1: *> \brief <b> ZHESV_RK computes the solution to system of linear equations A * X = B for SY matrices</b>
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 ZHESV_RK( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
22: * WORK, LWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDA, LDB, LWORK, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IPIV( * )
30: * COMPLEX*16 A( LDA, * ), B( LDB, * ), E( * ), WORK( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *> ZHESV_RK computes the solution to a complex system of linear
39: *> equations A * X = B, where A is an N-by-N Hermitian matrix
40: *> and X and B are N-by-NRHS matrices.
41: *>
42: *> The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
43: *> to factor A as
44: *> A = P*U*D*(U**H)*(P**T), if UPLO = 'U', or
45: *> A = P*L*D*(L**H)*(P**T), if UPLO = 'L',
46: *> where U (or L) is unit upper (or lower) triangular matrix,
47: *> U**H (or L**H) is the conjugate of U (or L), P is a permutation
48: *> matrix, P**T is the transpose of P, and D is Hermitian and block
49: *> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
50: *>
51: *> ZHETRF_RK is called to compute the factorization of a complex
52: *> Hermitian matrix. The factored form of A is then used to solve
53: *> the system of equations A * X = B by calling BLAS3 routine ZHETRS_3.
54: *> \endverbatim
55: *
56: * Arguments:
57: * ==========
58: *
59: *> \param[in] UPLO
60: *> \verbatim
61: *> UPLO is CHARACTER*1
62: *> Specifies whether the upper or lower triangular part of the
63: *> Hermitian matrix A is stored:
64: *> = 'U': Upper triangle of A is stored;
65: *> = 'L': Lower triangle of A is stored.
66: *> \endverbatim
67: *>
68: *> \param[in] N
69: *> \verbatim
70: *> N is INTEGER
71: *> The number of linear equations, i.e., the order of the
72: *> matrix A. N >= 0.
73: *> \endverbatim
74: *>
75: *> \param[in] NRHS
76: *> \verbatim
77: *> NRHS is INTEGER
78: *> The number of right hand sides, i.e., the number of columns
79: *> of the matrix B. NRHS >= 0.
80: *> \endverbatim
81: *>
82: *> \param[in,out] A
83: *> \verbatim
84: *> A is COMPLEX*16 array, dimension (LDA,N)
85: *> On entry, the Hermitian matrix A.
86: *> If UPLO = 'U': the leading N-by-N upper triangular part
87: *> of A contains the upper triangular part of the matrix A,
88: *> and the strictly lower triangular part of A is not
89: *> referenced.
90: *>
91: *> If UPLO = 'L': the leading N-by-N lower triangular part
92: *> of A contains the lower triangular part of the matrix A,
93: *> and the strictly upper triangular part of A is not
94: *> referenced.
95: *>
96: *> On exit, if INFO = 0, diagonal of the block diagonal
97: *> matrix D and factors U or L as computed by ZHETRF_RK:
98: *> a) ONLY diagonal elements of the Hermitian block diagonal
99: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
100: *> (superdiagonal (or subdiagonal) elements of D
101: *> are stored on exit in array E), and
102: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
103: *> If UPLO = 'L': factor L in the subdiagonal part of A.
104: *>
105: *> For more info see the description of ZHETRF_RK routine.
106: *> \endverbatim
107: *>
108: *> \param[in] LDA
109: *> \verbatim
110: *> LDA is INTEGER
111: *> The leading dimension of the array A. LDA >= max(1,N).
112: *> \endverbatim
113: *>
114: *> \param[out] E
115: *> \verbatim
116: *> E is COMPLEX*16 array, dimension (N)
117: *> On exit, contains the output computed by the factorization
118: *> routine ZHETRF_RK, i.e. the superdiagonal (or subdiagonal)
119: *> elements of the Hermitian block diagonal matrix D
120: *> with 1-by-1 or 2-by-2 diagonal blocks, where
121: *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
122: *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
123: *>
124: *> NOTE: For 1-by-1 diagonal block D(k), where
125: *> 1 <= k <= N, the element E(k) is set to 0 in both
126: *> UPLO = 'U' or UPLO = 'L' cases.
127: *>
128: *> For more info see the description of ZHETRF_RK routine.
129: *> \endverbatim
130: *>
131: *> \param[out] IPIV
132: *> \verbatim
133: *> IPIV is INTEGER array, dimension (N)
134: *> Details of the interchanges and the block structure of D,
135: *> as determined by ZHETRF_RK.
136: *>
137: *> For more info see the description of ZHETRF_RK routine.
138: *> \endverbatim
139: *>
140: *> \param[in,out] B
141: *> \verbatim
142: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
143: *> On entry, the N-by-NRHS right hand side matrix B.
144: *> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
145: *> \endverbatim
146: *>
147: *> \param[in] LDB
148: *> \verbatim
149: *> LDB is INTEGER
150: *> The leading dimension of the array B. LDB >= max(1,N).
151: *> \endverbatim
152: *>
153: *> \param[out] WORK
154: *> \verbatim
155: *> WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ).
156: *> Work array used in the factorization stage.
157: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
158: *> \endverbatim
159: *>
160: *> \param[in] LWORK
161: *> \verbatim
162: *> LWORK is INTEGER
163: *> The length of WORK. LWORK >= 1. For best performance
164: *> of factorization stage LWORK >= max(1,N*NB), where NB is
165: *> the optimal blocksize for ZHETRF_RK.
166: *>
167: *> If LWORK = -1, then a workspace query is assumed;
168: *> the routine only calculates the optimal size of the WORK
169: *> array for factorization stage, returns this value as
170: *> the first entry of the WORK array, and no error message
171: *> related to LWORK is issued by XERBLA.
172: *> \endverbatim
173: *>
174: *> \param[out] INFO
175: *> \verbatim
176: *> INFO is INTEGER
177: *> = 0: successful exit
178: *>
179: *> < 0: If INFO = -k, the k-th argument had an illegal value
180: *>
181: *> > 0: If INFO = k, the matrix A is singular, because:
182: *> If UPLO = 'U': column k in the upper
183: *> triangular part of A contains all zeros.
184: *> If UPLO = 'L': column k in the lower
185: *> triangular part of A contains all zeros.
186: *>
187: *> Therefore D(k,k) is exactly zero, and superdiagonal
188: *> elements of column k of U (or subdiagonal elements of
189: *> column k of L ) are all zeros. The factorization has
190: *> been completed, but the block diagonal matrix D is
191: *> exactly singular, and division by zero will occur if
192: *> it is used to solve a system of equations.
193: *>
194: *> NOTE: INFO only stores the first occurrence of
195: *> a singularity, any subsequent occurrence of singularity
196: *> is not stored in INFO even though the factorization
197: *> always completes.
198: *> \endverbatim
199: *
200: * Authors:
201: * ========
202: *
203: *> \author Univ. of Tennessee
204: *> \author Univ. of California Berkeley
205: *> \author Univ. of Colorado Denver
206: *> \author NAG Ltd.
207: *
208: *> \ingroup complex16HEsolve
209: *
210: *> \par Contributors:
211: * ==================
212: *>
213: *> \verbatim
214: *>
215: *> December 2016, Igor Kozachenko,
216: *> Computer Science Division,
217: *> University of California, Berkeley
218: *>
219: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
220: *> School of Mathematics,
221: *> University of Manchester
222: *>
223: *> \endverbatim
224: *
225: * =====================================================================
226: SUBROUTINE ZHESV_RK( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, WORK,
227: $ LWORK, INFO )
228: *
229: * -- LAPACK driver routine --
230: * -- LAPACK is a software package provided by Univ. of Tennessee, --
231: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
232: *
233: * .. Scalar Arguments ..
234: CHARACTER UPLO
235: INTEGER INFO, LDA, LDB, LWORK, N, NRHS
236: * ..
237: * .. Array Arguments ..
238: INTEGER IPIV( * )
239: COMPLEX*16 A( LDA, * ), B( LDB, * ), E( * ), WORK( * )
240: * ..
241: *
242: * =====================================================================
243: *
244: * .. Local Scalars ..
245: LOGICAL LQUERY
246: INTEGER LWKOPT
247: * ..
248: * .. External Functions ..
249: LOGICAL LSAME
250: EXTERNAL LSAME
251: * ..
252: * .. External Subroutines ..
253: EXTERNAL XERBLA, ZHETRF_RK, ZHETRS_3
254: * ..
255: * .. Intrinsic Functions ..
256: INTRINSIC MAX
257: * ..
258: * .. Executable Statements ..
259: *
260: * Test the input parameters.
261: *
262: INFO = 0
263: LQUERY = ( LWORK.EQ.-1 )
264: IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
265: INFO = -1
266: ELSE IF( N.LT.0 ) THEN
267: INFO = -2
268: ELSE IF( NRHS.LT.0 ) THEN
269: INFO = -3
270: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
271: INFO = -5
272: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
273: INFO = -9
274: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
275: INFO = -11
276: END IF
277: *
278: IF( INFO.EQ.0 ) THEN
279: IF( N.EQ.0 ) THEN
280: LWKOPT = 1
281: ELSE
282: CALL ZHETRF_RK( UPLO, N, A, LDA, E, IPIV, WORK, -1, INFO )
283: LWKOPT = INT( DBLE( WORK( 1 ) ) )
284: END IF
285: WORK( 1 ) = LWKOPT
286: END IF
287: *
288: IF( INFO.NE.0 ) THEN
289: CALL XERBLA( 'ZHESV_RK ', -INFO )
290: RETURN
291: ELSE IF( LQUERY ) THEN
292: RETURN
293: END IF
294: *
295: * Compute the factorization A = P*U*D*(U**H)*(P**T) or
296: * A = P*U*D*(U**H)*(P**T).
297: *
298: CALL ZHETRF_RK( UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO )
299: *
300: IF( INFO.EQ.0 ) THEN
301: *
302: * Solve the system A*X = B with BLAS3 solver, overwriting B with X.
303: *
304: CALL ZHETRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO )
305: *
306: END IF
307: *
308: WORK( 1 ) = LWKOPT
309: *
310: RETURN
311: *
312: * End of ZHESV_RK
313: *
314: END
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