1*> \brief \b ZLAED7 used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
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/zlaed7.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
22*                          LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM,
23*                          GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK,
24*                          INFO )
25*
26*       .. Scalar Arguments ..
27*       INTEGER            CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ,
28*      $                   TLVLS
29*       DOUBLE PRECISION   RHO
30*       ..
31*       .. Array Arguments ..
32*       INTEGER            GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
33*      $                   IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
34*       DOUBLE PRECISION   D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * )
35*       COMPLEX*16         Q( LDQ, * ), WORK( * )
36*       ..
37*
38*
39*> \par Purpose:
40*  =============
41*>
42*> \verbatim
43*>
44*> ZLAED7 computes the updated eigensystem of a diagonal
45*> matrix after modification by a rank-one symmetric matrix. This
46*> routine is used only for the eigenproblem which requires all
47*> eigenvalues and optionally eigenvectors of a dense or banded
48*> Hermitian matrix that has been reduced to tridiagonal form.
49*>
50*>   T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)
51*>
52*>   where Z = Q**Hu, u is a vector of length N with ones in the
53*>   CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
54*>
55*>    The eigenvectors of the original matrix are stored in Q, and the
56*>    eigenvalues are in D.  The algorithm consists of three stages:
57*>
58*>       The first stage consists of deflating the size of the problem
59*>       when there are multiple eigenvalues or if there is a zero in
60*>       the Z vector.  For each such occurrence the dimension of the
61*>       secular equation problem is reduced by one.  This stage is
62*>       performed by the routine DLAED2.
63*>
64*>       The second stage consists of calculating the updated
65*>       eigenvalues. This is done by finding the roots of the secular
66*>       equation via the routine DLAED4 (as called by SLAED3).
67*>       This routine also calculates the eigenvectors of the current
68*>       problem.
69*>
70*>       The final stage consists of computing the updated eigenvectors
71*>       directly using the updated eigenvalues.  The eigenvectors for
72*>       the current problem are multiplied with the eigenvectors from
73*>       the overall problem.
74*> \endverbatim
75*
76*  Arguments:
77*  ==========
78*
79*> \param[in] N
80*> \verbatim
81*>          N is INTEGER
82*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
83*> \endverbatim
84*>
85*> \param[in] CUTPNT
86*> \verbatim
87*>          CUTPNT is INTEGER
88*>         Contains the location of the last eigenvalue in the leading
89*>         sub-matrix.  min(1,N) <= CUTPNT <= N.
90*> \endverbatim
91*>
92*> \param[in] QSIZ
93*> \verbatim
94*>          QSIZ is INTEGER
95*>         The dimension of the unitary matrix used to reduce
96*>         the full matrix to tridiagonal form.  QSIZ >= N.
97*> \endverbatim
98*>
99*> \param[in] TLVLS
100*> \verbatim
101*>          TLVLS is INTEGER
102*>         The total number of merging levels in the overall divide and
103*>         conquer tree.
104*> \endverbatim
105*>
106*> \param[in] CURLVL
107*> \verbatim
108*>          CURLVL is INTEGER
109*>         The current level in the overall merge routine,
110*>         0 <= curlvl <= tlvls.
111*> \endverbatim
112*>
113*> \param[in] CURPBM
114*> \verbatim
115*>          CURPBM is INTEGER
116*>         The current problem in the current level in the overall
117*>         merge routine (counting from upper left to lower right).
118*> \endverbatim
119*>
120*> \param[in,out] D
121*> \verbatim
122*>          D is DOUBLE PRECISION array, dimension (N)
123*>         On entry, the eigenvalues of the rank-1-perturbed matrix.
124*>         On exit, the eigenvalues of the repaired matrix.
125*> \endverbatim
126*>
127*> \param[in,out] Q
128*> \verbatim
129*>          Q is COMPLEX*16 array, dimension (LDQ,N)
130*>         On entry, the eigenvectors of the rank-1-perturbed matrix.
131*>         On exit, the eigenvectors of the repaired tridiagonal matrix.
132*> \endverbatim
133*>
134*> \param[in] LDQ
135*> \verbatim
136*>          LDQ is INTEGER
137*>         The leading dimension of the array Q.  LDQ >= max(1,N).
138*> \endverbatim
139*>
140*> \param[in] RHO
141*> \verbatim
142*>          RHO is DOUBLE PRECISION
143*>         Contains the subdiagonal element used to create the rank-1
144*>         modification.
145*> \endverbatim
146*>
147*> \param[out] INDXQ
148*> \verbatim
149*>          INDXQ is INTEGER array, dimension (N)
150*>         This contains the permutation which will reintegrate the
151*>         subproblem just solved back into sorted order,
152*>         ie. D( INDXQ( I = 1, N ) ) will be in ascending order.
153*> \endverbatim
154*>
155*> \param[out] IWORK
156*> \verbatim
157*>          IWORK is INTEGER array, dimension (4*N)
158*> \endverbatim
159*>
160*> \param[out] RWORK
161*> \verbatim
162*>          RWORK is DOUBLE PRECISION array,
163*>                                 dimension (3*N+2*QSIZ*N)
164*> \endverbatim
165*>
166*> \param[out] WORK
167*> \verbatim
168*>          WORK is COMPLEX*16 array, dimension (QSIZ*N)
169*> \endverbatim
170*>
171*> \param[in,out] QSTORE
172*> \verbatim
173*>          QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
174*>         Stores eigenvectors of submatrices encountered during
175*>         divide and conquer, packed together. QPTR points to
176*>         beginning of the submatrices.
177*> \endverbatim
178*>
179*> \param[in,out] QPTR
180*> \verbatim
181*>          QPTR is INTEGER array, dimension (N+2)
182*>         List of indices pointing to beginning of submatrices stored
183*>         in QSTORE. The submatrices are numbered starting at the
184*>         bottom left of the divide and conquer tree, from left to
185*>         right and bottom to top.
186*> \endverbatim
187*>
188*> \param[in] PRMPTR
189*> \verbatim
190*>          PRMPTR is INTEGER array, dimension (N lg N)
191*>         Contains a list of pointers which indicate where in PERM a
192*>         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
193*>         indicates the size of the permutation and also the size of
194*>         the full, non-deflated problem.
195*> \endverbatim
196*>
197*> \param[in] PERM
198*> \verbatim
199*>          PERM is INTEGER array, dimension (N lg N)
200*>         Contains the permutations (from deflation and sorting) to be
201*>         applied to each eigenblock.
202*> \endverbatim
203*>
204*> \param[in] GIVPTR
205*> \verbatim
206*>          GIVPTR is INTEGER array, dimension (N lg N)
207*>         Contains a list of pointers which indicate where in GIVCOL a
208*>         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
209*>         indicates the number of Givens rotations.
210*> \endverbatim
211*>
212*> \param[in] GIVCOL
213*> \verbatim
214*>          GIVCOL is INTEGER array, dimension (2, N lg N)
215*>         Each pair of numbers indicates a pair of columns to take place
216*>         in a Givens rotation.
217*> \endverbatim
218*>
219*> \param[in] GIVNUM
220*> \verbatim
221*>          GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
222*>         Each number indicates the S value to be used in the
223*>         corresponding Givens rotation.
224*> \endverbatim
225*>
226*> \param[out] INFO
227*> \verbatim
228*>          INFO is INTEGER
229*>          = 0:  successful exit.
230*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
231*>          > 0:  if INFO = 1, an eigenvalue did not converge
232*> \endverbatim
233*
234*  Authors:
235*  ========
236*
237*> \author Univ. of Tennessee
238*> \author Univ. of California Berkeley
239*> \author Univ. of Colorado Denver
240*> \author NAG Ltd.
241*
242*> \ingroup complex16OTHERcomputational
243*
244*  =====================================================================
245      SUBROUTINE ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
246     $                   LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM,
247     $                   GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK,
248     $                   INFO )
249*
250*  -- LAPACK computational routine --
251*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
252*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
253*
254*     .. Scalar Arguments ..
255      INTEGER            CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ,
256     $                   TLVLS
257      DOUBLE PRECISION   RHO
258*     ..
259*     .. Array Arguments ..
260      INTEGER            GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
261     $                   IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
262      DOUBLE PRECISION   D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * )
263      COMPLEX*16         Q( LDQ, * ), WORK( * )
264*     ..
265*
266*  =====================================================================
267*
268*     .. Local Scalars ..
269      INTEGER            COLTYP, CURR, I, IDLMDA, INDX,
270     $                   INDXC, INDXP, IQ, IW, IZ, K, N1, N2, PTR
271*     ..
272*     .. External Subroutines ..
273      EXTERNAL           DLAED9, DLAEDA, DLAMRG, XERBLA, ZLACRM, ZLAED8
274*     ..
275*     .. Intrinsic Functions ..
276      INTRINSIC          MAX, MIN
277*     ..
278*     .. Executable Statements ..
279*
280*     Test the input parameters.
281*
282      INFO = 0
283*
284*     IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
285*        INFO = -1
286*     ELSE IF( N.LT.0 ) THEN
287      IF( N.LT.0 ) THEN
288         INFO = -1
289      ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN
290         INFO = -2
291      ELSE IF( QSIZ.LT.N ) THEN
292         INFO = -3
293      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
294         INFO = -9
295      END IF
296      IF( INFO.NE.0 ) THEN
297         CALL XERBLA( 'ZLAED7', -INFO )
298         RETURN
299      END IF
300*
301*     Quick return if possible
302*
303      IF( N.EQ.0 )
304     $   RETURN
305*
306*     The following values are for bookkeeping purposes only.  They are
307*     integer pointers which indicate the portion of the workspace
308*     used by a particular array in DLAED2 and SLAED3.
309*
310      IZ = 1
311      IDLMDA = IZ + N
312      IW = IDLMDA + N
313      IQ = IW + N
314*
315      INDX = 1
316      INDXC = INDX + N
317      COLTYP = INDXC + N
318      INDXP = COLTYP + N
319*
320*     Form the z-vector which consists of the last row of Q_1 and the
321*     first row of Q_2.
322*
323      PTR = 1 + 2**TLVLS
324      DO 10 I = 1, CURLVL - 1
325         PTR = PTR + 2**( TLVLS-I )
326   10 CONTINUE
327      CURR = PTR + CURPBM
328      CALL DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
329     $             GIVCOL, GIVNUM, QSTORE, QPTR, RWORK( IZ ),
330     $             RWORK( IZ+N ), INFO )
331*
332*     When solving the final problem, we no longer need the stored data,
333*     so we will overwrite the data from this level onto the previously
334*     used storage space.
335*
336      IF( CURLVL.EQ.TLVLS ) THEN
337         QPTR( CURR ) = 1
338         PRMPTR( CURR ) = 1
339         GIVPTR( CURR ) = 1
340      END IF
341*
342*     Sort and Deflate eigenvalues.
343*
344      CALL ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, RWORK( IZ ),
345     $             RWORK( IDLMDA ), WORK, QSIZ, RWORK( IW ),
346     $             IWORK( INDXP ), IWORK( INDX ), INDXQ,
347     $             PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ),
348     $             GIVCOL( 1, GIVPTR( CURR ) ),
349     $             GIVNUM( 1, GIVPTR( CURR ) ), INFO )
350      PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N
351      GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR )
352*
353*     Solve Secular Equation.
354*
355      IF( K.NE.0 ) THEN
356         CALL DLAED9( K, 1, K, N, D, RWORK( IQ ), K, RHO,
357     $                RWORK( IDLMDA ), RWORK( IW ),
358     $                QSTORE( QPTR( CURR ) ), K, INFO )
359         CALL ZLACRM( QSIZ, K, WORK, QSIZ, QSTORE( QPTR( CURR ) ), K, Q,
360     $                LDQ, RWORK( IQ ) )
361         QPTR( CURR+1 ) = QPTR( CURR ) + K**2
362         IF( INFO.NE.0 ) THEN
363            RETURN
364         END IF
365*
366*     Prepare the INDXQ sorting premutation.
367*
368         N1 = K
369         N2 = N - K
370         CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
371      ELSE
372         QPTR( CURR+1 ) = QPTR( CURR )
373         DO 20 I = 1, N
374            INDXQ( I ) = I
375   20    CONTINUE
376      END IF
377*
378      RETURN
379*
380*     End of ZLAED7
381*
382      END
383