LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
slaed8.f
Go to the documentation of this file.
1 *> \brief \b SLAED8 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 SLAED8 + 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/slaed8.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaed8.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
22 * CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
23 * GIVCOL, GIVNUM, INDXP, INDX, INFO )
24 *
25 * .. Scalar Arguments ..
26 * INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
27 * $ QSIZ
28 * REAL RHO
29 * ..
30 * .. Array Arguments ..
31 * INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
32 * $ INDXQ( * ), PERM( * )
33 * REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ),
34 * $ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> SLAED8 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[in] ICOMPQ
55 *> \verbatim
56 *> ICOMPQ is INTEGER
57 *> = 0: Compute eigenvalues only.
58 *> = 1: Compute eigenvectors of original dense symmetric matrix
59 *> also. On entry, Q contains the orthogonal matrix used
60 *> to reduce the original matrix to tridiagonal form.
61 *> \endverbatim
62 *>
63 *> \param[out] K
64 *> \verbatim
65 *> K is INTEGER
66 *> The number of non-deflated eigenvalues, and the order of the
67 *> related secular equation.
68 *> \endverbatim
69 *>
70 *> \param[in] N
71 *> \verbatim
72 *> N is INTEGER
73 *> The dimension of the symmetric tridiagonal matrix. N >= 0.
74 *> \endverbatim
75 *>
76 *> \param[in] QSIZ
77 *> \verbatim
78 *> QSIZ is INTEGER
79 *> The dimension of the orthogonal matrix used to reduce
80 *> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
81 *> \endverbatim
82 *>
83 *> \param[in,out] D
84 *> \verbatim
85 *> D is REAL array, dimension (N)
86 *> On entry, the eigenvalues of the two submatrices to be
87 *> combined. On exit, the trailing (N-K) updated eigenvalues
88 *> (those which were deflated) sorted into increasing order.
89 *> \endverbatim
90 *>
91 *> \param[in,out] Q
92 *> \verbatim
93 *> Q is REAL array, dimension (LDQ,N)
94 *> If ICOMPQ = 0, Q is not referenced. Otherwise,
95 *> on entry, Q contains the eigenvectors of the partially solved
96 *> system which has been previously updated in matrix
97 *> multiplies with other partially solved eigensystems.
98 *> On exit, Q contains the trailing (N-K) updated eigenvectors
99 *> (those which were deflated) in its last N-K columns.
100 *> \endverbatim
101 *>
102 *> \param[in] LDQ
103 *> \verbatim
104 *> LDQ is INTEGER
105 *> The leading dimension of the array Q. LDQ >= max(1,N).
106 *> \endverbatim
107 *>
108 *> \param[in] INDXQ
109 *> \verbatim
110 *> INDXQ is INTEGER array, dimension (N)
111 *> The permutation which separately sorts the two sub-problems
112 *> in D into ascending order. Note that elements in the second
113 *> half of this permutation must first have CUTPNT added to
114 *> their values in order to be accurate.
115 *> \endverbatim
116 *>
117 *> \param[in,out] RHO
118 *> \verbatim
119 *> RHO is REAL
120 *> On entry, the off-diagonal element associated with the rank-1
121 *> cut which originally split the two submatrices which are now
122 *> being recombined.
123 *> On exit, RHO has been modified to the value required by
124 *> SLAED3.
125 *> \endverbatim
126 *>
127 *> \param[in] CUTPNT
128 *> \verbatim
129 *> CUTPNT is INTEGER
130 *> The location of the last eigenvalue in the leading
131 *> sub-matrix. min(1,N) <= CUTPNT <= N.
132 *> \endverbatim
133 *>
134 *> \param[in] Z
135 *> \verbatim
136 *> Z is REAL array, dimension (N)
137 *> On entry, Z contains the updating vector (the last row of
138 *> the first sub-eigenvector matrix and the first row of the
139 *> second sub-eigenvector matrix).
140 *> On exit, the contents of Z are destroyed by the updating
141 *> process.
142 *> \endverbatim
143 *>
144 *> \param[out] DLAMDA
145 *> \verbatim
146 *> DLAMDA is REAL array, dimension (N)
147 *> A copy of the first K eigenvalues which will be used by
148 *> SLAED3 to form the secular equation.
149 *> \endverbatim
150 *>
151 *> \param[out] Q2
152 *> \verbatim
153 *> Q2 is REAL array, dimension (LDQ2,N)
154 *> If ICOMPQ = 0, Q2 is not referenced. Otherwise,
155 *> a copy of the first K eigenvectors which will be used by
156 *> SLAED7 in a matrix multiply (SGEMM) to update the new
157 *> eigenvectors.
158 *> \endverbatim
159 *>
160 *> \param[in] LDQ2
161 *> \verbatim
162 *> LDQ2 is INTEGER
163 *> The leading dimension of the array Q2. LDQ2 >= max(1,N).
164 *> \endverbatim
165 *>
166 *> \param[out] W
167 *> \verbatim
168 *> W is REAL array, dimension (N)
169 *> The first k values of the final deflation-altered z-vector and
170 *> will be passed to SLAED3.
171 *> \endverbatim
172 *>
173 *> \param[out] PERM
174 *> \verbatim
175 *> PERM is INTEGER array, dimension (N)
176 *> The permutations (from deflation and sorting) to be applied
177 *> to each eigenblock.
178 *> \endverbatim
179 *>
180 *> \param[out] GIVPTR
181 *> \verbatim
182 *> GIVPTR is INTEGER
183 *> The number of Givens rotations which took place in this
184 *> subproblem.
185 *> \endverbatim
186 *>
187 *> \param[out] GIVCOL
188 *> \verbatim
189 *> GIVCOL is INTEGER array, dimension (2, N)
190 *> Each pair of numbers indicates a pair of columns to take place
191 *> in a Givens rotation.
192 *> \endverbatim
193 *>
194 *> \param[out] GIVNUM
195 *> \verbatim
196 *> GIVNUM is REAL array, dimension (2, N)
197 *> Each number indicates the S value to be used in the
198 *> corresponding Givens rotation.
199 *> \endverbatim
200 *>
201 *> \param[out] INDXP
202 *> \verbatim
203 *> INDXP is INTEGER array, dimension (N)
204 *> The permutation used to place deflated values of D at the end
205 *> of the array. INDXP(1:K) points to the nondeflated D-values
206 *> and INDXP(K+1:N) points to the deflated eigenvalues.
207 *> \endverbatim
208 *>
209 *> \param[out] INDX
210 *> \verbatim
211 *> INDX is INTEGER array, dimension (N)
212 *> The permutation used to sort the contents of D into ascending
213 *> order.
214 *> \endverbatim
215 *>
216 *> \param[out] INFO
217 *> \verbatim
218 *> INFO is INTEGER
219 *> = 0: successful exit.
220 *> < 0: if INFO = -i, the i-th argument had an illegal value.
221 *> \endverbatim
222 *
223 * Authors:
224 * ========
225 *
226 *> \author Univ. of Tennessee
227 *> \author Univ. of California Berkeley
228 *> \author Univ. of Colorado Denver
229 *> \author NAG Ltd.
230 *
231 *> \date December 2016
232 *
233 *> \ingroup auxOTHERcomputational
234 *
235 *> \par Contributors:
236 * ==================
237 *>
238 *> Jeff Rutter, Computer Science Division, University of California
239 *> at Berkeley, USA
240 *
241 * =====================================================================
242  SUBROUTINE slaed8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
243  $ CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
244  $ GIVCOL, GIVNUM, INDXP, INDX, INFO )
245 *
246 * -- LAPACK computational routine (version 3.7.0) --
247 * -- LAPACK is a software package provided by Univ. of Tennessee, --
248 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
249 * December 2016
250 *
251 * .. Scalar Arguments ..
252  INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
253  $ qsiz
254  REAL RHO
255 * ..
256 * .. Array Arguments ..
257  INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
258  $ indxq( * ), perm( * )
259  REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ),
260  $ q( ldq, * ), q2( ldq2, * ), w( * ), z( * )
261 * ..
262 *
263 * =====================================================================
264 *
265 * .. Parameters ..
266  REAL MONE, ZERO, ONE, TWO, EIGHT
267  parameter( mone = -1.0e0, zero = 0.0e0, one = 1.0e0,
268  $ two = 2.0e0, eight = 8.0e0 )
269 * ..
270 * .. Local Scalars ..
271 *
272  INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
273  REAL C, EPS, S, T, TAU, TOL
274 * ..
275 * .. External Functions ..
276  INTEGER ISAMAX
277  REAL SLAMCH, SLAPY2
278  EXTERNAL isamax, slamch, slapy2
279 * ..
280 * .. External Subroutines ..
281  EXTERNAL scopy, slacpy, slamrg, srot, sscal, xerbla
282 * ..
283 * .. Intrinsic Functions ..
284  INTRINSIC abs, max, min, sqrt
285 * ..
286 * .. Executable Statements ..
287 *
288 * Test the input parameters.
289 *
290  info = 0
291 *
292  IF( icompq.LT.0 .OR. icompq.GT.1 ) THEN
293  info = -1
294  ELSE IF( n.LT.0 ) THEN
295  info = -3
296  ELSE IF( icompq.EQ.1 .AND. qsiz.LT.n ) THEN
297  info = -4
298  ELSE IF( ldq.LT.max( 1, n ) ) THEN
299  info = -7
300  ELSE IF( cutpnt.LT.min( 1, n ) .OR. cutpnt.GT.n ) THEN
301  info = -10
302  ELSE IF( ldq2.LT.max( 1, n ) ) THEN
303  info = -14
304  END IF
305  IF( info.NE.0 ) THEN
306  CALL xerbla( 'SLAED8', -info )
307  RETURN
308  END IF
309 *
310 * Need to initialize GIVPTR to O here in case of quick exit
311 * to prevent an unspecified code behavior (usually sigfault)
312 * when IWORK array on entry to *stedc is not zeroed
313 * (or at least some IWORK entries which used in *laed7 for GIVPTR).
314 *
315  givptr = 0
316 *
317 * Quick return if possible
318 *
319  IF( n.EQ.0 )
320  $ RETURN
321 *
322  n1 = cutpnt
323  n2 = n - n1
324  n1p1 = n1 + 1
325 *
326  IF( rho.LT.zero ) THEN
327  CALL sscal( n2, mone, z( n1p1 ), 1 )
328  END IF
329 *
330 * Normalize z so that norm(z) = 1
331 *
332  t = one / sqrt( two )
333  DO 10 j = 1, n
334  indx( j ) = j
335  10 CONTINUE
336  CALL sscal( n, t, z, 1 )
337  rho = abs( two*rho )
338 *
339 * Sort the eigenvalues into increasing order
340 *
341  DO 20 i = cutpnt + 1, n
342  indxq( i ) = indxq( i ) + cutpnt
343  20 CONTINUE
344  DO 30 i = 1, n
345  dlamda( i ) = d( indxq( i ) )
346  w( i ) = z( indxq( i ) )
347  30 CONTINUE
348  i = 1
349  j = cutpnt + 1
350  CALL slamrg( n1, n2, dlamda, 1, 1, indx )
351  DO 40 i = 1, n
352  d( i ) = dlamda( indx( i ) )
353  z( i ) = w( indx( i ) )
354  40 CONTINUE
355 *
356 * Calculate the allowable deflation tolerence
357 *
358  imax = isamax( n, z, 1 )
359  jmax = isamax( n, d, 1 )
360  eps = slamch( 'Epsilon' )
361  tol = eight*eps*abs( d( jmax ) )
362 *
363 * If the rank-1 modifier is small enough, no more needs to be done
364 * except to reorganize Q so that its columns correspond with the
365 * elements in D.
366 *
367  IF( rho*abs( z( imax ) ).LE.tol ) THEN
368  k = 0
369  IF( icompq.EQ.0 ) THEN
370  DO 50 j = 1, n
371  perm( j ) = indxq( indx( j ) )
372  50 CONTINUE
373  ELSE
374  DO 60 j = 1, n
375  perm( j ) = indxq( indx( j ) )
376  CALL scopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
377  60 CONTINUE
378  CALL slacpy( 'A', qsiz, n, q2( 1, 1 ), ldq2, q( 1, 1 ),
379  $ ldq )
380  END IF
381  RETURN
382  END IF
383 *
384 * If there are multiple eigenvalues then the problem deflates. Here
385 * the number of equal eigenvalues are found. As each equal
386 * eigenvalue is found, an elementary reflector is computed to rotate
387 * the corresponding eigensubspace so that the corresponding
388 * components of Z are zero in this new basis.
389 *
390  k = 0
391  k2 = n + 1
392  DO 70 j = 1, n
393  IF( rho*abs( z( j ) ).LE.tol ) THEN
394 *
395 * Deflate due to small z component.
396 *
397  k2 = k2 - 1
398  indxp( k2 ) = j
399  IF( j.EQ.n )
400  $ GO TO 110
401  ELSE
402  jlam = j
403  GO TO 80
404  END IF
405  70 CONTINUE
406  80 CONTINUE
407  j = j + 1
408  IF( j.GT.n )
409  $ GO TO 100
410  IF( rho*abs( z( j ) ).LE.tol ) THEN
411 *
412 * Deflate due to small z component.
413 *
414  k2 = k2 - 1
415  indxp( k2 ) = j
416  ELSE
417 *
418 * Check if eigenvalues are close enough to allow deflation.
419 *
420  s = z( jlam )
421  c = z( j )
422 *
423 * Find sqrt(a**2+b**2) without overflow or
424 * destructive underflow.
425 *
426  tau = slapy2( c, s )
427  t = d( j ) - d( jlam )
428  c = c / tau
429  s = -s / tau
430  IF( abs( t*c*s ).LE.tol ) THEN
431 *
432 * Deflation is possible.
433 *
434  z( j ) = tau
435  z( jlam ) = zero
436 *
437 * Record the appropriate Givens rotation
438 *
439  givptr = givptr + 1
440  givcol( 1, givptr ) = indxq( indx( jlam ) )
441  givcol( 2, givptr ) = indxq( indx( j ) )
442  givnum( 1, givptr ) = c
443  givnum( 2, givptr ) = s
444  IF( icompq.EQ.1 ) THEN
445  CALL srot( qsiz, q( 1, indxq( indx( jlam ) ) ), 1,
446  $ q( 1, indxq( indx( j ) ) ), 1, c, s )
447  END IF
448  t = d( jlam )*c*c + d( j )*s*s
449  d( j ) = d( jlam )*s*s + d( j )*c*c
450  d( jlam ) = t
451  k2 = k2 - 1
452  i = 1
453  90 CONTINUE
454  IF( k2+i.LE.n ) THEN
455  IF( d( jlam ).LT.d( indxp( k2+i ) ) ) THEN
456  indxp( k2+i-1 ) = indxp( k2+i )
457  indxp( k2+i ) = jlam
458  i = i + 1
459  GO TO 90
460  ELSE
461  indxp( k2+i-1 ) = jlam
462  END IF
463  ELSE
464  indxp( k2+i-1 ) = jlam
465  END IF
466  jlam = j
467  ELSE
468  k = k + 1
469  w( k ) = z( jlam )
470  dlamda( k ) = d( jlam )
471  indxp( k ) = jlam
472  jlam = j
473  END IF
474  END IF
475  GO TO 80
476  100 CONTINUE
477 *
478 * Record the last eigenvalue.
479 *
480  k = k + 1
481  w( k ) = z( jlam )
482  dlamda( k ) = d( jlam )
483  indxp( k ) = jlam
484 *
485  110 CONTINUE
486 *
487 * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
488 * and Q2 respectively. The eigenvalues/vectors which were not
489 * deflated go into the first K slots of DLAMDA and Q2 respectively,
490 * while those which were deflated go into the last N - K slots.
491 *
492  IF( icompq.EQ.0 ) THEN
493  DO 120 j = 1, n
494  jp = indxp( j )
495  dlamda( j ) = d( jp )
496  perm( j ) = indxq( indx( jp ) )
497  120 CONTINUE
498  ELSE
499  DO 130 j = 1, n
500  jp = indxp( j )
501  dlamda( j ) = d( jp )
502  perm( j ) = indxq( indx( jp ) )
503  CALL scopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
504  130 CONTINUE
505  END IF
506 *
507 * The deflated eigenvalues and their corresponding vectors go back
508 * into the last N - K slots of D and Q respectively.
509 *
510  IF( k.LT.n ) THEN
511  IF( icompq.EQ.0 ) THEN
512  CALL scopy( n-k, dlamda( k+1 ), 1, d( k+1 ), 1 )
513  ELSE
514  CALL scopy( n-k, dlamda( k+1 ), 1, d( k+1 ), 1 )
515  CALL slacpy( 'A', qsiz, n-k, q2( 1, k+1 ), ldq2,
516  $ q( 1, k+1 ), ldq )
517  END IF
518  END IF
519 *
520  RETURN
521 *
522 * End of SLAED8
523 *
524  END
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:94
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:101
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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:245
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:81
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84