1*> \brief \b CSYCON_3
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE CSYCON_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*       REAL               ANORM, RCOND
28*       ..
29*       .. Array Arguments ..
30*       INTEGER            IPIV( * )
31*       COMPLEX            A( LDA, * ), E ( * ), WORK( * )
32*       ..
33*
34*
35*> \par Purpose:
36*  =============
37*>
38*> \verbatim
39*> CSYCON_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 CSYTRF_RK or CSYTRF_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 CSYTRS_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 array, dimension (LDA,N)
76*>          Diagonal of the block diagonal matrix D and factors U or L
77*>          as computed by CSYTRF_RK and CSYTRF_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 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 CSYTRF_RK or CSYTRF_BK.
111*> \endverbatim
112*>
113*> \param[in] ANORM
114*> \verbatim
115*>          ANORM is REAL
116*>          The 1-norm of the original matrix A.
117*> \endverbatim
118*>
119*> \param[out] RCOND
120*> \verbatim
121*>          RCOND is REAL
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 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 complexSYcomputational
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 CSYCON_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      REAL               ANORM, RCOND
175*     ..
176*     .. Array Arguments ..
177      INTEGER            IPIV( * )
178      COMPLEX            A( LDA, * ), E( * ), WORK( * )
179*     ..
180*
181*  =====================================================================
182*
183*     .. Parameters ..
184      REAL               ONE, ZERO
185      PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
186      COMPLEX            CZERO
187      PARAMETER          ( CZERO = ( 0.0E+0, 0.0E+0 ) )
188*     ..
189*     .. Local Scalars ..
190      LOGICAL            UPPER
191      INTEGER            I, KASE
192      REAL               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           CLACN2, CSYTRS_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( 'CSYCON_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 CLACN2( 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 CSYTRS_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 CSYCON_3
278*
279      END
280