LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
claed7.f
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1 *> \brief \b CLAED7 used by CSTEDC. 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 CLAED7 + 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/claed7.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/claed7.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CLAED7( 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 * REAL RHO
30 * ..
31 * .. Array Arguments ..
32 * INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
33 * $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
34 * REAL D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * )
35 * COMPLEX Q( LDQ, * ), WORK( * )
36 * ..
37 *
38 *
39 *> \par Purpose:
40 * =============
41 *>
42 *> \verbatim
43 *>
44 *> CLAED7 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 SLAED2.
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 SLAED4 (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 REAL 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 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 REAL
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 REAL array,
163 *> dimension (3*N+2*QSIZ*N)
164 *> \endverbatim
165 *>
166 *> \param[out] WORK
167 *> \verbatim
168 *> WORK is COMPLEX array, dimension (QSIZ*N)
169 *> \endverbatim
170 *>
171 *> \param[in,out] QSTORE
172 *> \verbatim
173 *> QSTORE is REAL 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 REAL 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 complexOTHERcomputational
243 *
244 * =====================================================================
245  SUBROUTINE claed7( 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  REAL RHO
258 * ..
259 * .. Array Arguments ..
260  INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
261  $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
262  REAL D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * )
263  COMPLEX 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 clacrm, claed8, slaed9, slaeda, slamrg, xerbla
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( 'CLAED7', -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 SLAED2 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 slaeda( 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 claed8( 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 slaed9( k, 1, k, n, d, rwork( iq ), k, rho,
357  $ rwork( idlmda ), rwork( iw ),
358  $ qstore( qptr( curr ) ), k, info )
359  CALL clacrm( 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 slamrg( 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 CLAED7
381 *
382  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slamrg(N1, N2, A, STRD1, STRD2, INDEX)
SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition: slamrg.f:99
subroutine slaed9(K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W, S, LDS, INFO)
SLAED9 used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors....
Definition: slaed9.f:156
subroutine slaeda(N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO)
SLAEDA used by SSTEDC. Computes the Z vector determining the rank-one modification of the diagonal ma...
Definition: slaeda.f:166
subroutine clacrm(M, N, A, LDA, B, LDB, C, LDC, RWORK)
CLACRM multiplies a complex matrix by a square real matrix.
Definition: clacrm.f:114
subroutine claed7(N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK, INFO)
CLAED7 used by CSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a ...
Definition: claed7.f:249
subroutine claed8(K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM, INFO)
CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition: claed8.f:228