1*> \brief \b SSYGV
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssygv.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SSYGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
22*                         LWORK, INFO )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          JOBZ, UPLO
26*       INTEGER            INFO, ITYPE, LDA, LDB, LWORK, N
27*       ..
28*       .. Array Arguments ..
29*       REAL               A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
30*       ..
31*
32*
33*> \par Purpose:
34*  =============
35*>
36*> \verbatim
37*>
38*> SSYGV computes all the eigenvalues, and optionally, the eigenvectors
39*> of a real generalized symmetric-definite eigenproblem, of the form
40*> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
41*> Here A and B are assumed to be symmetric and B is also
42*> positive definite.
43*> \endverbatim
44*
45*  Arguments:
46*  ==========
47*
48*> \param[in] ITYPE
49*> \verbatim
50*>          ITYPE is INTEGER
51*>          Specifies the problem type to be solved:
52*>          = 1:  A*x = (lambda)*B*x
53*>          = 2:  A*B*x = (lambda)*x
54*>          = 3:  B*A*x = (lambda)*x
55*> \endverbatim
56*>
57*> \param[in] JOBZ
58*> \verbatim
59*>          JOBZ is CHARACTER*1
60*>          = 'N':  Compute eigenvalues only;
61*>          = 'V':  Compute eigenvalues and eigenvectors.
62*> \endverbatim
63*>
64*> \param[in] UPLO
65*> \verbatim
66*>          UPLO is CHARACTER*1
67*>          = 'U':  Upper triangles of A and B are stored;
68*>          = 'L':  Lower triangles of A and B are stored.
69*> \endverbatim
70*>
71*> \param[in] N
72*> \verbatim
73*>          N is INTEGER
74*>          The order of the matrices A and B.  N >= 0.
75*> \endverbatim
76*>
77*> \param[in,out] A
78*> \verbatim
79*>          A is REAL array, dimension (LDA, N)
80*>          On entry, the symmetric matrix A.  If UPLO = 'U', the
81*>          leading N-by-N upper triangular part of A contains the
82*>          upper triangular part of the matrix A.  If UPLO = 'L',
83*>          the leading N-by-N lower triangular part of A contains
84*>          the lower triangular part of the matrix A.
85*>
86*>          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
87*>          matrix Z of eigenvectors.  The eigenvectors are normalized
88*>          as follows:
89*>          if ITYPE = 1 or 2, Z**T*B*Z = I;
90*>          if ITYPE = 3, Z**T*inv(B)*Z = I.
91*>          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
92*>          or the lower triangle (if UPLO='L') of A, including the
93*>          diagonal, is destroyed.
94*> \endverbatim
95*>
96*> \param[in] LDA
97*> \verbatim
98*>          LDA is INTEGER
99*>          The leading dimension of the array A.  LDA >= max(1,N).
100*> \endverbatim
101*>
102*> \param[in,out] B
103*> \verbatim
104*>          B is REAL array, dimension (LDB, N)
105*>          On entry, the symmetric positive definite matrix B.
106*>          If UPLO = 'U', the leading N-by-N upper triangular part of B
107*>          contains the upper triangular part of the matrix B.
108*>          If UPLO = 'L', the leading N-by-N lower triangular part of B
109*>          contains the lower triangular part of the matrix B.
110*>
111*>          On exit, if INFO <= N, the part of B containing the matrix is
112*>          overwritten by the triangular factor U or L from the Cholesky
113*>          factorization B = U**T*U or B = L*L**T.
114*> \endverbatim
115*>
116*> \param[in] LDB
117*> \verbatim
118*>          LDB is INTEGER
119*>          The leading dimension of the array B.  LDB >= max(1,N).
120*> \endverbatim
121*>
122*> \param[out] W
123*> \verbatim
124*>          W is REAL array, dimension (N)
125*>          If INFO = 0, the eigenvalues in ascending order.
126*> \endverbatim
127*>
128*> \param[out] WORK
129*> \verbatim
130*>          WORK is REAL array, dimension (MAX(1,LWORK))
131*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
132*> \endverbatim
133*>
134*> \param[in] LWORK
135*> \verbatim
136*>          LWORK is INTEGER
137*>          The length of the array WORK.  LWORK >= max(1,3*N-1).
138*>          For optimal efficiency, LWORK >= (NB+2)*N,
139*>          where NB is the blocksize for SSYTRD returned by ILAENV.
140*>
141*>          If LWORK = -1, then a workspace query is assumed; the routine
142*>          only calculates the optimal size of the WORK array, returns
143*>          this value as the first entry of the WORK array, and no error
144*>          message related to LWORK is issued by XERBLA.
145*> \endverbatim
146*>
147*> \param[out] INFO
148*> \verbatim
149*>          INFO is INTEGER
150*>          = 0:  successful exit
151*>          < 0:  if INFO = -i, the i-th argument had an illegal value
152*>          > 0:  SPOTRF or SSYEV returned an error code:
153*>             <= N:  if INFO = i, SSYEV failed to converge;
154*>                    i off-diagonal elements of an intermediate
155*>                    tridiagonal form did not converge to zero;
156*>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
157*>                    minor of order i of B is not positive definite.
158*>                    The factorization of B could not be completed and
159*>                    no eigenvalues or eigenvectors were computed.
160*> \endverbatim
161*
162*  Authors:
163*  ========
164*
165*> \author Univ. of Tennessee
166*> \author Univ. of California Berkeley
167*> \author Univ. of Colorado Denver
168*> \author NAG Ltd.
169*
170*> \date November 2015
171*
172*> \ingroup realSYeigen
173*
174*  =====================================================================
175      SUBROUTINE SSYGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
176     $                  LWORK, INFO )
177*
178*  -- LAPACK driver routine (version 3.6.0) --
179*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
180*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181*     November 2015
182*
183*     .. Scalar Arguments ..
184      CHARACTER          JOBZ, UPLO
185      INTEGER            INFO, ITYPE, LDA, LDB, LWORK, N
186*     ..
187*     .. Array Arguments ..
188      REAL               A( LDA, * ), B( LDB, * ), W( * ), WORK( * )
189*     ..
190*
191*  =====================================================================
192*
193*     .. Parameters ..
194      REAL               ONE
195      PARAMETER          ( ONE = 1.0E+0 )
196*     ..
197*     .. Local Scalars ..
198      LOGICAL            LQUERY, UPPER, WANTZ
199      CHARACTER          TRANS
200      INTEGER            LWKMIN, LWKOPT, NB, NEIG
201*     ..
202*     .. External Functions ..
203      LOGICAL            LSAME
204      INTEGER            ILAENV
205      EXTERNAL           ILAENV, LSAME
206*     ..
207*     .. External Subroutines ..
208      EXTERNAL           SPOTRF, SSYEV, SSYGST, STRMM, STRSM, XERBLA
209*     ..
210*     .. Intrinsic Functions ..
211      INTRINSIC          MAX
212*     ..
213*     .. Executable Statements ..
214*
215*     Test the input parameters.
216*
217      WANTZ = LSAME( JOBZ, 'V' )
218      UPPER = LSAME( UPLO, 'U' )
219      LQUERY = ( LWORK.EQ.-1 )
220*
221      INFO = 0
222      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
223         INFO = -1
224      ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
225         INFO = -2
226      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
227         INFO = -3
228      ELSE IF( N.LT.0 ) THEN
229         INFO = -4
230      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
231         INFO = -6
232      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
233         INFO = -8
234      END IF
235*
236      IF( INFO.EQ.0 ) THEN
237         LWKMIN = MAX( 1, 3*N - 1 )
238         NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 )
239         LWKOPT = MAX( LWKMIN, ( NB + 2 )*N )
240         WORK( 1 ) = LWKOPT
241*
242         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
243            INFO = -11
244         END IF
245      END IF
246*
247      IF( INFO.NE.0 ) THEN
248         CALL XERBLA( 'SSYGV ', -INFO )
249         RETURN
250      ELSE IF( LQUERY ) THEN
251         RETURN
252      END IF
253*
254*     Quick return if possible
255*
256      IF( N.EQ.0 )
257     $   RETURN
258*
259*     Form a Cholesky factorization of B.
260*
261      CALL SPOTRF( UPLO, N, B, LDB, INFO )
262      IF( INFO.NE.0 ) THEN
263         INFO = N + INFO
264         RETURN
265      END IF
266*
267*     Transform problem to standard eigenvalue problem and solve.
268*
269      CALL SSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
270      CALL SSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO )
271*
272      IF( WANTZ ) THEN
273*
274*        Backtransform eigenvectors to the original problem.
275*
276         NEIG = N
277         IF( INFO.GT.0 )
278     $      NEIG = INFO - 1
279         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
280*
281*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
282*           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
283*
284            IF( UPPER ) THEN
285               TRANS = 'N'
286            ELSE
287               TRANS = 'T'
288            END IF
289*
290            CALL STRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE,
291     $                  B, LDB, A, LDA )
292*
293         ELSE IF( ITYPE.EQ.3 ) THEN
294*
295*           For B*A*x=(lambda)*x;
296*           backtransform eigenvectors: x = L*y or U**T*y
297*
298            IF( UPPER ) THEN
299               TRANS = 'T'
300            ELSE
301               TRANS = 'N'
302            END IF
303*
304            CALL STRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE,
305     $                  B, LDB, A, LDA )
306         END IF
307      END IF
308*
309      WORK( 1 ) = LWKOPT
310      RETURN
311*
312*     End of SSYGV
313*
314      END
315