1*> \brief \b DLAED1 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
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/dlaed1.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
22*                          INFO )
23*
24*       .. Scalar Arguments ..
25*       INTEGER            CUTPNT, INFO, LDQ, N
26*       DOUBLE PRECISION   RHO
27*       ..
28*       .. Array Arguments ..
29*       INTEGER            INDXQ( * ), IWORK( * )
30*       DOUBLE PRECISION   D( * ), Q( LDQ, * ), WORK( * )
31*       ..
32*
33*
34*> \par Purpose:
35*  =============
36*>
37*> \verbatim
38*>
39*> DLAED1 computes the updated eigensystem of a diagonal
40*> matrix after modification by a rank-one symmetric matrix.  This
41*> routine is used only for the eigenproblem which requires all
42*> eigenvalues and eigenvectors of a tridiagonal matrix.  DLAED7 handles
43*> the case in which eigenvalues only or eigenvalues and eigenvectors
44*> of a full symmetric matrix (which was reduced to tridiagonal form)
45*> are desired.
46*>
47*>   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
48*>
49*>    where Z = Q**T*u, u is a vector of length N with ones in the
50*>    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
51*>
52*>    The eigenvectors of the original matrix are stored in Q, and the
53*>    eigenvalues are in D.  The algorithm consists of three stages:
54*>
55*>       The first stage consists of deflating the size of the problem
56*>       when there are multiple eigenvalues or if there is a zero in
57*>       the Z vector.  For each such occurrence the dimension of the
58*>       secular equation problem is reduced by one.  This stage is
59*>       performed by the routine DLAED2.
60*>
61*>       The second stage consists of calculating the updated
62*>       eigenvalues. This is done by finding the roots of the secular
63*>       equation via the routine DLAED4 (as called by DLAED3).
64*>       This routine also calculates the eigenvectors of the current
65*>       problem.
66*>
67*>       The final stage consists of computing the updated eigenvectors
68*>       directly using the updated eigenvalues.  The eigenvectors for
69*>       the current problem are multiplied with the eigenvectors from
70*>       the overall problem.
71*> \endverbatim
72*
73*  Arguments:
74*  ==========
75*
76*> \param[in] N
77*> \verbatim
78*>          N is INTEGER
79*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
80*> \endverbatim
81*>
82*> \param[in,out] D
83*> \verbatim
84*>          D is DOUBLE PRECISION array, dimension (N)
85*>         On entry, the eigenvalues of the rank-1-perturbed matrix.
86*>         On exit, the eigenvalues of the repaired matrix.
87*> \endverbatim
88*>
89*> \param[in,out] Q
90*> \verbatim
91*>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
92*>         On entry, the eigenvectors of the rank-1-perturbed matrix.
93*>         On exit, the eigenvectors of the repaired tridiagonal matrix.
94*> \endverbatim
95*>
96*> \param[in] LDQ
97*> \verbatim
98*>          LDQ is INTEGER
99*>         The leading dimension of the array Q.  LDQ >= max(1,N).
100*> \endverbatim
101*>
102*> \param[in,out] INDXQ
103*> \verbatim
104*>          INDXQ is INTEGER array, dimension (N)
105*>         On entry, the permutation which separately sorts the two
106*>         subproblems in D into ascending order.
107*>         On exit, the permutation which will reintegrate the
108*>         subproblems back into sorted order,
109*>         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
110*> \endverbatim
111*>
112*> \param[in] RHO
113*> \verbatim
114*>          RHO is DOUBLE PRECISION
115*>         The subdiagonal entry used to create the rank-1 modification.
116*> \endverbatim
117*>
118*> \param[in] CUTPNT
119*> \verbatim
120*>          CUTPNT is INTEGER
121*>         The location of the last eigenvalue in the leading sub-matrix.
122*>         min(1,N) <= CUTPNT <= N/2.
123*> \endverbatim
124*>
125*> \param[out] WORK
126*> \verbatim
127*>          WORK is DOUBLE PRECISION array, dimension (4*N + N**2)
128*> \endverbatim
129*>
130*> \param[out] IWORK
131*> \verbatim
132*>          IWORK is INTEGER array, dimension (4*N)
133*> \endverbatim
134*>
135*> \param[out] INFO
136*> \verbatim
137*>          INFO is INTEGER
138*>          = 0:  successful exit.
139*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
140*>          > 0:  if INFO = 1, an eigenvalue did not converge
141*> \endverbatim
142*
143*  Authors:
144*  ========
145*
146*> \author Univ. of Tennessee
147*> \author Univ. of California Berkeley
148*> \author Univ. of Colorado Denver
149*> \author NAG Ltd.
150*
151*> \ingroup auxOTHERcomputational
152*
153*> \par Contributors:
154*  ==================
155*>
156*> Jeff Rutter, Computer Science Division, University of California
157*> at Berkeley, USA \n
158*>  Modified by Francoise Tisseur, University of Tennessee
159*>
160*  =====================================================================
161      SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
162     $                   INFO )
163*
164*  -- LAPACK computational routine --
165*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
166*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
167*
168*     .. Scalar Arguments ..
169      INTEGER            CUTPNT, INFO, LDQ, N
170      DOUBLE PRECISION   RHO
171*     ..
172*     .. Array Arguments ..
173      INTEGER            INDXQ( * ), IWORK( * )
174      DOUBLE PRECISION   D( * ), Q( LDQ, * ), WORK( * )
175*     ..
176*
177*  =====================================================================
178*
179*     .. Local Scalars ..
180      INTEGER            COLTYP, I, IDLMDA, INDX, INDXC, INDXP, IQ2, IS,
181     $                   IW, IZ, K, N1, N2, ZPP1
182*     ..
183*     .. External Subroutines ..
184      EXTERNAL           DCOPY, DLAED2, DLAED3, DLAMRG, XERBLA
185*     ..
186*     .. Intrinsic Functions ..
187      INTRINSIC          MAX, MIN
188*     ..
189*     .. Executable Statements ..
190*
191*     Test the input parameters.
192*
193      INFO = 0
194*
195      IF( N.LT.0 ) THEN
196         INFO = -1
197      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
198         INFO = -4
199      ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
200         INFO = -7
201      END IF
202      IF( INFO.NE.0 ) THEN
203         CALL XERBLA( 'DLAED1', -INFO )
204         RETURN
205      END IF
206*
207*     Quick return if possible
208*
209      IF( N.EQ.0 )
210     $   RETURN
211*
212*     The following values are integer pointers which indicate
213*     the portion of the workspace
214*     used by a particular array in DLAED2 and DLAED3.
215*
216      IZ = 1
217      IDLMDA = IZ + N
218      IW = IDLMDA + N
219      IQ2 = IW + N
220*
221      INDX = 1
222      INDXC = INDX + N
223      COLTYP = INDXC + N
224      INDXP = COLTYP + N
225*
226*
227*     Form the z-vector which consists of the last row of Q_1 and the
228*     first row of Q_2.
229*
230      CALL DCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
231      ZPP1 = CUTPNT + 1
232      CALL DCOPY( N-CUTPNT, Q( ZPP1, ZPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 )
233*
234*     Deflate eigenvalues.
235*
236      CALL DLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
237     $             WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
238     $             IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
239     $             IWORK( COLTYP ), INFO )
240*
241      IF( INFO.NE.0 )
242     $   GO TO 20
243*
244*     Solve Secular Equation.
245*
246      IF( K.NE.0 ) THEN
247         IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
248     $        ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
249         CALL DLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
250     $                WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
251     $                WORK( IW ), WORK( IS ), INFO )
252         IF( INFO.NE.0 )
253     $      GO TO 20
254*
255*     Prepare the INDXQ sorting permutation.
256*
257         N1 = K
258         N2 = N - K
259         CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
260      ELSE
261         DO 10 I = 1, N
262            INDXQ( I ) = I
263   10    CONTINUE
264      END IF
265*
266   20 CONTINUE
267      RETURN
268*
269*     End of DLAED1
270*
271      END
272