1*> \brief \b SLAED7 used by sstedc. 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
9*> Download SLAED7 + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaed7.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaed7.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaed7.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
22*                          LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
23*                          PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
24*                          INFO )
25*
26*       .. Scalar Arguments ..
27*       INTEGER            CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
28*      $                   QSIZ, TLVLS
29*       REAL               RHO
30*       ..
31*       .. Array Arguments ..
32*       INTEGER            GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
33*      $                   IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
34*       REAL               D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
35*      $                   QSTORE( * ), WORK( * )
36*       ..
37*
38*
39*> \par Purpose:
40*  =============
41*>
42*> \verbatim
43*>
44*> SLAED7 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 symmetric matrix
48*> that has been reduced to tridiagonal form.  SLAED1 handles
49*> the case in which all eigenvalues and eigenvectors of a symmetric
50*> tridiagonal matrix are desired.
51*>
52*>   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
53*>
54*>    where Z = Q**Tu, u is a vector of length N with ones in the
55*>    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
56*>
57*>    The eigenvectors of the original matrix are stored in Q, and the
58*>    eigenvalues are in D.  The algorithm consists of three stages:
59*>
60*>       The first stage consists of deflating the size of the problem
61*>       when there are multiple eigenvalues or if there is a zero in
62*>       the Z vector.  For each such occurrence the dimension of the
63*>       secular equation problem is reduced by one.  This stage is
64*>       performed by the routine SLAED8.
65*>
66*>       The second stage consists of calculating the updated
67*>       eigenvalues. This is done by finding the roots of the secular
68*>       equation via the routine SLAED4 (as called by SLAED9).
69*>       This routine also calculates the eigenvectors of the current
70*>       problem.
71*>
72*>       The final stage consists of computing the updated eigenvectors
73*>       directly using the updated eigenvalues.  The eigenvectors for
74*>       the current problem are multiplied with the eigenvectors from
75*>       the overall problem.
76*> \endverbatim
77*
78*  Arguments:
79*  ==========
80*
81*> \param[in] ICOMPQ
82*> \verbatim
83*>          ICOMPQ is INTEGER
84*>          = 0:  Compute eigenvalues only.
85*>          = 1:  Compute eigenvectors of original dense symmetric matrix
86*>                also.  On entry, Q contains the orthogonal matrix used
87*>                to reduce the original matrix to tridiagonal form.
88*> \endverbatim
89*>
90*> \param[in] N
91*> \verbatim
92*>          N is INTEGER
93*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
94*> \endverbatim
95*>
96*> \param[in] QSIZ
97*> \verbatim
98*>          QSIZ is INTEGER
99*>         The dimension of the orthogonal matrix used to reduce
100*>         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
101*> \endverbatim
102*>
103*> \param[in] TLVLS
104*> \verbatim
105*>          TLVLS is INTEGER
106*>         The total number of merging levels in the overall divide and
107*>         conquer tree.
108*> \endverbatim
109*>
110*> \param[in] CURLVL
111*> \verbatim
112*>          CURLVL is INTEGER
113*>         The current level in the overall merge routine,
114*>         0 <= CURLVL <= TLVLS.
115*> \endverbatim
116*>
117*> \param[in] CURPBM
118*> \verbatim
119*>          CURPBM is INTEGER
120*>         The current problem in the current level in the overall
121*>         merge routine (counting from upper left to lower right).
122*> \endverbatim
123*>
124*> \param[in,out] D
125*> \verbatim
126*>          D is REAL array, dimension (N)
127*>         On entry, the eigenvalues of the rank-1-perturbed matrix.
128*>         On exit, the eigenvalues of the repaired matrix.
129*> \endverbatim
130*>
131*> \param[in,out] Q
132*> \verbatim
133*>          Q is REAL array, dimension (LDQ, N)
134*>         On entry, the eigenvectors of the rank-1-perturbed matrix.
135*>         On exit, the eigenvectors of the repaired tridiagonal matrix.
136*> \endverbatim
137*>
138*> \param[in] LDQ
139*> \verbatim
140*>          LDQ is INTEGER
141*>         The leading dimension of the array Q.  LDQ >= max(1,N).
142*> \endverbatim
143*>
144*> \param[out] INDXQ
145*> \verbatim
146*>          INDXQ is INTEGER array, dimension (N)
147*>         The permutation which will reintegrate the subproblem just
148*>         solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
149*>         will be in ascending order.
150*> \endverbatim
151*>
152*> \param[in] RHO
153*> \verbatim
154*>          RHO is REAL
155*>         The subdiagonal element used to create the rank-1
156*>         modification.
157*> \endverbatim
158*>
159*> \param[in] CUTPNT
160*> \verbatim
161*>          CUTPNT is INTEGER
162*>         Contains the location of the last eigenvalue in the leading
163*>         sub-matrix.  min(1,N) <= CUTPNT <= N.
164*> \endverbatim
165*>
166*> \param[in,out] QSTORE
167*> \verbatim
168*>          QSTORE is REAL array, dimension (N**2+1)
169*>         Stores eigenvectors of submatrices encountered during
170*>         divide and conquer, packed together. QPTR points to
171*>         beginning of the submatrices.
172*> \endverbatim
173*>
174*> \param[in,out] QPTR
175*> \verbatim
176*>          QPTR is INTEGER array, dimension (N+2)
177*>         List of indices pointing to beginning of submatrices stored
178*>         in QSTORE. The submatrices are numbered starting at the
179*>         bottom left of the divide and conquer tree, from left to
180*>         right and bottom to top.
181*> \endverbatim
182*>
183*> \param[in] PRMPTR
184*> \verbatim
185*>          PRMPTR is INTEGER array, dimension (N lg N)
186*>         Contains a list of pointers which indicate where in PERM a
187*>         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
188*>         indicates the size of the permutation and also the size of
189*>         the full, non-deflated problem.
190*> \endverbatim
191*>
192*> \param[in] PERM
193*> \verbatim
194*>          PERM is INTEGER array, dimension (N lg N)
195*>         Contains the permutations (from deflation and sorting) to be
196*>         applied to each eigenblock.
197*> \endverbatim
198*>
199*> \param[in] GIVPTR
200*> \verbatim
201*>          GIVPTR is INTEGER array, dimension (N lg N)
202*>         Contains a list of pointers which indicate where in GIVCOL a
203*>         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
204*>         indicates the number of Givens rotations.
205*> \endverbatim
206*>
207*> \param[in] GIVCOL
208*> \verbatim
209*>          GIVCOL is INTEGER array, dimension (2, N lg N)
210*>         Each pair of numbers indicates a pair of columns to take place
211*>         in a Givens rotation.
212*> \endverbatim
213*>
214*> \param[in] GIVNUM
215*> \verbatim
216*>          GIVNUM is REAL array, dimension (2, N lg N)
217*>         Each number indicates the S value to be used in the
218*>         corresponding Givens rotation.
219*> \endverbatim
220*>
221*> \param[out] WORK
222*> \verbatim
223*>          WORK is REAL array, dimension (3*N+2*QSIZ*N)
224*> \endverbatim
225*>
226*> \param[out] IWORK
227*> \verbatim
228*>          IWORK is INTEGER array, dimension (4*N)
229*> \endverbatim
230*>
231*> \param[out] INFO
232*> \verbatim
233*>          INFO is INTEGER
234*>          = 0:  successful exit.
235*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
236*>          > 0:  if INFO = 1, an eigenvalue did not converge
237*> \endverbatim
238*
239*  Authors:
240*  ========
241*
242*> \author Univ. of Tennessee
243*> \author Univ. of California Berkeley
244*> \author Univ. of Colorado Denver
245*> \author NAG Ltd.
246*
247*> \date June 2016
248*
249*> \ingroup auxOTHERcomputational
250*
251*> \par Contributors:
252*  ==================
253*>
254*> Jeff Rutter, Computer Science Division, University of California
255*> at Berkeley, USA
256*
257*  =====================================================================
258      SUBROUTINE SLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
259     $                   LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
260     $                   PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
261     $                   INFO )
262*
263*  -- LAPACK computational routine (version 3.7.0) --
264*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
265*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
266*     June 2016
267*
268*     .. Scalar Arguments ..
269      INTEGER            CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
270     $                   QSIZ, TLVLS
271      REAL               RHO
272*     ..
273*     .. Array Arguments ..
274      INTEGER            GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
275     $                   IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
276      REAL               D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
277     $                   QSTORE( * ), WORK( * )
278*     ..
279*
280*  =====================================================================
281*
282*     .. Parameters ..
283      REAL               ONE, ZERO
284      PARAMETER          ( ONE = 1.0E0, ZERO = 0.0E0 )
285*     ..
286*     .. Local Scalars ..
287      INTEGER            COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP,
288     $                   IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR
289*     ..
290*     .. External Subroutines ..
291      EXTERNAL           SGEMM, SLAED8, SLAED9, SLAEDA, SLAMRG, XERBLA
292*     ..
293*     .. Intrinsic Functions ..
294      INTRINSIC          MAX, MIN
295*     ..
296*     .. Executable Statements ..
297*
298*     Test the input parameters.
299*
300      INFO = 0
301*
302      IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
303         INFO = -1
304      ELSE IF( N.LT.0 ) THEN
305         INFO = -2
306      ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
307         INFO = -3
308      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
309         INFO = -9
310      ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN
311         INFO = -12
312      END IF
313      IF( INFO.NE.0 ) THEN
314         CALL XERBLA( 'SLAED7', -INFO )
315         RETURN
316      END IF
317*
318*     Quick return if possible
319*
320      IF( N.EQ.0 )
321     $   RETURN
322*
323*     The following values are for bookkeeping purposes only.  They are
324*     integer pointers which indicate the portion of the workspace
325*     used by a particular array in SLAED8 and SLAED9.
326*
327      IF( ICOMPQ.EQ.1 ) THEN
328         LDQ2 = QSIZ
329      ELSE
330         LDQ2 = N
331      END IF
332*
333      IZ = 1
334      IDLMDA = IZ + N
335      IW = IDLMDA + N
336      IQ2 = IW + N
337      IS = IQ2 + N*LDQ2
338*
339      INDX = 1
340      INDXC = INDX + N
341      COLTYP = INDXC + N
342      INDXP = COLTYP + N
343*
344*     Form the z-vector which consists of the last row of Q_1 and the
345*     first row of Q_2.
346*
347      PTR = 1 + 2**TLVLS
348      DO 10 I = 1, CURLVL - 1
349         PTR = PTR + 2**( TLVLS-I )
350   10 CONTINUE
351      CURR = PTR + CURPBM
352      CALL SLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
353     $             GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ),
354     $             WORK( IZ+N ), INFO )
355*
356*     When solving the final problem, we no longer need the stored data,
357*     so we will overwrite the data from this level onto the previously
358*     used storage space.
359*
360      IF( CURLVL.EQ.TLVLS ) THEN
361         QPTR( CURR ) = 1
362         PRMPTR( CURR ) = 1
363         GIVPTR( CURR ) = 1
364      END IF
365*
366*     Sort and Deflate eigenvalues.
367*
368      CALL SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
369     $             WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2,
370     $             WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ),
371     $             GIVCOL( 1, GIVPTR( CURR ) ),
372     $             GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ),
373     $             IWORK( INDX ), INFO )
374      PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N
375      GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR )
376*
377*     Solve Secular Equation.
378*
379      IF( K.NE.0 ) THEN
380         CALL SLAED9( K, 1, K, N, D, WORK( IS ), K, RHO, WORK( IDLMDA ),
381     $                WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO )
382         IF( INFO.NE.0 )
383     $      GO TO 30
384         IF( ICOMPQ.EQ.1 ) THEN
385            CALL SGEMM( 'N', 'N', QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2,
386     $                  QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ )
387         END IF
388         QPTR( CURR+1 ) = QPTR( CURR ) + K**2
389*
390*     Prepare the INDXQ sorting permutation.
391*
392         N1 = K
393         N2 = N - K
394         CALL SLAMRG( N1, N2, D, 1, -1, INDXQ )
395      ELSE
396         QPTR( CURR+1 ) = QPTR( CURR )
397         DO 20 I = 1, N
398            INDXQ( I ) = I
399   20    CONTINUE
400      END IF
401*
402   30 CONTINUE
403      RETURN
404*
405*     End of SLAED7
406*
407      END
408