LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
zlaed7.f
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1 *> \brief \b ZLAED7 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 *
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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 *> \date June 2016
243 *
244 *> \ingroup complex16OTHERcomputational
245 *
246 * =====================================================================
247  SUBROUTINE zlaed7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
248  $ ldq, rho, indxq, qstore, qptr, prmptr, perm,
249  $ givptr, givcol, givnum, work, rwork, iwork,
250  $ info )
251 *
252 * -- LAPACK computational routine (version 3.6.1) --
253 * -- LAPACK is a software package provided by Univ. of Tennessee, --
254 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
255 * June 2016
256 *
257 * .. Scalar Arguments ..
258  INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ,
259  $ tlvls
260  DOUBLE PRECISION RHO
261 * ..
262 * .. Array Arguments ..
263  INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
264  $ iwork( * ), perm( * ), prmptr( * ), qptr( * )
265  DOUBLE PRECISION D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * )
266  COMPLEX*16 Q( ldq, * ), WORK( * )
267 * ..
268 *
269 * =====================================================================
270 *
271 * .. Local Scalars ..
272  INTEGER COLTYP, CURR, I, IDLMDA, INDX,
273  $ indxc, indxp, iq, iw, iz, k, n1, n2, ptr
274 * ..
275 * .. External Subroutines ..
276  EXTERNAL dlaed9, dlaeda, dlamrg, xerbla, zlacrm, zlaed8
277 * ..
278 * .. Intrinsic Functions ..
279  INTRINSIC max, min
280 * ..
281 * .. Executable Statements ..
282 *
283 * Test the input parameters.
284 *
285  info = 0
286 *
287 * IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
288 * INFO = -1
289 * ELSE IF( N.LT.0 ) THEN
290  IF( n.LT.0 ) THEN
291  info = -1
292  ELSE IF( min( 1, n ).GT.cutpnt .OR. n.LT.cutpnt ) THEN
293  info = -2
294  ELSE IF( qsiz.LT.n ) THEN
295  info = -3
296  ELSE IF( ldq.LT.max( 1, n ) ) THEN
297  info = -9
298  END IF
299  IF( info.NE.0 ) THEN
300  CALL xerbla( 'ZLAED7', -info )
301  RETURN
302  END IF
303 *
304 * Quick return if possible
305 *
306  IF( n.EQ.0 )
307  $ RETURN
308 *
309 * The following values are for bookkeeping purposes only. They are
310 * integer pointers which indicate the portion of the workspace
311 * used by a particular array in DLAED2 and SLAED3.
312 *
313  iz = 1
314  idlmda = iz + n
315  iw = idlmda + n
316  iq = iw + n
317 *
318  indx = 1
319  indxc = indx + n
320  coltyp = indxc + n
321  indxp = coltyp + n
322 *
323 * Form the z-vector which consists of the last row of Q_1 and the
324 * first row of Q_2.
325 *
326  ptr = 1 + 2**tlvls
327  DO 10 i = 1, curlvl - 1
328  ptr = ptr + 2**( tlvls-i )
329  10 CONTINUE
330  curr = ptr + curpbm
331  CALL dlaeda( n, tlvls, curlvl, curpbm, prmptr, perm, givptr,
332  $ givcol, givnum, qstore, qptr, rwork( iz ),
333  $ rwork( iz+n ), info )
334 *
335 * When solving the final problem, we no longer need the stored data,
336 * so we will overwrite the data from this level onto the previously
337 * used storage space.
338 *
339  IF( curlvl.EQ.tlvls ) THEN
340  qptr( curr ) = 1
341  prmptr( curr ) = 1
342  givptr( curr ) = 1
343  END IF
344 *
345 * Sort and Deflate eigenvalues.
346 *
347  CALL zlaed8( k, n, qsiz, q, ldq, d, rho, cutpnt, rwork( iz ),
348  $ rwork( idlmda ), work, qsiz, rwork( iw ),
349  $ iwork( indxp ), iwork( indx ), indxq,
350  $ perm( prmptr( curr ) ), givptr( curr+1 ),
351  $ givcol( 1, givptr( curr ) ),
352  $ givnum( 1, givptr( curr ) ), info )
353  prmptr( curr+1 ) = prmptr( curr ) + n
354  givptr( curr+1 ) = givptr( curr+1 ) + givptr( curr )
355 *
356 * Solve Secular Equation.
357 *
358  IF( k.NE.0 ) THEN
359  CALL dlaed9( k, 1, k, n, d, rwork( iq ), k, rho,
360  $ rwork( idlmda ), rwork( iw ),
361  $ qstore( qptr( curr ) ), k, info )
362  CALL zlacrm( qsiz, k, work, qsiz, qstore( qptr( curr ) ), k, q,
363  $ ldq, rwork( iq ) )
364  qptr( curr+1 ) = qptr( curr ) + k**2
365  IF( info.NE.0 ) THEN
366  RETURN
367  END IF
368 *
369 * Prepare the INDXQ sorting premutation.
370 *
371  n1 = k
372  n2 = n - k
373  CALL dlamrg( n1, n2, d, 1, -1, indxq )
374  ELSE
375  qptr( curr+1 ) = qptr( curr )
376  DO 20 i = 1, n
377  indxq( i ) = i
378  20 CONTINUE
379  END IF
380 *
381  RETURN
382 *
383 * End of ZLAED7
384 *
385  END
subroutine dlaed9(K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W, S, LDS, INFO)
DLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
Definition: dlaed9.f:158
subroutine zlaed8(K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM, INFO)
ZLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition: zlaed8.f:230
subroutine zlaed7(N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK, INFO)
ZLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a ...
Definition: zlaed7.f:251
subroutine dlamrg(N1, N2, A, DTRD1, DTRD2, INDEX)
DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition: dlamrg.f:101
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dlaeda(N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO)
DLAEDA used by sstedc. Computes the Z vector determining the rank-one modification of the diagonal ma...
Definition: dlaeda.f:168
subroutine zlacrm(M, N, A, LDA, B, LDB, C, LDC, RWORK)
ZLACRM multiplies a complex matrix by a square real matrix.
Definition: zlacrm.f:116