 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zpbcon()

 subroutine zpbcon ( character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO )

ZPBCON

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Purpose:
``` ZPBCON estimates the reciprocal of the condition number (in the
1-norm) of a complex Hermitian positive definite band matrix using
the Cholesky factorization A = U**H*U or A = L*L**H computed by
ZPBTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangular factor stored in AB; = 'L': Lower triangular factor stored in AB.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] KD ``` KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. KD >= 0.``` [in] AB ``` AB is COMPLEX*16 array, dimension (LDAB,N) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, stored in the first KD+1 rows of the array. The j-th column of U or L is stored in the j-th column of the array AB as follows: if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j; if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.``` [in] ANORM ``` ANORM is DOUBLE PRECISION The 1-norm (or infinity-norm) of the Hermitian band matrix A.``` [out] RCOND ``` RCOND is DOUBLE PRECISION The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (2*N)` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date
December 2016

Definition at line 135 of file zpbcon.f.

135 *
136 * -- LAPACK computational routine (version 3.7.0) --
137 * -- LAPACK is a software package provided by Univ. of Tennessee, --
138 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
139 * December 2016
140 *
141 * .. Scalar Arguments ..
142  CHARACTER uplo
143  INTEGER info, kd, ldab, n
144  DOUBLE PRECISION anorm, rcond
145 * ..
146 * .. Array Arguments ..
147  DOUBLE PRECISION rwork( * )
148  COMPLEX*16 ab( ldab, * ), work( * )
149 * ..
150 *
151 * =====================================================================
152 *
153 * .. Parameters ..
154  DOUBLE PRECISION one, zero
155  parameter( one = 1.0d+0, zero = 0.0d+0 )
156 * ..
157 * .. Local Scalars ..
158  LOGICAL upper
159  CHARACTER normin
160  INTEGER ix, kase
161  DOUBLE PRECISION ainvnm, scale, scalel, scaleu, smlnum
162  COMPLEX*16 zdum
163 * ..
164 * .. Local Arrays ..
165  INTEGER isave( 3 )
166 * ..
167 * .. External Functions ..
168  LOGICAL lsame
169  INTEGER izamax
170  DOUBLE PRECISION dlamch
171  EXTERNAL lsame, izamax, dlamch
172 * ..
173 * .. External Subroutines ..
174  EXTERNAL xerbla, zdrscl, zlacn2, zlatbs
175 * ..
176 * .. Intrinsic Functions ..
177  INTRINSIC abs, dble, dimag
178 * ..
179 * .. Statement Functions ..
180  DOUBLE PRECISION cabs1
181 * ..
182 * .. Statement Function definitions ..
183  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
184 * ..
185 * .. Executable Statements ..
186 *
187 * Test the input parameters.
188 *
189  info = 0
190  upper = lsame( uplo, 'U' )
191  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
192  info = -1
193  ELSE IF( n.LT.0 ) THEN
194  info = -2
195  ELSE IF( kd.LT.0 ) THEN
196  info = -3
197  ELSE IF( ldab.LT.kd+1 ) THEN
198  info = -5
199  ELSE IF( anorm.LT.zero ) THEN
200  info = -6
201  END IF
202  IF( info.NE.0 ) THEN
203  CALL xerbla( 'ZPBCON', -info )
204  RETURN
205  END IF
206 *
207 * Quick return if possible
208 *
209  rcond = zero
210  IF( n.EQ.0 ) THEN
211  rcond = one
212  RETURN
213  ELSE IF( anorm.EQ.zero ) THEN
214  RETURN
215  END IF
216 *
217  smlnum = dlamch( 'Safe minimum' )
218 *
219 * Estimate the 1-norm of the inverse.
220 *
221  kase = 0
222  normin = 'N'
223  10 CONTINUE
224  CALL zlacn2( n, work( n+1 ), work, ainvnm, kase, isave )
225  IF( kase.NE.0 ) THEN
226  IF( upper ) THEN
227 *
228 * Multiply by inv(U**H).
229 *
230  CALL zlatbs( 'Upper', 'Conjugate transpose', 'Non-unit',
231  \$ normin, n, kd, ab, ldab, work, scalel, rwork,
232  \$ info )
233  normin = 'Y'
234 *
235 * Multiply by inv(U).
236 *
237  CALL zlatbs( 'Upper', 'No transpose', 'Non-unit', normin, n,
238  \$ kd, ab, ldab, work, scaleu, rwork, info )
239  ELSE
240 *
241 * Multiply by inv(L).
242 *
243  CALL zlatbs( 'Lower', 'No transpose', 'Non-unit', normin, n,
244  \$ kd, ab, ldab, work, scalel, rwork, info )
245  normin = 'Y'
246 *
247 * Multiply by inv(L**H).
248 *
249  CALL zlatbs( 'Lower', 'Conjugate transpose', 'Non-unit',
250  \$ normin, n, kd, ab, ldab, work, scaleu, rwork,
251  \$ info )
252  END IF
253 *
254 * Multiply by 1/SCALE if doing so will not cause overflow.
255 *
256  scale = scalel*scaleu
257  IF( scale.NE.one ) THEN
258  ix = izamax( n, work, 1 )
259  IF( scale.LT.cabs1( work( ix ) )*smlnum .OR. scale.EQ.zero )
260  \$ GO TO 20
261  CALL zdrscl( n, scale, work, 1 )
262  END IF
263  GO TO 10
264  END IF
265 *
266 * Compute the estimate of the reciprocal condition number.
267 *
268  IF( ainvnm.NE.zero )
269  \$ rcond = ( one / ainvnm ) / anorm
270 *
271  20 CONTINUE
272 *
273  RETURN
274 *
275 * End of ZPBCON
276 *
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zlacn2(N, V, X, EST, KASE, ISAVE)
ZLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: zlacn2.f:135
subroutine zdrscl(N, SA, SX, INCX)
ZDRSCL multiplies a vector by the reciprocal of a real scalar.
Definition: zdrscl.f:86
subroutine zlatbs(UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
ZLATBS solves a triangular banded system of equations.
Definition: zlatbs.f:245
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:73
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