1*> \brief \b SSPGST
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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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sspgst.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SSPGST( ITYPE, UPLO, N, AP, BP, INFO )
22*
23*       .. Scalar Arguments ..
24*       CHARACTER          UPLO
25*       INTEGER            INFO, ITYPE, N
26*       ..
27*       .. Array Arguments ..
28*       REAL               AP( * ), BP( * )
29*       ..
30*
31*
32*> \par Purpose:
33*  =============
34*>
35*> \verbatim
36*>
37*> SSPGST reduces a real symmetric-definite generalized eigenproblem
38*> to standard form, using packed storage.
39*>
40*> If ITYPE = 1, the problem is A*x = lambda*B*x,
41*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
42*>
43*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
44*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
45*>
46*> B must have been previously factorized as U**T*U or L*L**T by SPPTRF.
47*> \endverbatim
48*
49*  Arguments:
50*  ==========
51*
52*> \param[in] ITYPE
53*> \verbatim
54*>          ITYPE is INTEGER
55*>          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
56*>          = 2 or 3: compute U*A*U**T or L**T*A*L.
57*> \endverbatim
58*>
59*> \param[in] UPLO
60*> \verbatim
61*>          UPLO is CHARACTER*1
62*>          = 'U':  Upper triangle of A is stored and B is factored as
63*>                  U**T*U;
64*>          = 'L':  Lower triangle of A is stored and B is factored as
65*>                  L*L**T.
66*> \endverbatim
67*>
68*> \param[in] N
69*> \verbatim
70*>          N is INTEGER
71*>          The order of the matrices A and B.  N >= 0.
72*> \endverbatim
73*>
74*> \param[in,out] AP
75*> \verbatim
76*>          AP is REAL array, dimension (N*(N+1)/2)
77*>          On entry, the upper or lower triangle of the symmetric matrix
78*>          A, packed columnwise in a linear array.  The j-th column of A
79*>          is stored in the array AP as follows:
80*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
81*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
82*>
83*>          On exit, if INFO = 0, the transformed matrix, stored in the
84*>          same format as A.
85*> \endverbatim
86*>
87*> \param[in] BP
88*> \verbatim
89*>          BP is REAL array, dimension (N*(N+1)/2)
90*>          The triangular factor from the Cholesky factorization of B,
91*>          stored in the same format as A, as returned by SPPTRF.
92*> \endverbatim
93*>
94*> \param[out] INFO
95*> \verbatim
96*>          INFO is INTEGER
97*>          = 0:  successful exit
98*>          < 0:  if INFO = -i, the i-th argument had an illegal value
99*> \endverbatim
100*
101*  Authors:
102*  ========
103*
104*> \author Univ. of Tennessee
105*> \author Univ. of California Berkeley
106*> \author Univ. of Colorado Denver
107*> \author NAG Ltd.
108*
109*> \ingroup realOTHERcomputational
110*
111*  =====================================================================
112      SUBROUTINE SSPGST( ITYPE, UPLO, N, AP, BP, INFO )
113*
114*  -- LAPACK computational routine --
115*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
116*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
117*
118*     .. Scalar Arguments ..
119      CHARACTER          UPLO
120      INTEGER            INFO, ITYPE, N
121*     ..
122*     .. Array Arguments ..
123      REAL               AP( * ), BP( * )
124*     ..
125*
126*  =====================================================================
127*
128*     .. Parameters ..
129      REAL               ONE, HALF
130      PARAMETER          ( ONE = 1.0, HALF = 0.5 )
131*     ..
132*     .. Local Scalars ..
133      LOGICAL            UPPER
134      INTEGER            J, J1, J1J1, JJ, K, K1, K1K1, KK
135      REAL               AJJ, AKK, BJJ, BKK, CT
136*     ..
137*     .. External Subroutines ..
138      EXTERNAL           SAXPY, SSCAL, SSPMV, SSPR2, STPMV, STPSV,
139     $                   XERBLA
140*     ..
141*     .. External Functions ..
142      LOGICAL            LSAME
143      REAL               SDOT
144      EXTERNAL           LSAME, SDOT
145*     ..
146*     .. Executable Statements ..
147*
148*     Test the input parameters.
149*
150      INFO = 0
151      UPPER = LSAME( UPLO, 'U' )
152      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
153         INFO = -1
154      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
155         INFO = -2
156      ELSE IF( N.LT.0 ) THEN
157         INFO = -3
158      END IF
159      IF( INFO.NE.0 ) THEN
160         CALL XERBLA( 'SSPGST', -INFO )
161         RETURN
162      END IF
163*
164      IF( ITYPE.EQ.1 ) THEN
165         IF( UPPER ) THEN
166*
167*           Compute inv(U**T)*A*inv(U)
168*
169*           J1 and JJ are the indices of A(1,j) and A(j,j)
170*
171            JJ = 0
172            DO 10 J = 1, N
173               J1 = JJ + 1
174               JJ = JJ + J
175*
176*              Compute the j-th column of the upper triangle of A
177*
178               BJJ = BP( JJ )
179               CALL STPSV( UPLO, 'Transpose', 'Nonunit', J, BP,
180     $                     AP( J1 ), 1 )
181               CALL SSPMV( UPLO, J-1, -ONE, AP, BP( J1 ), 1, ONE,
182     $                     AP( J1 ), 1 )
183               CALL SSCAL( J-1, ONE / BJJ, AP( J1 ), 1 )
184               AP( JJ ) = ( AP( JJ )-SDOT( J-1, AP( J1 ), 1, BP( J1 ),
185     $                    1 ) ) / BJJ
186   10       CONTINUE
187         ELSE
188*
189*           Compute inv(L)*A*inv(L**T)
190*
191*           KK and K1K1 are the indices of A(k,k) and A(k+1,k+1)
192*
193            KK = 1
194            DO 20 K = 1, N
195               K1K1 = KK + N - K + 1
196*
197*              Update the lower triangle of A(k:n,k:n)
198*
199               AKK = AP( KK )
200               BKK = BP( KK )
201               AKK = AKK / BKK**2
202               AP( KK ) = AKK
203               IF( K.LT.N ) THEN
204                  CALL SSCAL( N-K, ONE / BKK, AP( KK+1 ), 1 )
205                  CT = -HALF*AKK
206                  CALL SAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
207                  CALL SSPR2( UPLO, N-K, -ONE, AP( KK+1 ), 1,
208     $                        BP( KK+1 ), 1, AP( K1K1 ) )
209                  CALL SAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
210                  CALL STPSV( UPLO, 'No transpose', 'Non-unit', N-K,
211     $                        BP( K1K1 ), AP( KK+1 ), 1 )
212               END IF
213               KK = K1K1
214   20       CONTINUE
215         END IF
216      ELSE
217         IF( UPPER ) THEN
218*
219*           Compute U*A*U**T
220*
221*           K1 and KK are the indices of A(1,k) and A(k,k)
222*
223            KK = 0
224            DO 30 K = 1, N
225               K1 = KK + 1
226               KK = KK + K
227*
228*              Update the upper triangle of A(1:k,1:k)
229*
230               AKK = AP( KK )
231               BKK = BP( KK )
232               CALL STPMV( UPLO, 'No transpose', 'Non-unit', K-1, BP,
233     $                     AP( K1 ), 1 )
234               CT = HALF*AKK
235               CALL SAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
236               CALL SSPR2( UPLO, K-1, ONE, AP( K1 ), 1, BP( K1 ), 1,
237     $                     AP )
238               CALL SAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
239               CALL SSCAL( K-1, BKK, AP( K1 ), 1 )
240               AP( KK ) = AKK*BKK**2
241   30       CONTINUE
242         ELSE
243*
244*           Compute L**T *A*L
245*
246*           JJ and J1J1 are the indices of A(j,j) and A(j+1,j+1)
247*
248            JJ = 1
249            DO 40 J = 1, N
250               J1J1 = JJ + N - J + 1
251*
252*              Compute the j-th column of the lower triangle of A
253*
254               AJJ = AP( JJ )
255               BJJ = BP( JJ )
256               AP( JJ ) = AJJ*BJJ + SDOT( N-J, AP( JJ+1 ), 1,
257     $                    BP( JJ+1 ), 1 )
258               CALL SSCAL( N-J, BJJ, AP( JJ+1 ), 1 )
259               CALL SSPMV( UPLO, N-J, ONE, AP( J1J1 ), BP( JJ+1 ), 1,
260     $                     ONE, AP( JJ+1 ), 1 )
261               CALL STPMV( UPLO, 'Transpose', 'Non-unit', N-J+1,
262     $                     BP( JJ ), AP( JJ ), 1 )
263               JJ = J1J1
264   40       CONTINUE
265         END IF
266      END IF
267      RETURN
268*
269*     End of SSPGST
270*
271      END
272