reciprocal condition number.
299 *
300  IF( ainvnm.NE.zero )
301  \$ rcond = ( one / ainvnm ) / anorm
302 *
303  40 CONTINUE
304  RETURN
305 *
306 * End of DGBCON
307 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function ddot(N, DX, INCX, DY, INCY)
DDOT
Definition: ddot.f:82
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dlatbs(UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
DLATBS solves a triangular banded system of equations.
Definition: dlatbs.f:242
subroutine drscl(N, SA, SX, INCX)
DRSCL multiplies a vector by the reciprocal of a real scalar.
Definition: drscl.f:84
subroutine dlacn2(N, V, X, ISGN, EST, KASE, ISAVE)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: dlacn2.f:136
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