LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
zsycon_3.f
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1 *> \brief \b ZSYCON_3
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZSYCON_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND,
22 * WORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, LDA, N
27 * DOUBLE PRECISION ANORM, RCOND
28 * ..
29 * .. Array Arguments ..
30 * INTEGER IPIV( * )
31 * COMPLEX*16 A( LDA, * ), E ( * ), WORK( * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *> ZSYCON_3 estimates the reciprocal of the condition number (in the
40 *> 1-norm) of a complex symmetric matrix A using the factorization
41 *> computed by ZSYTRF_RK or ZSYTRF_BK:
42 *>
43 *> A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
44 *>
45 *> where U (or L) is unit upper (or lower) triangular matrix,
46 *> U**T (or L**T) is the transpose of U (or L), P is a permutation
47 *> matrix, P**T is the transpose of P, and D is symmetric and block
48 *> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
49 *>
50 *> An estimate is obtained for norm(inv(A)), and the reciprocal of the
51 *> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
52 *> This routine uses BLAS3 solver ZSYTRS_3.
53 *> \endverbatim
54 *
55 * Arguments:
56 * ==========
57 *
58 *> \param[in] UPLO
59 *> \verbatim
60 *> UPLO is CHARACTER*1
61 *> Specifies whether the details of the factorization are
62 *> stored as an upper or lower triangular matrix:
63 *> = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T);
64 *> = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).
65 *> \endverbatim
66 *>
67 *> \param[in] N
68 *> \verbatim
69 *> N is INTEGER
70 *> The order of the matrix A. N >= 0.
71 *> \endverbatim
72 *>
73 *> \param[in] A
74 *> \verbatim
75 *> A is COMPLEX*16 array, dimension (LDA,N)
76 *> Diagonal of the block diagonal matrix D and factors U or L
77 *> as computed by ZSYTRF_RK and ZSYTRF_BK:
78 *> a) ONLY diagonal elements of the symmetric block diagonal
79 *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
80 *> (superdiagonal (or subdiagonal) elements of D
81 *> should be provided on entry in array E), and
82 *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
83 *> If UPLO = 'L': factor L in the subdiagonal part of A.
84 *> \endverbatim
85 *>
86 *> \param[in] LDA
87 *> \verbatim
88 *> LDA is INTEGER
89 *> The leading dimension of the array A. LDA >= max(1,N).
90 *> \endverbatim
91 *>
92 *> \param[in] E
93 *> \verbatim
94 *> E is COMPLEX*16 array, dimension (N)
95 *> On entry, contains the superdiagonal (or subdiagonal)
96 *> elements of the symmetric block diagonal matrix D
97 *> with 1-by-1 or 2-by-2 diagonal blocks, where
98 *> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
99 *> If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
100 *>
101 *> NOTE: For 1-by-1 diagonal block D(k), where
102 *> 1 <= k <= N, the element E(k) is not referenced in both
103 *> UPLO = 'U' or UPLO = 'L' cases.
104 *> \endverbatim
105 *>
106 *> \param[in] IPIV
107 *> \verbatim
108 *> IPIV is INTEGER array, dimension (N)
109 *> Details of the interchanges and the block structure of D
110 *> as determined by ZSYTRF_RK or ZSYTRF_BK.
111 *> \endverbatim
112 *>
113 *> \param[in] ANORM
114 *> \verbatim
115 *> ANORM is DOUBLE PRECISION
116 *> The 1-norm of the original matrix A.
117 *> \endverbatim
118 *>
119 *> \param[out] RCOND
120 *> \verbatim
121 *> RCOND is DOUBLE PRECISION
122 *> The reciprocal of the condition number of the matrix A,
123 *> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
124 *> estimate of the 1-norm of inv(A) computed in this routine.
125 *> \endverbatim
126 *>
127 *> \param[out] WORK
128 *> \verbatim
129 *> WORK is COMPLEX*16 array, dimension (2*N)
130 *> \endverbatim
131 *>
132 *> \param[out] INFO
133 *> \verbatim
134 *> INFO is INTEGER
135 *> = 0: successful exit
136 *> < 0: if INFO = -i, the i-th argument had an illegal value
137 *> \endverbatim
138 *
139 * Authors:
140 * ========
141 *
142 *> \author Univ. of Tennessee
143 *> \author Univ. of California Berkeley
144 *> \author Univ. of Colorado Denver
145 *> \author NAG Ltd.
146 *
147 *> \ingroup complex16SYcomputational
148 *
149 *> \par Contributors:
150 * ==================
151 *> \verbatim
152 *>
153 *> June 2017, Igor Kozachenko,
154 *> Computer Science Division,
155 *> University of California, Berkeley
156 *>
157 *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
158 *> School of Mathematics,
159 *> University of Manchester
160 *>
161 *> \endverbatim
162 *
163 * =====================================================================
164  SUBROUTINE zsycon_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND,
165  \$ WORK, INFO )
166 *
167 * -- LAPACK computational routine --
168 * -- LAPACK is a software package provided by Univ. of Tennessee, --
169 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
170 *
171 * .. Scalar Arguments ..
172  CHARACTER UPLO
173  INTEGER INFO, LDA, N
174  DOUBLE PRECISION ANORM, RCOND
175 * ..
176 * .. Array Arguments ..
177  INTEGER IPIV( * )
178  COMPLEX*16 A( LDA, * ), E( * ), WORK( * )
179 * ..
180 *
181 * =====================================================================
182 *
183 * .. Parameters ..
184  DOUBLE PRECISION ONE, ZERO
185  parameter( one = 1.0d+0, zero = 0.0d+0 )
186  COMPLEX*16 CZERO
187  parameter( czero = ( 0.0d+0, 0.0d+0 ) )
188 * ..
189 * .. Local Scalars ..
190  LOGICAL UPPER
191  INTEGER I, KASE
192  DOUBLE PRECISION AINVNM
193 * ..
194 * .. Local Arrays ..
195  INTEGER ISAVE( 3 )
196 * ..
197 * .. External Functions ..
198  LOGICAL LSAME
199  EXTERNAL lsame
200 * ..
201 * .. External Subroutines ..
202  EXTERNAL zlacn2, zsytrs_3, xerbla
203 * ..
204 * .. Intrinsic Functions ..
205  INTRINSIC max
206 * ..
207 * .. Executable Statements ..
208 *
209 * Test the input parameters.
210 *
211  info = 0
212  upper = lsame( uplo, 'U' )
213  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
214  info = -1
215  ELSE IF( n.LT.0 ) THEN
216  info = -2
217  ELSE IF( lda.LT.max( 1, n ) ) THEN
218  info = -4
219  ELSE IF( anorm.LT.zero ) THEN
220  info = -7
221  END IF
222  IF( info.NE.0 ) THEN
223  CALL xerbla( 'ZSYCON_3', -info )
224  RETURN
225  END IF
226 *
227 * Quick return if possible
228 *
229  rcond = zero
230  IF( n.EQ.0 ) THEN
231  rcond = one
232  RETURN
233  ELSE IF( anorm.LE.zero ) THEN
234  RETURN
235  END IF
236 *
237 * Check that the diagonal matrix D is nonsingular.
238 *
239  IF( upper ) THEN
240 *
241 * Upper triangular storage: examine D from bottom to top
242 *
243  DO i = n, 1, -1
244  IF( ipiv( i ).GT.0 .AND. a( i, i ).EQ.czero )
245  \$ RETURN
246  END DO
247  ELSE
248 *
249 * Lower triangular storage: examine D from top to bottom.
250 *
251  DO i = 1, n
252  IF( ipiv( i ).GT.0 .AND. a( i, i ).EQ.czero )
253  \$ RETURN
254  END DO
255  END IF
256 *
257 * Estimate the 1-norm of the inverse.
258 *
259  kase = 0
260  30 CONTINUE
261  CALL zlacn2( n, work( n+1 ), work, ainvnm, kase, isave )
262  IF( kase.NE.0 ) THEN
263 *
264 * Multiply by inv(L*D*L**T) or inv(U*D*U**T).
265 *
266  CALL zsytrs_3( uplo, n, 1, a, lda, e, ipiv, work, n, info )
267  GO TO 30
268  END IF
269 *
270 * Compute the estimate of the reciprocal condition number.
271 *
272  IF( ainvnm.NE.zero )
273  \$ rcond = ( one / ainvnm ) / anorm
274 *
275  RETURN
276 *
277 * End of ZSYCON_3
278 *
279  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 zsytrs_3(UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
ZSYTRS_3
Definition: zsytrs_3.f:165
subroutine zsycon_3(UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, INFO)
ZSYCON_3
Definition: zsycon_3.f:166