Annotation of rpl/lapack/lapack/zhecon_3.f, revision 1.6
1.1 bertrand 1: *> \brief \b ZHECON_3
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZHECON_3 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhecon_3.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhecon_3.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhecon_3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHECON_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND,
1.5 bertrand 22: * WORK, INFO )
1.1 bertrand 23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDA, N
27: * DOUBLE PRECISION ANORM, RCOND
28: * ..
29: * .. Array Arguments ..
1.5 bertrand 30: * INTEGER IPIV( * )
1.1 bertrand 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
1.3 bertrand 98: *> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
1.1 bertrand 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: *>
1.3 bertrand 153: *> June 2017, Igor Kozachenko,
1.1 bertrand 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: *
1.6 ! bertrand 167: * -- LAPACK computational routine --
1.1 bertrand 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 ..
1.3 bertrand 184: DOUBLE PRECISION ONE, ZERO
1.1 bertrand 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
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