LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
zpbcon.f
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1 *> \brief \b ZPBCON
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZPBCON( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK,
22 * RWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, KD, LDAB, N
27 * DOUBLE PRECISION ANORM, RCOND
28 * ..
29 * .. Array Arguments ..
30 * DOUBLE PRECISION RWORK( * )
31 * COMPLEX*16 AB( LDAB, * ), WORK( * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> ZPBCON estimates the reciprocal of the condition number (in the
41 *> 1-norm) of a complex Hermitian positive definite band matrix using
42 *> the Cholesky factorization A = U**H*U or A = L*L**H computed by
43 *> ZPBTRF.
44 *>
45 *> An estimate is obtained for norm(inv(A)), and the reciprocal of the
46 *> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
47 *> \endverbatim
48 *
49 * Arguments:
50 * ==========
51 *
52 *> \param[in] UPLO
53 *> \verbatim
54 *> UPLO is CHARACTER*1
55 *> = 'U': Upper triangular factor stored in AB;
56 *> = 'L': Lower triangular factor stored in AB.
57 *> \endverbatim
58 *>
59 *> \param[in] N
60 *> \verbatim
61 *> N is INTEGER
62 *> The order of the matrix A. N >= 0.
63 *> \endverbatim
64 *>
65 *> \param[in] KD
66 *> \verbatim
67 *> KD is INTEGER
68 *> The number of superdiagonals of the matrix A if UPLO = 'U',
69 *> or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
70 *> \endverbatim
71 *>
72 *> \param[in] AB
73 *> \verbatim
74 *> AB is COMPLEX*16 array, dimension (LDAB,N)
75 *> The triangular factor U or L from the Cholesky factorization
76 *> A = U**H*U or A = L*L**H of the band matrix A, stored in the
77 *> first KD+1 rows of the array. The j-th column of U or L is
78 *> stored in the j-th column of the array AB as follows:
79 *> if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
80 *> if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
81 *> \endverbatim
82 *>
83 *> \param[in] LDAB
84 *> \verbatim
85 *> LDAB is INTEGER
86 *> The leading dimension of the array AB. LDAB >= KD+1.
87 *> \endverbatim
88 *>
89 *> \param[in] ANORM
90 *> \verbatim
91 *> ANORM is DOUBLE PRECISION
92 *> The 1-norm (or infinity-norm) of the Hermitian band matrix A.
93 *> \endverbatim
94 *>
95 *> \param[out] RCOND
96 *> \verbatim
97 *> RCOND is DOUBLE PRECISION
98 *> The reciprocal of the condition number of the matrix A,
99 *> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
100 *> estimate of the 1-norm of inv(A) computed in this routine.
101 *> \endverbatim
102 *>
103 *> \param[out] WORK
104 *> \verbatim
105 *> WORK is COMPLEX*16 array, dimension (2*N)
106 *> \endverbatim
107 *>
108 *> \param[out] RWORK
109 *> \verbatim
110 *> RWORK is DOUBLE PRECISION array, dimension (N)
111 *> \endverbatim
112 *>
113 *> \param[out] INFO
114 *> \verbatim
115 *> INFO is INTEGER
116 *> = 0: successful exit
117 *> < 0: if INFO = -i, the i-th argument had an illegal value
118 *> \endverbatim
119 *
120 * Authors:
121 * ========
122 *
123 *> \author Univ. of Tennessee
124 *> \author Univ. of California Berkeley
125 *> \author Univ. of Colorado Denver
126 *> \author NAG Ltd.
127 *
128 *> \ingroup complex16OTHERcomputational
129 *
130 * =====================================================================
131  SUBROUTINE zpbcon( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK,
132  \$ RWORK, INFO )
133 *
134 * -- LAPACK computational routine --
135 * -- LAPACK is a software package provided by Univ. of Tennessee, --
136 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
137 *
138 * .. Scalar Arguments ..
139  CHARACTER UPLO
140  INTEGER INFO, KD, LDAB, N
141  DOUBLE PRECISION ANORM, RCOND
142 * ..
143 * .. Array Arguments ..
144  DOUBLE PRECISION RWORK( * )
145  COMPLEX*16 AB( LDAB, * ), WORK( * )
146 * ..
147 *
148 * =====================================================================
149 *
150 * .. Parameters ..
151  DOUBLE PRECISION ONE, ZERO
152  parameter( one = 1.0d+0, zero = 0.0d+0 )
153 * ..
154 * .. Local Scalars ..
155  LOGICAL UPPER
156  CHARACTER NORMIN
157  INTEGER IX, KASE
158  DOUBLE PRECISION AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
159  COMPLEX*16 ZDUM
160 * ..
161 * .. Local Arrays ..
162  INTEGER ISAVE( 3 )
163 * ..
164 * .. External Functions ..
165  LOGICAL LSAME
166  INTEGER IZAMAX
167  DOUBLE PRECISION DLAMCH
168  EXTERNAL lsame, izamax, dlamch
169 * ..
170 * .. External Subroutines ..
171  EXTERNAL xerbla, zdrscl, zlacn2, zlatbs
172 * ..
173 * .. Intrinsic Functions ..
174  INTRINSIC abs, dble, dimag
175 * ..
176 * .. Statement Functions ..
177  DOUBLE PRECISION CABS1
178 * ..
179 * .. Statement Function definitions ..
180  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
181 * ..
182 * .. Executable Statements ..
183 *
184 * Test the input parameters.
185 *
186  info = 0
187  upper = lsame( uplo, 'U' )
188  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
189  info = -1
190  ELSE IF( n.LT.0 ) THEN
191  info = -2
192  ELSE IF( kd.LT.0 ) THEN
193  info = -3
194  ELSE IF( ldab.LT.kd+1 ) THEN
195  info = -5
196  ELSE IF( anorm.LT.zero ) THEN
197  info = -6
198  END IF
199  IF( info.NE.0 ) THEN
200  CALL xerbla( 'ZPBCON', -info )
201  RETURN
202  END IF
203 *
204 * Quick return if possible
205 *
206  rcond = zero
207  IF( n.EQ.0 ) THEN
208  rcond = one
209  RETURN
210  ELSE IF( anorm.EQ.zero ) THEN
211  RETURN
212  END IF
213 *
214  smlnum = dlamch( 'Safe minimum' )
215 *
216 * Estimate the 1-norm of the inverse.
217 *
218  kase = 0
219  normin = 'N'
220  10 CONTINUE
221  CALL zlacn2( n, work( n+1 ), work, ainvnm, kase, isave )
222  IF( kase.NE.0 ) THEN
223  IF( upper ) THEN
224 *
225 * Multiply by inv(U**H).
226 *
227  CALL zlatbs( 'Upper', 'Conjugate transpose', 'Non-unit',
228  \$ normin, n, kd, ab, ldab, work, scalel, rwork,
229  \$ info )
230  normin = 'Y'
231 *
232 * Multiply by inv(U).
233 *
234  CALL zlatbs( 'Upper', 'No transpose', 'Non-unit', normin, n,
235  \$ kd, ab, ldab, work, scaleu, rwork, info )
236  ELSE
237 *
238 * Multiply by inv(L).
239 *
240  CALL zlatbs( 'Lower', 'No transpose', 'Non-unit', normin, n,
241  \$ kd, ab, ldab, work, scalel, rwork, info )
242  normin = 'Y'
243 *
244 * Multiply by inv(L**H).
245 *
246  CALL zlatbs( 'Lower', 'Conjugate transpose', 'Non-unit',
247  \$ normin, n, kd, ab, ldab, work, scaleu, rwork,
248  \$ info )
249  END IF
250 *
251 * Multiply by 1/SCALE if doing so will not cause overflow.
252 *
253  scale = scalel*scaleu
254  IF( scale.NE.one ) THEN
255  ix = izamax( n, work, 1 )
256  IF( scale.LT.cabs1( work( ix ) )*smlnum .OR. scale.EQ.zero )
257  \$ GO TO 20
258  CALL zdrscl( n, scale, work, 1 )
259  END IF
260  GO TO 10
261  END IF
262 *
263 * Compute the estimate of the reciprocal condition number.
264 *
265  IF( ainvnm.NE.zero )
266  \$ rcond = ( one / ainvnm ) / anorm
267 *
268  20 CONTINUE
269 *
270  RETURN
271 *
272 * End of ZPBCON
273 *
274  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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:133
subroutine zdrscl(N, SA, SX, INCX)
ZDRSCL multiplies a vector by the reciprocal of a real scalar.
Definition: zdrscl.f:84
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:243
subroutine zpbcon(UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, RWORK, INFO)
ZPBCON
Definition: zpbcon.f:133