LAPACK  3.10.0 LAPACK: Linear Algebra PACKage
cstein.f
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1 *> \brief \b CSTEIN
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
22 * IWORK, IFAIL, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER INFO, LDZ, M, N
26 * ..
27 * .. Array Arguments ..
28 * INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
29 * \$ IWORK( * )
30 * REAL D( * ), E( * ), W( * ), WORK( * )
31 * COMPLEX Z( LDZ, * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> CSTEIN computes the eigenvectors of a real symmetric tridiagonal
41 *> matrix T corresponding to specified eigenvalues, using inverse
42 *> iteration.
43 *>
44 *> The maximum number of iterations allowed for each eigenvector is
45 *> specified by an internal parameter MAXITS (currently set to 5).
46 *>
47 *> Although the eigenvectors are real, they are stored in a complex
48 *> array, which may be passed to CUNMTR or CUPMTR for back
49 *> transformation to the eigenvectors of a complex Hermitian matrix
50 *> which was reduced to tridiagonal form.
51 *>
52 *> \endverbatim
53 *
54 * Arguments:
55 * ==========
56 *
57 *> \param[in] N
58 *> \verbatim
59 *> N is INTEGER
60 *> The order of the matrix. N >= 0.
61 *> \endverbatim
62 *>
63 *> \param[in] D
64 *> \verbatim
65 *> D is REAL array, dimension (N)
66 *> The n diagonal elements of the tridiagonal matrix T.
67 *> \endverbatim
68 *>
69 *> \param[in] E
70 *> \verbatim
71 *> E is REAL array, dimension (N-1)
72 *> The (n-1) subdiagonal elements of the tridiagonal matrix
73 *> T, stored in elements 1 to N-1.
74 *> \endverbatim
75 *>
76 *> \param[in] M
77 *> \verbatim
78 *> M is INTEGER
79 *> The number of eigenvectors to be found. 0 <= M <= N.
80 *> \endverbatim
81 *>
82 *> \param[in] W
83 *> \verbatim
84 *> W is REAL array, dimension (N)
85 *> The first M elements of W contain the eigenvalues for
86 *> which eigenvectors are to be computed. The eigenvalues
87 *> should be grouped by split-off block and ordered from
88 *> smallest to largest within the block. ( The output array
89 *> W from SSTEBZ with ORDER = 'B' is expected here. )
90 *> \endverbatim
91 *>
92 *> \param[in] IBLOCK
93 *> \verbatim
94 *> IBLOCK is INTEGER array, dimension (N)
95 *> The submatrix indices associated with the corresponding
96 *> eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
97 *> the first submatrix from the top, =2 if W(i) belongs to
98 *> the second submatrix, etc. ( The output array IBLOCK
99 *> from SSTEBZ is expected here. )
100 *> \endverbatim
101 *>
102 *> \param[in] ISPLIT
103 *> \verbatim
104 *> ISPLIT is INTEGER array, dimension (N)
105 *> The splitting points, at which T breaks up into submatrices.
106 *> The first submatrix consists of rows/columns 1 to
107 *> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
108 *> through ISPLIT( 2 ), etc.
109 *> ( The output array ISPLIT from SSTEBZ is expected here. )
110 *> \endverbatim
111 *>
112 *> \param[out] Z
113 *> \verbatim
114 *> Z is COMPLEX array, dimension (LDZ, M)
115 *> The computed eigenvectors. The eigenvector associated
116 *> with the eigenvalue W(i) is stored in the i-th column of
117 *> Z. Any vector which fails to converge is set to its current
118 *> iterate after MAXITS iterations.
119 *> The imaginary parts of the eigenvectors are set to zero.
120 *> \endverbatim
121 *>
122 *> \param[in] LDZ
123 *> \verbatim
124 *> LDZ is INTEGER
125 *> The leading dimension of the array Z. LDZ >= max(1,N).
126 *> \endverbatim
127 *>
128 *> \param[out] WORK
129 *> \verbatim
130 *> WORK is REAL array, dimension (5*N)
131 *> \endverbatim
132 *>
133 *> \param[out] IWORK
134 *> \verbatim
135 *> IWORK is INTEGER array, dimension (N)
136 *> \endverbatim
137 *>
138 *> \param[out] IFAIL
139 *> \verbatim
140 *> IFAIL is INTEGER array, dimension (M)
141 *> On normal exit, all elements of IFAIL are zero.
142 *> If one or more eigenvectors fail to converge after
143 *> MAXITS iterations, then their indices are stored in
144 *> array IFAIL.
145 *> \endverbatim
146 *>
147 *> \param[out] INFO
148 *> \verbatim
149 *> INFO is INTEGER
150 *> = 0: successful exit
151 *> < 0: if INFO = -i, the i-th argument had an illegal value
152 *> > 0: if INFO = i, then i eigenvectors failed to converge
153 *> in MAXITS iterations. Their indices are stored in
154 *> array IFAIL.
155 *> \endverbatim
156 *
157 *> \par Internal Parameters:
158 * =========================
159 *>
160 *> \verbatim
161 *> MAXITS INTEGER, default = 5
162 *> The maximum number of iterations performed.
163 *>
164 *> EXTRA INTEGER, default = 2
165 *> The number of iterations performed after norm growth
166 *> criterion is satisfied, should be at least 1.
167 *> \endverbatim
168 *
169 * Authors:
170 * ========
171 *
172 *> \author Univ. of Tennessee
173 *> \author Univ. of California Berkeley
174 *> \author Univ. of Colorado Denver
175 *> \author NAG Ltd.
176 *
177 *> \ingroup complexOTHERcomputational
178 *
179 * =====================================================================
180  SUBROUTINE cstein( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
181  \$ IWORK, IFAIL, INFO )
182 *
183 * -- LAPACK computational routine --
184 * -- LAPACK is a software package provided by Univ. of Tennessee, --
185 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186 *
187 * .. Scalar Arguments ..
188  INTEGER INFO, LDZ, M, N
189 * ..
190 * .. Array Arguments ..
191  INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
192  \$ iwork( * )
193  REAL D( * ), E( * ), W( * ), WORK( * )
194  COMPLEX Z( LDZ, * )
195 * ..
196 *
197 * =====================================================================
198 *
199 * .. Parameters ..
200  COMPLEX CZERO, CONE
201  parameter( czero = ( 0.0e+0, 0.0e+0 ),
202  \$ cone = ( 1.0e+0, 0.0e+0 ) )
203  REAL ZERO, ONE, TEN, ODM3, ODM1
204  parameter( zero = 0.0e+0, one = 1.0e+0, ten = 1.0e+1,
205  \$ odm3 = 1.0e-3, odm1 = 1.0e-1 )
206  INTEGER MAXITS, EXTRA
207  parameter( maxits = 5, extra = 2 )
208 * ..
209 * .. Local Scalars ..
210  INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
211  \$ indrv2, indrv3, indrv4, indrv5, its, j, j1,
212  \$ jblk, jmax, jr, nblk, nrmchk
213  REAL CTR, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
214  \$ scl, sep, stpcrt, tol, xj, xjm
215 * ..
216 * .. Local Arrays ..
217  INTEGER ISEED( 4 )
218 * ..
219 * .. External Functions ..
220  INTEGER ISAMAX
221  REAL SLAMCH, SNRM2
222  EXTERNAL isamax, slamch, snrm2
223 * ..
224 * .. External Subroutines ..
225  EXTERNAL scopy, slagtf, slagts, slarnv, sscal, xerbla
226 * ..
227 * .. Intrinsic Functions ..
228  INTRINSIC abs, cmplx, max, real, sqrt
229 * ..
230 * .. Executable Statements ..
231 *
232 * Test the input parameters.
233 *
234  info = 0
235  DO 10 i = 1, m
236  ifail( i ) = 0
237  10 CONTINUE
238 *
239  IF( n.LT.0 ) THEN
240  info = -1
241  ELSE IF( m.LT.0 .OR. m.GT.n ) THEN
242  info = -4
243  ELSE IF( ldz.LT.max( 1, n ) ) THEN
244  info = -9
245  ELSE
246  DO 20 j = 2, m
247  IF( iblock( j ).LT.iblock( j-1 ) ) THEN
248  info = -6
249  GO TO 30
250  END IF
251  IF( iblock( j ).EQ.iblock( j-1 ) .AND. w( j ).LT.w( j-1 ) )
252  \$ THEN
253  info = -5
254  GO TO 30
255  END IF
256  20 CONTINUE
257  30 CONTINUE
258  END IF
259 *
260  IF( info.NE.0 ) THEN
261  CALL xerbla( 'CSTEIN', -info )
262  RETURN
263  END IF
264 *
265 * Quick return if possible
266 *
267  IF( n.EQ.0 .OR. m.EQ.0 ) THEN
268  RETURN
269  ELSE IF( n.EQ.1 ) THEN
270  z( 1, 1 ) = cone
271  RETURN
272  END IF
273 *
274 * Get machine constants.
275 *
276  eps = slamch( 'Precision' )
277 *
278 * Initialize seed for random number generator SLARNV.
279 *
280  DO 40 i = 1, 4
281  iseed( i ) = 1
282  40 CONTINUE
283 *
284 * Initialize pointers.
285 *
286  indrv1 = 0
287  indrv2 = indrv1 + n
288  indrv3 = indrv2 + n
289  indrv4 = indrv3 + n
290  indrv5 = indrv4 + n
291 *
292 * Compute eigenvectors of matrix blocks.
293 *
294  j1 = 1
295  DO 180 nblk = 1, iblock( m )
296 *
297 * Find starting and ending indices of block nblk.
298 *
299  IF( nblk.EQ.1 ) THEN
300  b1 = 1
301  ELSE
302  b1 = isplit( nblk-1 ) + 1
303  END IF
304  bn = isplit( nblk )
305  blksiz = bn - b1 + 1
306  IF( blksiz.EQ.1 )
307  \$ GO TO 60
308  gpind = j1
309 *
310 * Compute reorthogonalization criterion and stopping criterion.
311 *
312  onenrm = abs( d( b1 ) ) + abs( e( b1 ) )
313  onenrm = max( onenrm, abs( d( bn ) )+abs( e( bn-1 ) ) )
314  DO 50 i = b1 + 1, bn - 1
315  onenrm = max( onenrm, abs( d( i ) )+abs( e( i-1 ) )+
316  \$ abs( e( i ) ) )
317  50 CONTINUE
318  ortol = odm3*onenrm
319 *
320  stpcrt = sqrt( odm1 / blksiz )
321 *
322 * Loop through eigenvalues of block nblk.
323 *
324  60 CONTINUE
325  jblk = 0
326  DO 170 j = j1, m
327  IF( iblock( j ).NE.nblk ) THEN
328  j1 = j
329  GO TO 180
330  END IF
331  jblk = jblk + 1
332  xj = w( j )
333 *
334 * Skip all the work if the block size is one.
335 *
336  IF( blksiz.EQ.1 ) THEN
337  work( indrv1+1 ) = one
338  GO TO 140
339  END IF
340 *
341 * If eigenvalues j and j-1 are too close, add a relatively
342 * small perturbation.
343 *
344  IF( jblk.GT.1 ) THEN
345  eps1 = abs( eps*xj )
346  pertol = ten*eps1
347  sep = xj - xjm
348  IF( sep.LT.pertol )
349  \$ xj = xjm + pertol
350  END IF
351 *
352  its = 0
353  nrmchk = 0
354 *
355 * Get random starting vector.
356 *
357  CALL slarnv( 2, iseed, blksiz, work( indrv1+1 ) )
358 *
359 * Copy the matrix T so it won't be destroyed in factorization.
360 *
361  CALL scopy( blksiz, d( b1 ), 1, work( indrv4+1 ), 1 )
362  CALL scopy( blksiz-1, e( b1 ), 1, work( indrv2+2 ), 1 )
363  CALL scopy( blksiz-1, e( b1 ), 1, work( indrv3+1 ), 1 )
364 *
365 * Compute LU factors with partial pivoting ( PT = LU )
366 *
367  tol = zero
368  CALL slagtf( blksiz, work( indrv4+1 ), xj, work( indrv2+2 ),
369  \$ work( indrv3+1 ), tol, work( indrv5+1 ), iwork,
370  \$ iinfo )
371 *
372 * Update iteration count.
373 *
374  70 CONTINUE
375  its = its + 1
376  IF( its.GT.maxits )
377  \$ GO TO 120
378 *
379 * Normalize and scale the righthand side vector Pb.
380 *
381  jmax = isamax( blksiz, work( indrv1+1 ), 1 )
382  scl = blksiz*onenrm*max( eps,
383  \$ abs( work( indrv4+blksiz ) ) ) /
384  \$ abs( work( indrv1+jmax ) )
385  CALL sscal( blksiz, scl, work( indrv1+1 ), 1 )
386 *
387 * Solve the system LU = Pb.
388 *
389  CALL slagts( -1, blksiz, work( indrv4+1 ), work( indrv2+2 ),
390  \$ work( indrv3+1 ), work( indrv5+1 ), iwork,
391  \$ work( indrv1+1 ), tol, iinfo )
392 *
393 * Reorthogonalize by modified Gram-Schmidt if eigenvalues are
394 * close enough.
395 *
396  IF( jblk.EQ.1 )
397  \$ GO TO 110
398  IF( abs( xj-xjm ).GT.ortol )
399  \$ gpind = j
400  IF( gpind.NE.j ) THEN
401  DO 100 i = gpind, j - 1
402  ctr = zero
403  DO 80 jr = 1, blksiz
404  ctr = ctr + work( indrv1+jr )*
405  \$ real( z( b1-1+jr, i ) )
406  80 CONTINUE
407  DO 90 jr = 1, blksiz
408  work( indrv1+jr ) = work( indrv1+jr ) -
409  \$ ctr*real( z( b1-1+jr, i ) )
410  90 CONTINUE
411  100 CONTINUE
412  END IF
413 *
414 * Check the infinity norm of the iterate.
415 *
416  110 CONTINUE
417  jmax = isamax( blksiz, work( indrv1+1 ), 1 )
418  nrm = abs( work( indrv1+jmax ) )
419 *
420 * Continue for additional iterations after norm reaches
421 * stopping criterion.
422 *
423  IF( nrm.LT.stpcrt )
424  \$ GO TO 70
425  nrmchk = nrmchk + 1
426  IF( nrmchk.LT.extra+1 )
427  \$ GO TO 70
428 *
429  GO TO 130
430 *
431 * If stopping criterion was not satisfied, update info and
432 * store eigenvector number in array ifail.
433 *
434  120 CONTINUE
435  info = info + 1
436  ifail( info ) = j
437 *
438 * Accept iterate as jth eigenvector.
439 *
440  130 CONTINUE
441  scl = one / snrm2( blksiz, work( indrv1+1 ), 1 )
442  jmax = isamax( blksiz, work( indrv1+1 ), 1 )
443  IF( work( indrv1+jmax ).LT.zero )
444  \$ scl = -scl
445  CALL sscal( blksiz, scl, work( indrv1+1 ), 1 )
446  140 CONTINUE
447  DO 150 i = 1, n
448  z( i, j ) = czero
449  150 CONTINUE
450  DO 160 i = 1, blksiz
451  z( b1+i-1, j ) = cmplx( work( indrv1+i ), zero )
452  160 CONTINUE
453 *
454 * Save the shift to check eigenvalue spacing at next
455 * iteration.
456 *
457  xjm = xj
458 *
459  170 CONTINUE
460  180 CONTINUE
461 *
462  RETURN
463 *
464 * End of CSTEIN
465 *
466  END
subroutine slarnv(IDIST, ISEED, N, X)
SLARNV returns a vector of random numbers from a uniform or normal distribution.
Definition: slarnv.f:97
subroutine slagts(JOB, N, A, B, C, D, IN, Y, TOL, INFO)
SLAGTS solves the system of equations (T-λI)x = y or (T-λI)Tx = y,where T is a general tridiagonal ma...
Definition: slagts.f:161
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slagtf(N, A, LAMBDA, B, C, TOL, D, IN, INFO)
SLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix,...
Definition: slagtf.f:156
subroutine cstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
CSTEIN
Definition: cstein.f:182
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79