LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
slaed7.f
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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
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11 *> [TGZ]</a>
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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 *> \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 slaed7( 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  REAL RHO
269 * ..
270 * .. Array Arguments ..
271  INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
272  $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
273  REAL D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
274  $ qstore( * ), work( * )
275 * ..
276 *
277 * =====================================================================
278 *
279 * .. Parameters ..
280  REAL ONE, ZERO
281  PARAMETER ( ONE = 1.0e0, zero = 0.0e0 )
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 sgemm, slaed8, slaed9, slaeda, slamrg, 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( 'SLAED7', -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 SLAED8 and SLAED9.
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 slaeda( 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 slaed8( 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 slaed9( 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 sgemm( '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 slamrg( 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 SLAED7
403 *
404  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 slaed8(ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL, GIVNUM, INDXP, INDX, INFO)
SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition: slaed8.f:243
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 slaed7(ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, INFO)
SLAED7 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a ...
Definition: slaed7.f:260
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187