LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dtbcon()

 subroutine dtbcon ( character NORM, character UPLO, character DIAG, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

DTBCON

Purpose:
``` DTBCON estimates the reciprocal of the condition number of a
triangular band matrix A, in either the 1-norm or the infinity-norm.

The norm of A is computed and an estimate is obtained for
norm(inv(A)), then 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] UPLO ``` UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular.``` [in] DIAG ``` DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] KD ``` KD is INTEGER The number of superdiagonals or subdiagonals of the triangular band matrix A. KD >= 0.``` [in] AB ``` AB is DOUBLE PRECISION array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). If DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be 1.``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.``` [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 141 of file dtbcon.f.

143 *
144 * -- LAPACK computational routine --
145 * -- LAPACK is a software package provided by Univ. of Tennessee, --
146 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
147 *
148 * .. Scalar Arguments ..
149  CHARACTER DIAG, NORM, UPLO
150  INTEGER INFO, KD, LDAB, N
151  DOUBLE PRECISION RCOND
152 * ..
153 * .. Array Arguments ..
154  INTEGER IWORK( * )
155  DOUBLE PRECISION AB( LDAB, * ), WORK( * )
156 * ..
157 *
158 * =====================================================================
159 *
160 * .. Parameters ..
161  DOUBLE PRECISION ONE, ZERO
162  parameter( one = 1.0d+0, zero = 0.0d+0 )
163 * ..
164 * .. Local Scalars ..
165  LOGICAL NOUNIT, ONENRM, UPPER
166  CHARACTER NORMIN
167  INTEGER IX, KASE, KASE1
168  DOUBLE PRECISION AINVNM, ANORM, SCALE, SMLNUM, XNORM
169 * ..
170 * .. Local Arrays ..
171  INTEGER ISAVE( 3 )
172 * ..
173 * .. External Functions ..
174  LOGICAL LSAME
175  INTEGER IDAMAX
176  DOUBLE PRECISION DLAMCH, DLANTB
177  EXTERNAL lsame, idamax, dlamch, dlantb
178 * ..
179 * .. External Subroutines ..
180  EXTERNAL dlacn2, dlatbs, drscl, xerbla
181 * ..
182 * .. Intrinsic Functions ..
183  INTRINSIC abs, dble, max
184 * ..
185 * .. Executable Statements ..
186 *
187 * Test the input parameters.
188 *
189  info = 0
190  upper = lsame( uplo, 'U' )
191  onenrm = norm.EQ.'1' .OR. lsame( norm, 'O' )
192  nounit = lsame( diag, 'N' )
193 *
194  IF( .NOT.onenrm .AND. .NOT.lsame( norm, 'I' ) ) THEN
195  info = -1
196  ELSE IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
197  info = -2
198  ELSE IF( .NOT.nounit .AND. .NOT.lsame( diag, 'U' ) ) THEN
199  info = -3
200  ELSE IF( n.LT.0 ) THEN
201  info = -4
202  ELSE IF( kd.LT.0 ) THEN
203  info = -5
204  ELSE IF( ldab.LT.kd+1 ) THEN
205  info = -7
206  END IF
207  IF( info.NE.0 ) THEN
208  CALL xerbla( 'DTBCON', -info )
209  RETURN
210  END IF
211 *
212 * Quick return if possible
213 *
214  IF( n.EQ.0 ) THEN
215  rcond = one
216  RETURN
217  END IF
218 *
219  rcond = zero
220  smlnum = dlamch( 'Safe minimum' )*dble( max( 1, n ) )
221 *
222 * Compute the norm of the triangular matrix A.
223 *
224  anorm = dlantb( norm, uplo, diag, n, kd, ab, ldab, work )
225 *
226 * Continue only if ANORM > 0.
227 *
228  IF( anorm.GT.zero ) THEN
229 *
230 * Estimate the norm of the inverse of A.
231 *
232  ainvnm = zero
233  normin = 'N'
234  IF( onenrm ) THEN
235  kase1 = 1
236  ELSE
237  kase1 = 2
238  END IF
239  kase = 0
240  10 CONTINUE
241  CALL dlacn2( n, work( n+1 ), work, iwork, ainvnm, kase, isave )
242  IF( kase.NE.0 ) THEN
243  IF( kase.EQ.kase1 ) THEN
244 *
245 * Multiply by inv(A).
246 *
247  CALL dlatbs( uplo, 'No transpose', diag, normin, n, kd,
248  \$ ab, ldab, work, scale, work( 2*n+1 ), info )
249  ELSE
250 *
251 * Multiply by inv(A**T).
252 *
253  CALL dlatbs( uplo, 'Transpose', diag, normin, n, kd, ab,
254  \$ ldab, work, scale, work( 2*n+1 ), info )
255  END IF
256  normin = 'Y'
257 *
258 * Multiply by 1/SCALE if doing so will not cause overflow.
259 *
260  IF( scale.NE.one ) THEN
261  ix = idamax( n, work, 1 )
262  xnorm = abs( work( ix ) )
263  IF( scale.LT.xnorm*smlnum .OR. scale.EQ.zero )
264  \$ GO TO 20
265  CALL drscl( n, scale, work, 1 )
266  END IF
267  GO TO 10
268  END IF
269 *
270 * Compute the estimate of the reciprocal condition number.
271 *
272  IF( ainvnm.NE.zero )
273  \$ rcond = ( one / anorm ) / ainvnm
274  END IF
275 *
276  20 CONTINUE
277  RETURN
278 *
279 * End of DTBCON
280 *
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
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
double precision function dlantb(NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
DLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlantb.f:140
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