LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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zhecon_3.f
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1*> \brief \b ZHECON_3
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZHECON_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*> ZHECON_3 estimates the reciprocal of the condition number (in the
40*> 1-norm) of a complex Hermitian matrix A using the factorization
41*> computed by ZHETRF_RK or ZHETRF_BK:
42*>
43*> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
44*>
45*> where U (or L) is unit upper (or lower) triangular matrix,
46*> U**H (or L**H) is the conjugate of U (or L), P is a permutation
47*> matrix, P**T is the transpose of P, and D is Hermitian 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 ZHETRS_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**H)*(P**T);
64*> = 'L': Lower triangular, form is A = P*L*D*(L**H)*(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 ZHETRF_RK and ZHETRF_BK:
78*> a) ONLY diagonal elements of the Hermitian 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 Hermitian 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 ZHETRF_RK or ZHETRF_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 complex16HEcomputational
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 zhecon_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* ..
187* .. Local Scalars ..
188 LOGICAL UPPER
189 INTEGER I, KASE
190 DOUBLE PRECISION AINVNM
191* ..
192* .. Local Arrays ..
193 INTEGER ISAVE( 3 )
194* ..
195* .. External Functions ..
196 LOGICAL LSAME
197 EXTERNAL lsame
198* ..
199* .. External Subroutines ..
200 EXTERNAL zhetrs_3, zlacn2, xerbla
201* ..
202* .. Intrinsic Functions ..
203 INTRINSIC max
204* ..
205* .. Executable Statements ..
206*
207* Test the input parameters.
208*
209 info = 0
210 upper = lsame( uplo, 'U' )
211 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
212 info = -1
213 ELSE IF( n.LT.0 ) THEN
214 info = -2
215 ELSE IF( lda.LT.max( 1, n ) ) THEN
216 info = -4
217 ELSE IF( anorm.LT.zero ) THEN
218 info = -7
219 END IF
220 IF( info.NE.0 ) THEN
221 CALL xerbla( 'ZHECON_3', -info )
222 RETURN
223 END IF
224*
225* Quick return if possible
226*
227 rcond = zero
228 IF( n.EQ.0 ) THEN
229 rcond = one
230 RETURN
231 ELSE IF( anorm.LE.zero ) THEN
232 RETURN
233 END IF
234*
235* Check that the diagonal matrix D is nonsingular.
236*
237 IF( upper ) THEN
238*
239* Upper triangular storage: examine D from bottom to top
240*
241 DO i = n, 1, -1
242 IF( ipiv( i ).GT.0 .AND. a( i, i ).EQ.zero )
243 \$ RETURN
244 END DO
245 ELSE
246*
247* Lower triangular storage: examine D from top to bottom.
248*
249 DO i = 1, n
250 IF( ipiv( i ).GT.0 .AND. a( i, i ).EQ.zero )
251 \$ RETURN
252 END DO
253 END IF
254*
255* Estimate the 1-norm of the inverse.
256*
257 kase = 0
258 30 CONTINUE
259 CALL zlacn2( n, work( n+1 ), work, ainvnm, kase, isave )
260 IF( kase.NE.0 ) THEN
261*
262* Multiply by inv(L*D*L**H) or inv(U*D*U**H).
263*
264 CALL zhetrs_3( uplo, n, 1, a, lda, e, ipiv, work, n, info )
265 GO TO 30
266 END IF
267*
268* Compute the estimate of the reciprocal condition number.
269*
270 IF( ainvnm.NE.zero )
271 \$ rcond = ( one / ainvnm ) / anorm
272*
273 RETURN
274*
275* End of ZHECON_3
276*
277 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zhecon_3(UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, INFO)
ZHECON_3
Definition: zhecon_3.f:166
subroutine zhetrs_3(UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
ZHETRS_3
Definition: zhetrs_3.f:165
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