LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
slar1v.f
Go to the documentation of this file.
1 *> \brief \b SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SLAR1V + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slar1v.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slar1v.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slar1v.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD,
22 * PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
23 * R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
24 *
25 * .. Scalar Arguments ..
26 * LOGICAL WANTNC
27 * INTEGER B1, BN, N, NEGCNT, R
28 * REAL GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
29 * $ RQCORR, ZTZ
30 * ..
31 * .. Array Arguments ..
32 * INTEGER ISUPPZ( * )
33 * REAL D( * ), L( * ), LD( * ), LLD( * ),
34 * $ WORK( * )
35 * REAL Z( * )
36 * ..
37 *
38 *
39 *> \par Purpose:
40 * =============
41 *>
42 *> \verbatim
43 *>
44 *> SLAR1V computes the (scaled) r-th column of the inverse of
45 *> the sumbmatrix in rows B1 through BN of the tridiagonal matrix
46 *> L D L**T - sigma I. When sigma is close to an eigenvalue, the
47 *> computed vector is an accurate eigenvector. Usually, r corresponds
48 *> to the index where the eigenvector is largest in magnitude.
49 *> The following steps accomplish this computation :
50 *> (a) Stationary qd transform, L D L**T - sigma I = L(+) D(+) L(+)**T,
51 *> (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
52 *> (c) Computation of the diagonal elements of the inverse of
53 *> L D L**T - sigma I by combining the above transforms, and choosing
54 *> r as the index where the diagonal of the inverse is (one of the)
55 *> largest in magnitude.
56 *> (d) Computation of the (scaled) r-th column of the inverse using the
57 *> twisted factorization obtained by combining the top part of the
58 *> the stationary and the bottom part of the progressive transform.
59 *> \endverbatim
60 *
61 * Arguments:
62 * ==========
63 *
64 *> \param[in] N
65 *> \verbatim
66 *> N is INTEGER
67 *> The order of the matrix L D L**T.
68 *> \endverbatim
69 *>
70 *> \param[in] B1
71 *> \verbatim
72 *> B1 is INTEGER
73 *> First index of the submatrix of L D L**T.
74 *> \endverbatim
75 *>
76 *> \param[in] BN
77 *> \verbatim
78 *> BN is INTEGER
79 *> Last index of the submatrix of L D L**T.
80 *> \endverbatim
81 *>
82 *> \param[in] LAMBDA
83 *> \verbatim
84 *> LAMBDA is REAL
85 *> The shift. In order to compute an accurate eigenvector,
86 *> LAMBDA should be a good approximation to an eigenvalue
87 *> of L D L**T.
88 *> \endverbatim
89 *>
90 *> \param[in] L
91 *> \verbatim
92 *> L is REAL array, dimension (N-1)
93 *> The (n-1) subdiagonal elements of the unit bidiagonal matrix
94 *> L, in elements 1 to N-1.
95 *> \endverbatim
96 *>
97 *> \param[in] D
98 *> \verbatim
99 *> D is REAL array, dimension (N)
100 *> The n diagonal elements of the diagonal matrix D.
101 *> \endverbatim
102 *>
103 *> \param[in] LD
104 *> \verbatim
105 *> LD is REAL array, dimension (N-1)
106 *> The n-1 elements L(i)*D(i).
107 *> \endverbatim
108 *>
109 *> \param[in] LLD
110 *> \verbatim
111 *> LLD is REAL array, dimension (N-1)
112 *> The n-1 elements L(i)*L(i)*D(i).
113 *> \endverbatim
114 *>
115 *> \param[in] PIVMIN
116 *> \verbatim
117 *> PIVMIN is REAL
118 *> The minimum pivot in the Sturm sequence.
119 *> \endverbatim
120 *>
121 *> \param[in] GAPTOL
122 *> \verbatim
123 *> GAPTOL is REAL
124 *> Tolerance that indicates when eigenvector entries are negligible
125 *> w.r.t. their contribution to the residual.
126 *> \endverbatim
127 *>
128 *> \param[in,out] Z
129 *> \verbatim
130 *> Z is REAL array, dimension (N)
131 *> On input, all entries of Z must be set to 0.
132 *> On output, Z contains the (scaled) r-th column of the
133 *> inverse. The scaling is such that Z(R) equals 1.
134 *> \endverbatim
135 *>
136 *> \param[in] WANTNC
137 *> \verbatim
138 *> WANTNC is LOGICAL
139 *> Specifies whether NEGCNT has to be computed.
140 *> \endverbatim
141 *>
142 *> \param[out] NEGCNT
143 *> \verbatim
144 *> NEGCNT is INTEGER
145 *> If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
146 *> in the matrix factorization L D L**T, and NEGCNT = -1 otherwise.
147 *> \endverbatim
148 *>
149 *> \param[out] ZTZ
150 *> \verbatim
151 *> ZTZ is REAL
152 *> The square of the 2-norm of Z.
153 *> \endverbatim
154 *>
155 *> \param[out] MINGMA
156 *> \verbatim
157 *> MINGMA is REAL
158 *> The reciprocal of the largest (in magnitude) diagonal
159 *> element of the inverse of L D L**T - sigma I.
160 *> \endverbatim
161 *>
162 *> \param[in,out] R
163 *> \verbatim
164 *> R is INTEGER
165 *> The twist index for the twisted factorization used to
166 *> compute Z.
167 *> On input, 0 <= R <= N. If R is input as 0, R is set to
168 *> the index where (L D L**T - sigma I)^{-1} is largest
169 *> in magnitude. If 1 <= R <= N, R is unchanged.
170 *> On output, R contains the twist index used to compute Z.
171 *> Ideally, R designates the position of the maximum entry in the
172 *> eigenvector.
173 *> \endverbatim
174 *>
175 *> \param[out] ISUPPZ
176 *> \verbatim
177 *> ISUPPZ is INTEGER array, dimension (2)
178 *> The support of the vector in Z, i.e., the vector Z is
179 *> nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).
180 *> \endverbatim
181 *>
182 *> \param[out] NRMINV
183 *> \verbatim
184 *> NRMINV is REAL
185 *> NRMINV = 1/SQRT( ZTZ )
186 *> \endverbatim
187 *>
188 *> \param[out] RESID
189 *> \verbatim
190 *> RESID is REAL
191 *> The residual of the FP vector.
192 *> RESID = ABS( MINGMA )/SQRT( ZTZ )
193 *> \endverbatim
194 *>
195 *> \param[out] RQCORR
196 *> \verbatim
197 *> RQCORR is REAL
198 *> The Rayleigh Quotient correction to LAMBDA.
199 *> RQCORR = MINGMA*TMP
200 *> \endverbatim
201 *>
202 *> \param[out] WORK
203 *> \verbatim
204 *> WORK is REAL array, dimension (4*N)
205 *> \endverbatim
206 *
207 * Authors:
208 * ========
209 *
210 *> \author Univ. of Tennessee
211 *> \author Univ. of California Berkeley
212 *> \author Univ. of Colorado Denver
213 *> \author NAG Ltd.
214 *
215 *> \date December 2016
216 *
217 *> \ingroup realOTHERauxiliary
218 *
219 *> \par Contributors:
220 * ==================
221 *>
222 *> Beresford Parlett, University of California, Berkeley, USA \n
223 *> Jim Demmel, University of California, Berkeley, USA \n
224 *> Inderjit Dhillon, University of Texas, Austin, USA \n
225 *> Osni Marques, LBNL/NERSC, USA \n
226 *> Christof Voemel, University of California, Berkeley, USA
227 *
228 * =====================================================================
229  SUBROUTINE slar1v( N, B1, BN, LAMBDA, D, L, LD, LLD,
230  $ PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
231  $ R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
232 *
233 * -- LAPACK auxiliary routine (version 3.7.0) --
234 * -- LAPACK is a software package provided by Univ. of Tennessee, --
235 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
236 * December 2016
237 *
238 * .. Scalar Arguments ..
239  LOGICAL WANTNC
240  INTEGER B1, BN, N, NEGCNT, R
241  REAL GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
242  $ rqcorr, ztz
243 * ..
244 * .. Array Arguments ..
245  INTEGER ISUPPZ( * )
246  REAL D( * ), L( * ), LD( * ), LLD( * ),
247  $ work( * )
248  REAL Z( * )
249 * ..
250 *
251 * =====================================================================
252 *
253 * .. Parameters ..
254  REAL ZERO, ONE
255  parameter( zero = 0.0e0, one = 1.0e0 )
256 
257 * ..
258 * .. Local Scalars ..
259  LOGICAL SAWNAN1, SAWNAN2
260  INTEGER I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1,
261  $ r2
262  REAL DMINUS, DPLUS, EPS, S, TMP
263 * ..
264 * .. External Functions ..
265  LOGICAL SISNAN
266  REAL SLAMCH
267  EXTERNAL sisnan, slamch
268 * ..
269 * .. Intrinsic Functions ..
270  INTRINSIC abs
271 * ..
272 * .. Executable Statements ..
273 *
274  eps = slamch( 'Precision' )
275 
276 
277  IF( r.EQ.0 ) THEN
278  r1 = b1
279  r2 = bn
280  ELSE
281  r1 = r
282  r2 = r
283  END IF
284 
285 * Storage for LPLUS
286  indlpl = 0
287 * Storage for UMINUS
288  indumn = n
289  inds = 2*n + 1
290  indp = 3*n + 1
291 
292  IF( b1.EQ.1 ) THEN
293  work( inds ) = zero
294  ELSE
295  work( inds+b1-1 ) = lld( b1-1 )
296  END IF
297 
298 *
299 * Compute the stationary transform (using the differential form)
300 * until the index R2.
301 *
302  sawnan1 = .false.
303  neg1 = 0
304  s = work( inds+b1-1 ) - lambda
305  DO 50 i = b1, r1 - 1
306  dplus = d( i ) + s
307  work( indlpl+i ) = ld( i ) / dplus
308  IF(dplus.LT.zero) neg1 = neg1 + 1
309  work( inds+i ) = s*work( indlpl+i )*l( i )
310  s = work( inds+i ) - lambda
311  50 CONTINUE
312  sawnan1 = sisnan( s )
313  IF( sawnan1 ) GOTO 60
314  DO 51 i = r1, r2 - 1
315  dplus = d( i ) + s
316  work( indlpl+i ) = ld( i ) / dplus
317  work( inds+i ) = s*work( indlpl+i )*l( i )
318  s = work( inds+i ) - lambda
319  51 CONTINUE
320  sawnan1 = sisnan( s )
321 *
322  60 CONTINUE
323  IF( sawnan1 ) THEN
324 * Runs a slower version of the above loop if a NaN is detected
325  neg1 = 0
326  s = work( inds+b1-1 ) - lambda
327  DO 70 i = b1, r1 - 1
328  dplus = d( i ) + s
329  IF(abs(dplus).LT.pivmin) dplus = -pivmin
330  work( indlpl+i ) = ld( i ) / dplus
331  IF(dplus.LT.zero) neg1 = neg1 + 1
332  work( inds+i ) = s*work( indlpl+i )*l( i )
333  IF( work( indlpl+i ).EQ.zero )
334  $ work( inds+i ) = lld( i )
335  s = work( inds+i ) - lambda
336  70 CONTINUE
337  DO 71 i = r1, r2 - 1
338  dplus = d( i ) + s
339  IF(abs(dplus).LT.pivmin) dplus = -pivmin
340  work( indlpl+i ) = ld( i ) / dplus
341  work( inds+i ) = s*work( indlpl+i )*l( i )
342  IF( work( indlpl+i ).EQ.zero )
343  $ work( inds+i ) = lld( i )
344  s = work( inds+i ) - lambda
345  71 CONTINUE
346  END IF
347 *
348 * Compute the progressive transform (using the differential form)
349 * until the index R1
350 *
351  sawnan2 = .false.
352  neg2 = 0
353  work( indp+bn-1 ) = d( bn ) - lambda
354  DO 80 i = bn - 1, r1, -1
355  dminus = lld( i ) + work( indp+i )
356  tmp = d( i ) / dminus
357  IF(dminus.LT.zero) neg2 = neg2 + 1
358  work( indumn+i ) = l( i )*tmp
359  work( indp+i-1 ) = work( indp+i )*tmp - lambda
360  80 CONTINUE
361  tmp = work( indp+r1-1 )
362  sawnan2 = sisnan( tmp )
363 
364  IF( sawnan2 ) THEN
365 * Runs a slower version of the above loop if a NaN is detected
366  neg2 = 0
367  DO 100 i = bn-1, r1, -1
368  dminus = lld( i ) + work( indp+i )
369  IF(abs(dminus).LT.pivmin) dminus = -pivmin
370  tmp = d( i ) / dminus
371  IF(dminus.LT.zero) neg2 = neg2 + 1
372  work( indumn+i ) = l( i )*tmp
373  work( indp+i-1 ) = work( indp+i )*tmp - lambda
374  IF( tmp.EQ.zero )
375  $ work( indp+i-1 ) = d( i ) - lambda
376  100 CONTINUE
377  END IF
378 *
379 * Find the index (from R1 to R2) of the largest (in magnitude)
380 * diagonal element of the inverse
381 *
382  mingma = work( inds+r1-1 ) + work( indp+r1-1 )
383  IF( mingma.LT.zero ) neg1 = neg1 + 1
384  IF( wantnc ) THEN
385  negcnt = neg1 + neg2
386  ELSE
387  negcnt = -1
388  ENDIF
389  IF( abs(mingma).EQ.zero )
390  $ mingma = eps*work( inds+r1-1 )
391  r = r1
392  DO 110 i = r1, r2 - 1
393  tmp = work( inds+i ) + work( indp+i )
394  IF( tmp.EQ.zero )
395  $ tmp = eps*work( inds+i )
396  IF( abs( tmp ).LE.abs( mingma ) ) THEN
397  mingma = tmp
398  r = i + 1
399  END IF
400  110 CONTINUE
401 *
402 * Compute the FP vector: solve N^T v = e_r
403 *
404  isuppz( 1 ) = b1
405  isuppz( 2 ) = bn
406  z( r ) = one
407  ztz = one
408 *
409 * Compute the FP vector upwards from R
410 *
411  IF( .NOT.sawnan1 .AND. .NOT.sawnan2 ) THEN
412  DO 210 i = r-1, b1, -1
413  z( i ) = -( work( indlpl+i )*z( i+1 ) )
414  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
415  $ THEN
416  z( i ) = zero
417  isuppz( 1 ) = i + 1
418  GOTO 220
419  ENDIF
420  ztz = ztz + z( i )*z( i )
421  210 CONTINUE
422  220 CONTINUE
423  ELSE
424 * Run slower loop if NaN occurred.
425  DO 230 i = r - 1, b1, -1
426  IF( z( i+1 ).EQ.zero ) THEN
427  z( i ) = -( ld( i+1 ) / ld( i ) )*z( i+2 )
428  ELSE
429  z( i ) = -( work( indlpl+i )*z( i+1 ) )
430  END IF
431  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
432  $ THEN
433  z( i ) = zero
434  isuppz( 1 ) = i + 1
435  GO TO 240
436  END IF
437  ztz = ztz + z( i )*z( i )
438  230 CONTINUE
439  240 CONTINUE
440  ENDIF
441 
442 * Compute the FP vector downwards from R in blocks of size BLKSIZ
443  IF( .NOT.sawnan1 .AND. .NOT.sawnan2 ) THEN
444  DO 250 i = r, bn-1
445  z( i+1 ) = -( work( indumn+i )*z( i ) )
446  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
447  $ THEN
448  z( i+1 ) = zero
449  isuppz( 2 ) = i
450  GO TO 260
451  END IF
452  ztz = ztz + z( i+1 )*z( i+1 )
453  250 CONTINUE
454  260 CONTINUE
455  ELSE
456 * Run slower loop if NaN occurred.
457  DO 270 i = r, bn - 1
458  IF( z( i ).EQ.zero ) THEN
459  z( i+1 ) = -( ld( i-1 ) / ld( i ) )*z( i-1 )
460  ELSE
461  z( i+1 ) = -( work( indumn+i )*z( i ) )
462  END IF
463  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
464  $ THEN
465  z( i+1 ) = zero
466  isuppz( 2 ) = i
467  GO TO 280
468  END IF
469  ztz = ztz + z( i+1 )*z( i+1 )
470  270 CONTINUE
471  280 CONTINUE
472  END IF
473 *
474 * Compute quantities for convergence test
475 *
476  tmp = one / ztz
477  nrminv = sqrt( tmp )
478  resid = abs( mingma )*nrminv
479  rqcorr = mingma*tmp
480 *
481 *
482  RETURN
483 *
484 * End of SLAR1V
485 *
486  END
subroutine slar1v(N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the...
Definition: slar1v.f:232