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zlaed8.f
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1 *> \brief \b ZLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. 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 ZLAED8 + 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/zlaed8.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
22 * Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
23 * GIVCOL, GIVNUM, INFO )
24 *
25 * .. Scalar Arguments ..
26 * INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
27 * DOUBLE PRECISION RHO
28 * ..
29 * .. Array Arguments ..
30 * INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
31 * $ INDXQ( * ), PERM( * )
32 * DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
33 * $ Z( * )
34 * COMPLEX*16 Q( LDQ, * ), Q2( LDQ2, * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> ZLAED8 merges the two sets of eigenvalues together into a single
44 *> sorted set. Then it tries to deflate the size of the problem.
45 *> There are two ways in which deflation can occur: when two or more
46 *> eigenvalues are close together or if there is a tiny element in the
47 *> Z vector. For each such occurrence the order of the related secular
48 *> equation problem is reduced by one.
49 *> \endverbatim
50 *
51 * Arguments:
52 * ==========
53 *
54 *> \param[out] K
55 *> \verbatim
56 *> K is INTEGER
57 *> Contains the number of non-deflated eigenvalues.
58 *> This is the order of the related secular equation.
59 *> \endverbatim
60 *>
61 *> \param[in] N
62 *> \verbatim
63 *> N is INTEGER
64 *> The dimension of the symmetric tridiagonal matrix. N >= 0.
65 *> \endverbatim
66 *>
67 *> \param[in] QSIZ
68 *> \verbatim
69 *> QSIZ is INTEGER
70 *> The dimension of the unitary matrix used to reduce
71 *> the dense or band matrix to tridiagonal form.
72 *> QSIZ >= N if ICOMPQ = 1.
73 *> \endverbatim
74 *>
75 *> \param[in,out] Q
76 *> \verbatim
77 *> Q is COMPLEX*16 array, dimension (LDQ,N)
78 *> On entry, Q contains the eigenvectors of the partially solved
79 *> system which has been previously updated in matrix
80 *> multiplies with other partially solved eigensystems.
81 *> On exit, Q contains the trailing (N-K) updated eigenvectors
82 *> (those which were deflated) in its last N-K columns.
83 *> \endverbatim
84 *>
85 *> \param[in] LDQ
86 *> \verbatim
87 *> LDQ is INTEGER
88 *> The leading dimension of the array Q. LDQ >= max( 1, N ).
89 *> \endverbatim
90 *>
91 *> \param[in,out] D
92 *> \verbatim
93 *> D is DOUBLE PRECISION array, dimension (N)
94 *> On entry, D contains the eigenvalues of the two submatrices to
95 *> be combined. On exit, D contains the trailing (N-K) updated
96 *> eigenvalues (those which were deflated) sorted into increasing
97 *> order.
98 *> \endverbatim
99 *>
100 *> \param[in,out] RHO
101 *> \verbatim
102 *> RHO is DOUBLE PRECISION
103 *> Contains the off diagonal element associated with the rank-1
104 *> cut which originally split the two submatrices which are now
105 *> being recombined. RHO is modified during the computation to
106 *> the value required by DLAED3.
107 *> \endverbatim
108 *>
109 *> \param[in] CUTPNT
110 *> \verbatim
111 *> CUTPNT is INTEGER
112 *> Contains the location of the last eigenvalue in the leading
113 *> sub-matrix. MIN(1,N) <= CUTPNT <= N.
114 *> \endverbatim
115 *>
116 *> \param[in] Z
117 *> \verbatim
118 *> Z is DOUBLE PRECISION array, dimension (N)
119 *> On input this vector contains the updating vector (the last
120 *> row of the first sub-eigenvector matrix and the first row of
121 *> the second sub-eigenvector matrix). The contents of Z are
122 *> destroyed during the updating process.
123 *> \endverbatim
124 *>
125 *> \param[out] DLAMDA
126 *> \verbatim
127 *> DLAMDA is DOUBLE PRECISION array, dimension (N)
128 *> Contains a copy of the first K eigenvalues which will be used
129 *> by DLAED3 to form the secular equation.
130 *> \endverbatim
131 *>
132 *> \param[out] Q2
133 *> \verbatim
134 *> Q2 is COMPLEX*16 array, dimension (LDQ2,N)
135 *> If ICOMPQ = 0, Q2 is not referenced. Otherwise,
136 *> Contains a copy of the first K eigenvectors which will be used
137 *> by DLAED7 in a matrix multiply (DGEMM) to update the new
138 *> eigenvectors.
139 *> \endverbatim
140 *>
141 *> \param[in] LDQ2
142 *> \verbatim
143 *> LDQ2 is INTEGER
144 *> The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
145 *> \endverbatim
146 *>
147 *> \param[out] W
148 *> \verbatim
149 *> W is DOUBLE PRECISION array, dimension (N)
150 *> This will hold the first k values of the final
151 *> deflation-altered z-vector and will be passed to DLAED3.
152 *> \endverbatim
153 *>
154 *> \param[out] INDXP
155 *> \verbatim
156 *> INDXP is INTEGER array, dimension (N)
157 *> This will contain the permutation used to place deflated
158 *> values of D at the end of the array. On output INDXP(1:K)
159 *> points to the nondeflated D-values and INDXP(K+1:N)
160 *> points to the deflated eigenvalues.
161 *> \endverbatim
162 *>
163 *> \param[out] INDX
164 *> \verbatim
165 *> INDX is INTEGER array, dimension (N)
166 *> This will contain the permutation used to sort the contents of
167 *> D into ascending order.
168 *> \endverbatim
169 *>
170 *> \param[in] INDXQ
171 *> \verbatim
172 *> INDXQ is INTEGER array, dimension (N)
173 *> This contains the permutation which separately sorts the two
174 *> sub-problems in D into ascending order. Note that elements in
175 *> the second half of this permutation must first have CUTPNT
176 *> added to their values in order to be accurate.
177 *> \endverbatim
178 *>
179 *> \param[out] PERM
180 *> \verbatim
181 *> PERM is INTEGER array, dimension (N)
182 *> Contains the permutations (from deflation and sorting) to be
183 *> applied to each eigenblock.
184 *> \endverbatim
185 *>
186 *> \param[out] GIVPTR
187 *> \verbatim
188 *> GIVPTR is INTEGER
189 *> Contains the number of Givens rotations which took place in
190 *> this subproblem.
191 *> \endverbatim
192 *>
193 *> \param[out] GIVCOL
194 *> \verbatim
195 *> GIVCOL is INTEGER array, dimension (2, N)
196 *> Each pair of numbers indicates a pair of columns to take place
197 *> in a Givens rotation.
198 *> \endverbatim
199 *>
200 *> \param[out] GIVNUM
201 *> \verbatim
202 *> GIVNUM is DOUBLE PRECISION array, dimension (2, N)
203 *> Each number indicates the S value to be used in the
204 *> corresponding Givens rotation.
205 *> \endverbatim
206 *>
207 *> \param[out] INFO
208 *> \verbatim
209 *> INFO is INTEGER
210 *> = 0: successful exit.
211 *> < 0: if INFO = -i, the i-th argument had an illegal value.
212 *> \endverbatim
213 *
214 * Authors:
215 * ========
216 *
217 *> \author Univ. of Tennessee
218 *> \author Univ. of California Berkeley
219 *> \author Univ. of Colorado Denver
220 *> \author NAG Ltd.
221 *
222 *> \date September 2012
223 *
224 *> \ingroup complex16OTHERcomputational
225 *
226 * =====================================================================
227  SUBROUTINE zlaed8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
228  $ q2, ldq2, w, indxp, indx, indxq, perm, givptr,
229  $ givcol, givnum, info )
230 *
231 * -- LAPACK computational routine (version 3.4.2) --
232 * -- LAPACK is a software package provided by Univ. of Tennessee, --
233 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
234 * September 2012
235 *
236 * .. Scalar Arguments ..
237  INTEGER cutpnt, givptr, info, k, ldq, ldq2, n, qsiz
238  DOUBLE PRECISION rho
239 * ..
240 * .. Array Arguments ..
241  INTEGER givcol( 2, * ), indx( * ), indxp( * ),
242  $ indxq( * ), perm( * )
243  DOUBLE PRECISION d( * ), dlamda( * ), givnum( 2, * ), w( * ),
244  $ z( * )
245  COMPLEX*16 q( ldq, * ), q2( ldq2, * )
246 * ..
247 *
248 * =====================================================================
249 *
250 * .. Parameters ..
251  DOUBLE PRECISION mone, zero, one, two, eight
252  parameter( mone = -1.0d0, zero = 0.0d0, one = 1.0d0,
253  $ two = 2.0d0, eight = 8.0d0 )
254 * ..
255 * .. Local Scalars ..
256  INTEGER i, imax, j, jlam, jmax, jp, k2, n1, n1p1, n2
257  DOUBLE PRECISION c, eps, s, t, tau, tol
258 * ..
259 * .. External Functions ..
260  INTEGER idamax
261  DOUBLE PRECISION dlamch, dlapy2
262  EXTERNAL idamax, dlamch, dlapy2
263 * ..
264 * .. External Subroutines ..
265  EXTERNAL dcopy, dlamrg, dscal, xerbla, zcopy, zdrot,
266  $ zlacpy
267 * ..
268 * .. Intrinsic Functions ..
269  INTRINSIC abs, max, min, sqrt
270 * ..
271 * .. Executable Statements ..
272 *
273 * Test the input parameters.
274 *
275  info = 0
276 *
277  IF( n.LT.0 ) THEN
278  info = -2
279  ELSE IF( qsiz.LT.n ) THEN
280  info = -3
281  ELSE IF( ldq.LT.max( 1, n ) ) THEN
282  info = -5
283  ELSE IF( cutpnt.LT.min( 1, n ) .OR. cutpnt.GT.n ) THEN
284  info = -8
285  ELSE IF( ldq2.LT.max( 1, n ) ) THEN
286  info = -12
287  END IF
288  IF( info.NE.0 ) THEN
289  CALL xerbla( 'ZLAED8', -info )
290  return
291  END IF
292 *
293 * Need to initialize GIVPTR to O here in case of quick exit
294 * to prevent an unspecified code behavior (usually sigfault)
295 * when IWORK array on entry to *stedc is not zeroed
296 * (or at least some IWORK entries which used in *laed7 for GIVPTR).
297 *
298  givptr = 0
299 *
300 * Quick return if possible
301 *
302  IF( n.EQ.0 )
303  $ return
304 *
305  n1 = cutpnt
306  n2 = n - n1
307  n1p1 = n1 + 1
308 *
309  IF( rho.LT.zero ) THEN
310  CALL dscal( n2, mone, z( n1p1 ), 1 )
311  END IF
312 *
313 * Normalize z so that norm(z) = 1
314 *
315  t = one / sqrt( two )
316  DO 10 j = 1, n
317  indx( j ) = j
318  10 continue
319  CALL dscal( n, t, z, 1 )
320  rho = abs( two*rho )
321 *
322 * Sort the eigenvalues into increasing order
323 *
324  DO 20 i = cutpnt + 1, n
325  indxq( i ) = indxq( i ) + cutpnt
326  20 continue
327  DO 30 i = 1, n
328  dlamda( i ) = d( indxq( i ) )
329  w( i ) = z( indxq( i ) )
330  30 continue
331  i = 1
332  j = cutpnt + 1
333  CALL dlamrg( n1, n2, dlamda, 1, 1, indx )
334  DO 40 i = 1, n
335  d( i ) = dlamda( indx( i ) )
336  z( i ) = w( indx( i ) )
337  40 continue
338 *
339 * Calculate the allowable deflation tolerance
340 *
341  imax = idamax( n, z, 1 )
342  jmax = idamax( n, d, 1 )
343  eps = dlamch( 'Epsilon' )
344  tol = eight*eps*abs( d( jmax ) )
345 *
346 * If the rank-1 modifier is small enough, no more needs to be done
347 * -- except to reorganize Q so that its columns correspond with the
348 * elements in D.
349 *
350  IF( rho*abs( z( imax ) ).LE.tol ) THEN
351  k = 0
352  DO 50 j = 1, n
353  perm( j ) = indxq( indx( j ) )
354  CALL zcopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
355  50 continue
356  CALL zlacpy( 'A', qsiz, n, q2( 1, 1 ), ldq2, q( 1, 1 ), ldq )
357  return
358  END IF
359 *
360 * If there are multiple eigenvalues then the problem deflates. Here
361 * the number of equal eigenvalues are found. As each equal
362 * eigenvalue is found, an elementary reflector is computed to rotate
363 * the corresponding eigensubspace so that the corresponding
364 * components of Z are zero in this new basis.
365 *
366  k = 0
367  k2 = n + 1
368  DO 60 j = 1, n
369  IF( rho*abs( z( j ) ).LE.tol ) THEN
370 *
371 * Deflate due to small z component.
372 *
373  k2 = k2 - 1
374  indxp( k2 ) = j
375  IF( j.EQ.n )
376  $ go to 100
377  ELSE
378  jlam = j
379  go to 70
380  END IF
381  60 continue
382  70 continue
383  j = j + 1
384  IF( j.GT.n )
385  $ go to 90
386  IF( rho*abs( z( j ) ).LE.tol ) THEN
387 *
388 * Deflate due to small z component.
389 *
390  k2 = k2 - 1
391  indxp( k2 ) = j
392  ELSE
393 *
394 * Check if eigenvalues are close enough to allow deflation.
395 *
396  s = z( jlam )
397  c = z( j )
398 *
399 * Find sqrt(a**2+b**2) without overflow or
400 * destructive underflow.
401 *
402  tau = dlapy2( c, s )
403  t = d( j ) - d( jlam )
404  c = c / tau
405  s = -s / tau
406  IF( abs( t*c*s ).LE.tol ) THEN
407 *
408 * Deflation is possible.
409 *
410  z( j ) = tau
411  z( jlam ) = zero
412 *
413 * Record the appropriate Givens rotation
414 *
415  givptr = givptr + 1
416  givcol( 1, givptr ) = indxq( indx( jlam ) )
417  givcol( 2, givptr ) = indxq( indx( j ) )
418  givnum( 1, givptr ) = c
419  givnum( 2, givptr ) = s
420  CALL zdrot( qsiz, q( 1, indxq( indx( jlam ) ) ), 1,
421  $ q( 1, indxq( indx( j ) ) ), 1, c, s )
422  t = d( jlam )*c*c + d( j )*s*s
423  d( j ) = d( jlam )*s*s + d( j )*c*c
424  d( jlam ) = t
425  k2 = k2 - 1
426  i = 1
427  80 continue
428  IF( k2+i.LE.n ) THEN
429  IF( d( jlam ).LT.d( indxp( k2+i ) ) ) THEN
430  indxp( k2+i-1 ) = indxp( k2+i )
431  indxp( k2+i ) = jlam
432  i = i + 1
433  go to 80
434  ELSE
435  indxp( k2+i-1 ) = jlam
436  END IF
437  ELSE
438  indxp( k2+i-1 ) = jlam
439  END IF
440  jlam = j
441  ELSE
442  k = k + 1
443  w( k ) = z( jlam )
444  dlamda( k ) = d( jlam )
445  indxp( k ) = jlam
446  jlam = j
447  END IF
448  END IF
449  go to 70
450  90 continue
451 *
452 * Record the last eigenvalue.
453 *
454  k = k + 1
455  w( k ) = z( jlam )
456  dlamda( k ) = d( jlam )
457  indxp( k ) = jlam
458 *
459  100 continue
460 *
461 * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
462 * and Q2 respectively. The eigenvalues/vectors which were not
463 * deflated go into the first K slots of DLAMDA and Q2 respectively,
464 * while those which were deflated go into the last N - K slots.
465 *
466  DO 110 j = 1, n
467  jp = indxp( j )
468  dlamda( j ) = d( jp )
469  perm( j ) = indxq( indx( jp ) )
470  CALL zcopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
471  110 continue
472 *
473 * The deflated eigenvalues and their corresponding vectors go back
474 * into the last N - K slots of D and Q respectively.
475 *
476  IF( k.LT.n ) THEN
477  CALL dcopy( n-k, dlamda( k+1 ), 1, d( k+1 ), 1 )
478  CALL zlacpy( 'A', qsiz, n-k, q2( 1, k+1 ), ldq2, q( 1, k+1 ),
479  $ ldq )
480  END IF
481 *
482  return
483 *
484 * End of ZLAED8
485 *
486  END