LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
dlaed2.f
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1 *> \brief \b DLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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13 *> [ZIP]</a>
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
22 * Q2, INDX, INDXC, INDXP, COLTYP, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER INFO, K, LDQ, N, N1
26 * DOUBLE PRECISION RHO
27 * ..
28 * .. Array Arguments ..
29 * INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
30 * $ INDXQ( * )
31 * DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
32 * $ W( * ), Z( * )
33 * ..
34 *
35 *
36 *> \par Purpose:
37 * =============
38 *>
39 *> \verbatim
40 *>
41 *> DLAED2 merges the two sets of eigenvalues together into a single
42 *> sorted set. Then it tries to deflate the size of the problem.
43 *> There are two ways in which deflation can occur: when two or more
44 *> eigenvalues are close together or if there is a tiny entry in the
45 *> Z vector. For each such occurrence the order of the related secular
46 *> equation problem is reduced by one.
47 *> \endverbatim
48 *
49 * Arguments:
50 * ==========
51 *
52 *> \param[out] K
53 *> \verbatim
54 *> K is INTEGER
55 *> The number of non-deflated eigenvalues, and the order of the
56 *> related secular equation. 0 <= K <=N.
57 *> \endverbatim
58 *>
59 *> \param[in] N
60 *> \verbatim
61 *> N is INTEGER
62 *> The dimension of the symmetric tridiagonal matrix. N >= 0.
63 *> \endverbatim
64 *>
65 *> \param[in] N1
66 *> \verbatim
67 *> N1 is INTEGER
68 *> The location of the last eigenvalue in the leading sub-matrix.
69 *> min(1,N) <= N1 <= N/2.
70 *> \endverbatim
71 *>
72 *> \param[in,out] D
73 *> \verbatim
74 *> D is DOUBLE PRECISION array, dimension (N)
75 *> On entry, D contains the eigenvalues of the two submatrices to
76 *> be combined.
77 *> On exit, D contains the trailing (N-K) updated eigenvalues
78 *> (those which were deflated) sorted into increasing order.
79 *> \endverbatim
80 *>
81 *> \param[in,out] Q
82 *> \verbatim
83 *> Q is DOUBLE PRECISION array, dimension (LDQ, N)
84 *> On entry, Q contains the eigenvectors of two submatrices in
85 *> the two square blocks with corners at (1,1), (N1,N1)
86 *> and (N1+1, N1+1), (N,N).
87 *> On exit, Q contains the trailing (N-K) updated eigenvectors
88 *> (those which were deflated) in its last N-K columns.
89 *> \endverbatim
90 *>
91 *> \param[in] LDQ
92 *> \verbatim
93 *> LDQ is INTEGER
94 *> The leading dimension of the array Q. LDQ >= max(1,N).
95 *> \endverbatim
96 *>
97 *> \param[in,out] INDXQ
98 *> \verbatim
99 *> INDXQ is INTEGER array, dimension (N)
100 *> The permutation which separately sorts the two sub-problems
101 *> in D into ascending order. Note that elements in the second
102 *> half of this permutation must first have N1 added to their
103 *> values. Destroyed on exit.
104 *> \endverbatim
105 *>
106 *> \param[in,out] RHO
107 *> \verbatim
108 *> RHO is DOUBLE PRECISION
109 *> On entry, the off-diagonal element associated with the rank-1
110 *> cut which originally split the two submatrices which are now
111 *> being recombined.
112 *> On exit, RHO has been modified to the value required by
113 *> DLAED3.
114 *> \endverbatim
115 *>
116 *> \param[in] Z
117 *> \verbatim
118 *> Z is DOUBLE PRECISION array, dimension (N)
119 *> On entry, Z 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).
122 *> On exit, the contents of Z have been destroyed by the updating
123 *> process.
124 *> \endverbatim
125 *>
126 *> \param[out] DLAMDA
127 *> \verbatim
128 *> DLAMDA is DOUBLE PRECISION array, dimension (N)
129 *> A copy of the first K eigenvalues which will be used by
130 *> DLAED3 to form the secular equation.
131 *> \endverbatim
132 *>
133 *> \param[out] W
134 *> \verbatim
135 *> W is DOUBLE PRECISION array, dimension (N)
136 *> The first k values of the final deflation-altered z-vector
137 *> which will be passed to DLAED3.
138 *> \endverbatim
139 *>
140 *> \param[out] Q2
141 *> \verbatim
142 *> Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
143 *> A copy of the first K eigenvectors which will be used by
144 *> DLAED3 in a matrix multiply (DGEMM) to solve for the new
145 *> eigenvectors.
146 *> \endverbatim
147 *>
148 *> \param[out] INDX
149 *> \verbatim
150 *> INDX is INTEGER array, dimension (N)
151 *> The permutation used to sort the contents of DLAMDA into
152 *> ascending order.
153 *> \endverbatim
154 *>
155 *> \param[out] INDXC
156 *> \verbatim
157 *> INDXC is INTEGER array, dimension (N)
158 *> The permutation used to arrange the columns of the deflated
159 *> Q matrix into three groups: the first group contains non-zero
160 *> elements only at and above N1, the second contains
161 *> non-zero elements only below N1, and the third is dense.
162 *> \endverbatim
163 *>
164 *> \param[out] INDXP
165 *> \verbatim
166 *> INDXP is INTEGER array, dimension (N)
167 *> The permutation used to place deflated values of D at the end
168 *> of the array. INDXP(1:K) points to the nondeflated D-values
169 *> and INDXP(K+1:N) points to the deflated eigenvalues.
170 *> \endverbatim
171 *>
172 *> \param[out] COLTYP
173 *> \verbatim
174 *> COLTYP is INTEGER array, dimension (N)
175 *> During execution, a label which will indicate which of the
176 *> following types a column in the Q2 matrix is:
177 *> 1 : non-zero in the upper half only;
178 *> 2 : dense;
179 *> 3 : non-zero in the lower half only;
180 *> 4 : deflated.
181 *> On exit, COLTYP(i) is the number of columns of type i,
182 *> for i=1 to 4 only.
183 *> \endverbatim
184 *>
185 *> \param[out] INFO
186 *> \verbatim
187 *> INFO is INTEGER
188 *> = 0: successful exit.
189 *> < 0: if INFO = -i, the i-th argument had an illegal value.
190 *> \endverbatim
191 *
192 * Authors:
193 * ========
194 *
195 *> \author Univ. of Tennessee
196 *> \author Univ. of California Berkeley
197 *> \author Univ. of Colorado Denver
198 *> \author NAG Ltd.
199 *
200 *> \date September 2012
201 *
202 *> \ingroup auxOTHERcomputational
203 *
204 *> \par Contributors:
205 * ==================
206 *>
207 *> Jeff Rutter, Computer Science Division, University of California
208 *> at Berkeley, USA \n
209 *> Modified by Francoise Tisseur, University of Tennessee
210 *>
211 * =====================================================================
212  SUBROUTINE dlaed2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
213  $ q2, indx, indxc, indxp, coltyp, info )
214 *
215 * -- LAPACK computational routine (version 3.4.2) --
216 * -- LAPACK is a software package provided by Univ. of Tennessee, --
217 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
218 * September 2012
219 *
220 * .. Scalar Arguments ..
221  INTEGER INFO, K, LDQ, N, N1
222  DOUBLE PRECISION RHO
223 * ..
224 * .. Array Arguments ..
225  INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
226  $ indxq( * )
227  DOUBLE PRECISION D( * ), DLAMDA( * ), Q( ldq, * ), Q2( * ),
228  $ w( * ), z( * )
229 * ..
230 *
231 * =====================================================================
232 *
233 * .. Parameters ..
234  DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
235  parameter ( mone = -1.0d0, zero = 0.0d0, one = 1.0d0,
236  $ two = 2.0d0, eight = 8.0d0 )
237 * ..
238 * .. Local Arrays ..
239  INTEGER CTOT( 4 ), PSM( 4 )
240 * ..
241 * .. Local Scalars ..
242  INTEGER CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
243  $ n2, nj, pj
244  DOUBLE PRECISION C, EPS, S, T, TAU, TOL
245 * ..
246 * .. External Functions ..
247  INTEGER IDAMAX
248  DOUBLE PRECISION DLAMCH, DLAPY2
249  EXTERNAL idamax, dlamch, dlapy2
250 * ..
251 * .. External Subroutines ..
252  EXTERNAL dcopy, dlacpy, dlamrg, drot, dscal, xerbla
253 * ..
254 * .. Intrinsic Functions ..
255  INTRINSIC abs, max, min, sqrt
256 * ..
257 * .. Executable Statements ..
258 *
259 * Test the input parameters.
260 *
261  info = 0
262 *
263  IF( n.LT.0 ) THEN
264  info = -2
265  ELSE IF( ldq.LT.max( 1, n ) ) THEN
266  info = -6
267  ELSE IF( min( 1, ( n / 2 ) ).GT.n1 .OR. ( n / 2 ).LT.n1 ) THEN
268  info = -3
269  END IF
270  IF( info.NE.0 ) THEN
271  CALL xerbla( 'DLAED2', -info )
272  RETURN
273  END IF
274 *
275 * Quick return if possible
276 *
277  IF( n.EQ.0 )
278  $ RETURN
279 *
280  n2 = n - n1
281  n1p1 = n1 + 1
282 *
283  IF( rho.LT.zero ) THEN
284  CALL dscal( n2, mone, z( n1p1 ), 1 )
285  END IF
286 *
287 * Normalize z so that norm(z) = 1. Since z is the concatenation of
288 * two normalized vectors, norm2(z) = sqrt(2).
289 *
290  t = one / sqrt( two )
291  CALL dscal( n, t, z, 1 )
292 *
293 * RHO = ABS( norm(z)**2 * RHO )
294 *
295  rho = abs( two*rho )
296 *
297 * Sort the eigenvalues into increasing order
298 *
299  DO 10 i = n1p1, n
300  indxq( i ) = indxq( i ) + n1
301  10 CONTINUE
302 *
303 * re-integrate the deflated parts from the last pass
304 *
305  DO 20 i = 1, n
306  dlamda( i ) = d( indxq( i ) )
307  20 CONTINUE
308  CALL dlamrg( n1, n2, dlamda, 1, 1, indxc )
309  DO 30 i = 1, n
310  indx( i ) = indxq( indxc( i ) )
311  30 CONTINUE
312 *
313 * Calculate the allowable deflation tolerance
314 *
315  imax = idamax( n, z, 1 )
316  jmax = idamax( n, d, 1 )
317  eps = dlamch( 'Epsilon' )
318  tol = eight*eps*max( abs( d( jmax ) ), abs( z( imax ) ) )
319 *
320 * If the rank-1 modifier is small enough, no more needs to be done
321 * except to reorganize Q so that its columns correspond with the
322 * elements in D.
323 *
324  IF( rho*abs( z( imax ) ).LE.tol ) THEN
325  k = 0
326  iq2 = 1
327  DO 40 j = 1, n
328  i = indx( j )
329  CALL dcopy( n, q( 1, i ), 1, q2( iq2 ), 1 )
330  dlamda( j ) = d( i )
331  iq2 = iq2 + n
332  40 CONTINUE
333  CALL dlacpy( 'A', n, n, q2, n, q, ldq )
334  CALL dcopy( n, dlamda, 1, d, 1 )
335  GO TO 190
336  END IF
337 *
338 * If there are multiple eigenvalues then the problem deflates. Here
339 * the number of equal eigenvalues are found. As each equal
340 * eigenvalue is found, an elementary reflector is computed to rotate
341 * the corresponding eigensubspace so that the corresponding
342 * components of Z are zero in this new basis.
343 *
344  DO 50 i = 1, n1
345  coltyp( i ) = 1
346  50 CONTINUE
347  DO 60 i = n1p1, n
348  coltyp( i ) = 3
349  60 CONTINUE
350 *
351 *
352  k = 0
353  k2 = n + 1
354  DO 70 j = 1, n
355  nj = indx( j )
356  IF( rho*abs( z( nj ) ).LE.tol ) THEN
357 *
358 * Deflate due to small z component.
359 *
360  k2 = k2 - 1
361  coltyp( nj ) = 4
362  indxp( k2 ) = nj
363  IF( j.EQ.n )
364  $ GO TO 100
365  ELSE
366  pj = nj
367  GO TO 80
368  END IF
369  70 CONTINUE
370  80 CONTINUE
371  j = j + 1
372  nj = indx( j )
373  IF( j.GT.n )
374  $ GO TO 100
375  IF( rho*abs( z( nj ) ).LE.tol ) THEN
376 *
377 * Deflate due to small z component.
378 *
379  k2 = k2 - 1
380  coltyp( nj ) = 4
381  indxp( k2 ) = nj
382  ELSE
383 *
384 * Check if eigenvalues are close enough to allow deflation.
385 *
386  s = z( pj )
387  c = z( nj )
388 *
389 * Find sqrt(a**2+b**2) without overflow or
390 * destructive underflow.
391 *
392  tau = dlapy2( c, s )
393  t = d( nj ) - d( pj )
394  c = c / tau
395  s = -s / tau
396  IF( abs( t*c*s ).LE.tol ) THEN
397 *
398 * Deflation is possible.
399 *
400  z( nj ) = tau
401  z( pj ) = zero
402  IF( coltyp( nj ).NE.coltyp( pj ) )
403  $ coltyp( nj ) = 2
404  coltyp( pj ) = 4
405  CALL drot( n, q( 1, pj ), 1, q( 1, nj ), 1, c, s )
406  t = d( pj )*c**2 + d( nj )*s**2
407  d( nj ) = d( pj )*s**2 + d( nj )*c**2
408  d( pj ) = t
409  k2 = k2 - 1
410  i = 1
411  90 CONTINUE
412  IF( k2+i.LE.n ) THEN
413  IF( d( pj ).LT.d( indxp( k2+i ) ) ) THEN
414  indxp( k2+i-1 ) = indxp( k2+i )
415  indxp( k2+i ) = pj
416  i = i + 1
417  GO TO 90
418  ELSE
419  indxp( k2+i-1 ) = pj
420  END IF
421  ELSE
422  indxp( k2+i-1 ) = pj
423  END IF
424  pj = nj
425  ELSE
426  k = k + 1
427  dlamda( k ) = d( pj )
428  w( k ) = z( pj )
429  indxp( k ) = pj
430  pj = nj
431  END IF
432  END IF
433  GO TO 80
434  100 CONTINUE
435 *
436 * Record the last eigenvalue.
437 *
438  k = k + 1
439  dlamda( k ) = d( pj )
440  w( k ) = z( pj )
441  indxp( k ) = pj
442 *
443 * Count up the total number of the various types of columns, then
444 * form a permutation which positions the four column types into
445 * four uniform groups (although one or more of these groups may be
446 * empty).
447 *
448  DO 110 j = 1, 4
449  ctot( j ) = 0
450  110 CONTINUE
451  DO 120 j = 1, n
452  ct = coltyp( j )
453  ctot( ct ) = ctot( ct ) + 1
454  120 CONTINUE
455 *
456 * PSM(*) = Position in SubMatrix (of types 1 through 4)
457 *
458  psm( 1 ) = 1
459  psm( 2 ) = 1 + ctot( 1 )
460  psm( 3 ) = psm( 2 ) + ctot( 2 )
461  psm( 4 ) = psm( 3 ) + ctot( 3 )
462  k = n - ctot( 4 )
463 *
464 * Fill out the INDXC array so that the permutation which it induces
465 * will place all type-1 columns first, all type-2 columns next,
466 * then all type-3's, and finally all type-4's.
467 *
468  DO 130 j = 1, n
469  js = indxp( j )
470  ct = coltyp( js )
471  indx( psm( ct ) ) = js
472  indxc( psm( ct ) ) = j
473  psm( ct ) = psm( ct ) + 1
474  130 CONTINUE
475 *
476 * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
477 * and Q2 respectively. The eigenvalues/vectors which were not
478 * deflated go into the first K slots of DLAMDA and Q2 respectively,
479 * while those which were deflated go into the last N - K slots.
480 *
481  i = 1
482  iq1 = 1
483  iq2 = 1 + ( ctot( 1 )+ctot( 2 ) )*n1
484  DO 140 j = 1, ctot( 1 )
485  js = indx( i )
486  CALL dcopy( n1, q( 1, js ), 1, q2( iq1 ), 1 )
487  z( i ) = d( js )
488  i = i + 1
489  iq1 = iq1 + n1
490  140 CONTINUE
491 *
492  DO 150 j = 1, ctot( 2 )
493  js = indx( i )
494  CALL dcopy( n1, q( 1, js ), 1, q2( iq1 ), 1 )
495  CALL dcopy( n2, q( n1+1, js ), 1, q2( iq2 ), 1 )
496  z( i ) = d( js )
497  i = i + 1
498  iq1 = iq1 + n1
499  iq2 = iq2 + n2
500  150 CONTINUE
501 *
502  DO 160 j = 1, ctot( 3 )
503  js = indx( i )
504  CALL dcopy( n2, q( n1+1, js ), 1, q2( iq2 ), 1 )
505  z( i ) = d( js )
506  i = i + 1
507  iq2 = iq2 + n2
508  160 CONTINUE
509 *
510  iq1 = iq2
511  DO 170 j = 1, ctot( 4 )
512  js = indx( i )
513  CALL dcopy( n, q( 1, js ), 1, q2( iq2 ), 1 )
514  iq2 = iq2 + n
515  z( i ) = d( js )
516  i = i + 1
517  170 CONTINUE
518 *
519 * The deflated eigenvalues and their corresponding vectors go back
520 * into the last N - K slots of D and Q respectively.
521 *
522  IF( k.LT.n ) THEN
523  CALL dlacpy( 'A', n, ctot( 4 ), q2( iq1 ), n,
524  $ q( 1, k+1 ), ldq )
525  CALL dcopy( n-k, z( k+1 ), 1, d( k+1 ), 1 )
526  END IF
527 *
528 * Copy CTOT into COLTYP for referencing in DLAED3.
529 *
530  DO 180 j = 1, 4
531  coltyp( j ) = ctot( j )
532  180 CONTINUE
533 *
534  190 CONTINUE
535  RETURN
536 *
537 * End of DLAED2
538 *
539  END
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:53
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:53
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 dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:55
subroutine dlaed2(K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W, Q2, INDX, INDXC, INDXP, COLTYP, INFO)
DLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition: dlaed2.f:214