1*> \brief \b DLAED7 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 dense.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download DLAED7 + dependencies
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11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed7.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed7.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DLAED7( 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*       DOUBLE PRECISION   RHO
30*       ..
31*       .. Array Arguments ..
32*       INTEGER            GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
33*      $                   IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
34*       DOUBLE PRECISION   D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
35*      $                   QSTORE( * ), WORK( * )
36*       ..
37*
38*
39*> \par Purpose:
40*  =============
41*>
42*> \verbatim
43*>
44*> DLAED7 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.  DLAED1 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 DLAED8.
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 DLAED4 (as called by DLAED9).
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*> \ingroup auxOTHERcomputational
248*
249*> \par Contributors:
250*  ==================
251*>
252*> Jeff Rutter, Computer Science Division, University of California
253*> at Berkeley, USA
254*
255*  =====================================================================
256      SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
257     $                   LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
258     $                   PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
259     $                   INFO )
260*
261*  -- LAPACK computational routine --
262*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
263*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
264*
265*     .. Scalar Arguments ..
266      INTEGER            CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
267     $                   QSIZ, TLVLS
268      DOUBLE PRECISION   RHO
269*     ..
270*     .. Array Arguments ..
271      INTEGER            GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
272     $                   IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
273      DOUBLE PRECISION   D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
274     $                   QSTORE( * ), WORK( * )
275*     ..
276*
277*  =====================================================================
278*
279*     .. Parameters ..
280      DOUBLE PRECISION   ONE, ZERO
281      PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0 )
282*     ..
283*     .. Local Scalars ..
284      INTEGER            COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP,
285     $                   IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR
286*     ..
287*     .. External Subroutines ..
288      EXTERNAL           DGEMM, DLAED8, DLAED9, DLAEDA, DLAMRG, XERBLA
289*     ..
290*     .. Intrinsic Functions ..
291      INTRINSIC          MAX, MIN
292*     ..
293*     .. Executable Statements ..
294*
295*     Test the input parameters.
296*
297      INFO = 0
298*
299      IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
300         INFO = -1
301      ELSE IF( N.LT.0 ) THEN
302         INFO = -2
303      ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
304         INFO = -3
305      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
306         INFO = -9
307      ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN
308         INFO = -12
309      END IF
310      IF( INFO.NE.0 ) THEN
311         CALL XERBLA( 'DLAED7', -INFO )
312         RETURN
313      END IF
314*
315*     Quick return if possible
316*
317      IF( N.EQ.0 )
318     $   RETURN
319*
320*     The following values are for bookkeeping purposes only.  They are
321*     integer pointers which indicate the portion of the workspace
322*     used by a particular array in DLAED8 and DLAED9.
323*
324      IF( ICOMPQ.EQ.1 ) THEN
325         LDQ2 = QSIZ
326      ELSE
327         LDQ2 = N
328      END IF
329*
330      IZ = 1
331      IDLMDA = IZ + N
332      IW = IDLMDA + N
333      IQ2 = IW + N
334      IS = IQ2 + N*LDQ2
335*
336      INDX = 1
337      INDXC = INDX + N
338      COLTYP = INDXC + N
339      INDXP = COLTYP + N
340*
341*     Form the z-vector which consists of the last row of Q_1 and the
342*     first row of Q_2.
343*
344      PTR = 1 + 2**TLVLS
345      DO 10 I = 1, CURLVL - 1
346         PTR = PTR + 2**( TLVLS-I )
347   10 CONTINUE
348      CURR = PTR + CURPBM
349      CALL DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
350     $             GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ),
351     $             WORK( IZ+N ), INFO )
352*
353*     When solving the final problem, we no longer need the stored data,
354*     so we will overwrite the data from this level onto the previously
355*     used storage space.
356*
357      IF( CURLVL.EQ.TLVLS ) THEN
358         QPTR( CURR ) = 1
359         PRMPTR( CURR ) = 1
360         GIVPTR( CURR ) = 1
361      END IF
362*
363*     Sort and Deflate eigenvalues.
364*
365      CALL DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
366     $             WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2,
367     $             WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ),
368     $             GIVCOL( 1, GIVPTR( CURR ) ),
369     $             GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ),
370     $             IWORK( INDX ), INFO )
371      PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N
372      GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR )
373*
374*     Solve Secular Equation.
375*
376      IF( K.NE.0 ) THEN
377         CALL DLAED9( K, 1, K, N, D, WORK( IS ), K, RHO, WORK( IDLMDA ),
378     $                WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO )
379         IF( INFO.NE.0 )
380     $      GO TO 30
381         IF( ICOMPQ.EQ.1 ) THEN
382            CALL DGEMM( 'N', 'N', QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2,
383     $                  QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ )
384         END IF
385         QPTR( CURR+1 ) = QPTR( CURR ) + K**2
386*
387*     Prepare the INDXQ sorting permutation.
388*
389         N1 = K
390         N2 = N - K
391         CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
392      ELSE
393         QPTR( CURR+1 ) = QPTR( CURR )
394         DO 20 I = 1, N
395            INDXQ( I ) = I
396   20    CONTINUE
397      END IF
398*
399   30 CONTINUE
400      RETURN
401*
402*     End of DLAED7
403*
404      END
405