1*> \brief <b> ZHESV computes the solution to system of linear equations A * X = B for HE matrices</b>
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
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7*
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17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE ZHESV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
22*                         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, * ), WORK( * )
31*       ..
32*
33*
34*> \par Purpose:
35*  =============
36*>
37*> \verbatim
38*>
39*> ZHESV computes the solution to a complex system of linear equations
40*>    A * X = B,
41*> where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
42*> matrices.
43*>
44*> The diagonal pivoting method is used to factor A as
45*>    A = U * D * U**H,  if UPLO = 'U', or
46*>    A = L * D * L**H,  if UPLO = 'L',
47*> where U (or L) is a product of permutation and unit upper (lower)
48*> triangular matrices, and D is Hermitian and block diagonal with
49*> 1-by-1 and 2-by-2 diagonal blocks.  The factored form of A is then
50*> used to solve the system of equations A * X = B.
51*> \endverbatim
52*
53*  Arguments:
54*  ==========
55*
56*> \param[in] UPLO
57*> \verbatim
58*>          UPLO is CHARACTER*1
59*>          = 'U':  Upper triangle of A is stored;
60*>          = 'L':  Lower triangle of A is stored.
61*> \endverbatim
62*>
63*> \param[in] N
64*> \verbatim
65*>          N is INTEGER
66*>          The number of linear equations, i.e., the order of the
67*>          matrix A.  N >= 0.
68*> \endverbatim
69*>
70*> \param[in] NRHS
71*> \verbatim
72*>          NRHS is INTEGER
73*>          The number of right hand sides, i.e., the number of columns
74*>          of the matrix B.  NRHS >= 0.
75*> \endverbatim
76*>
77*> \param[in,out] A
78*> \verbatim
79*>          A is COMPLEX*16 array, dimension (LDA,N)
80*>          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
81*>          N-by-N upper triangular part of A contains the upper
82*>          triangular part of the matrix A, and the strictly lower
83*>          triangular part of A is not referenced.  If UPLO = 'L', the
84*>          leading N-by-N lower triangular part of A contains the lower
85*>          triangular part of the matrix A, and the strictly upper
86*>          triangular part of A is not referenced.
87*>
88*>          On exit, if INFO = 0, the block diagonal matrix D and the
89*>          multipliers used to obtain the factor U or L from the
90*>          factorization A = U*D*U**H or A = L*D*L**H as computed by
91*>          ZHETRF.
92*> \endverbatim
93*>
94*> \param[in] LDA
95*> \verbatim
96*>          LDA is INTEGER
97*>          The leading dimension of the array A.  LDA >= max(1,N).
98*> \endverbatim
99*>
100*> \param[out] IPIV
101*> \verbatim
102*>          IPIV is INTEGER array, dimension (N)
103*>          Details of the interchanges and the block structure of D, as
104*>          determined by ZHETRF.  If IPIV(k) > 0, then rows and columns
105*>          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
106*>          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
107*>          then rows and columns k-1 and -IPIV(k) were interchanged and
108*>          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
109*>          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
110*>          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
111*>          diagonal block.
112*> \endverbatim
113*>
114*> \param[in,out] B
115*> \verbatim
116*>          B is COMPLEX*16 array, dimension (LDB,NRHS)
117*>          On entry, the N-by-NRHS right hand side matrix B.
118*>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
119*> \endverbatim
120*>
121*> \param[in] LDB
122*> \verbatim
123*>          LDB is INTEGER
124*>          The leading dimension of the array B.  LDB >= max(1,N).
125*> \endverbatim
126*>
127*> \param[out] WORK
128*> \verbatim
129*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
130*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
131*> \endverbatim
132*>
133*> \param[in] LWORK
134*> \verbatim
135*>          LWORK is INTEGER
136*>          The length of WORK.  LWORK >= 1, and for best performance
137*>          LWORK >= max(1,N*NB), where NB is the optimal blocksize for
138*>          ZHETRF.
139*>          for LWORK < N, TRS will be done with Level BLAS 2
140*>          for LWORK >= N, TRS will be done with Level BLAS 3
141*>
142*>          If LWORK = -1, then a workspace query is assumed; the routine
143*>          only calculates the optimal size of the WORK array, returns
144*>          this value as the first entry of the WORK array, and no error
145*>          message related to LWORK is issued by XERBLA.
146*> \endverbatim
147*>
148*> \param[out] INFO
149*> \verbatim
150*>          INFO is INTEGER
151*>          = 0: successful exit
152*>          < 0: if INFO = -i, the i-th argument had an illegal value
153*>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
154*>               has been completed, but the block diagonal matrix D is
155*>               exactly singular, so the solution could not be computed.
156*> \endverbatim
157*
158*  Authors:
159*  ========
160*
161*> \author Univ. of Tennessee
162*> \author Univ. of California Berkeley
163*> \author Univ. of Colorado Denver
164*> \author NAG Ltd.
165*
166*> \ingroup complex16HEsolve
167*
168*  =====================================================================
169      SUBROUTINE ZHESV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
170     $                  LWORK, INFO )
171*
172*  -- LAPACK driver routine --
173*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
174*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175*
176*     .. Scalar Arguments ..
177      CHARACTER          UPLO
178      INTEGER            INFO, LDA, LDB, LWORK, N, NRHS
179*     ..
180*     .. Array Arguments ..
181      INTEGER            IPIV( * )
182      COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
183*     ..
184*
185*  =====================================================================
186*
187*     .. Local Scalars ..
188      LOGICAL            LQUERY
189      INTEGER            LWKOPT, NB
190*     ..
191*     .. External Functions ..
192      LOGICAL            LSAME
193      INTEGER            ILAENV
194      EXTERNAL           LSAME, ILAENV
195*     ..
196*     .. External Subroutines ..
197      EXTERNAL           XERBLA, ZHETRF, ZHETRS, ZHETRS2
198*     ..
199*     .. Intrinsic Functions ..
200      INTRINSIC          MAX
201*     ..
202*     .. Executable Statements ..
203*
204*     Test the input parameters.
205*
206      INFO = 0
207      LQUERY = ( LWORK.EQ.-1 )
208      IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
209         INFO = -1
210      ELSE IF( N.LT.0 ) THEN
211         INFO = -2
212      ELSE IF( NRHS.LT.0 ) THEN
213         INFO = -3
214      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
215         INFO = -5
216      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
217         INFO = -8
218      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
219         INFO = -10
220      END IF
221*
222      IF( INFO.EQ.0 ) THEN
223         IF( N.EQ.0 ) THEN
224            LWKOPT = 1
225         ELSE
226            NB = ILAENV( 1, 'ZHETRF', UPLO, N, -1, -1, -1 )
227            LWKOPT = N*NB
228         END IF
229         WORK( 1 ) = LWKOPT
230      END IF
231*
232      IF( INFO.NE.0 ) THEN
233         CALL XERBLA( 'ZHESV ', -INFO )
234         RETURN
235      ELSE IF( LQUERY ) THEN
236         RETURN
237      END IF
238*
239*     Compute the factorization A = U*D*U**H or A = L*D*L**H.
240*
241      CALL ZHETRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
242      IF( INFO.EQ.0 ) THEN
243*
244*        Solve the system A*X = B, overwriting B with X.
245*
246         IF ( LWORK.LT.N ) THEN
247*
248*        Solve with TRS ( Use Level BLAS 2)
249*
250            CALL ZHETRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
251*
252         ELSE
253*
254*        Solve with TRS2 ( Use Level BLAS 3)
255*
256            CALL ZHETRS2( UPLO,N,NRHS,A,LDA,IPIV,B,LDB,WORK,INFO )
257*
258         END IF
259*
260      END IF
261*
262      WORK( 1 ) = LWKOPT
263*
264      RETURN
265*
266*     End of ZHESV
267*
268      END
269