LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ sstein()

 subroutine sstein ( integer N, real, dimension( * ) D, real, dimension( * ) E, integer M, real, dimension( * ) W, integer, dimension( * ) IBLOCK, integer, dimension( * ) ISPLIT, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

SSTEIN

Purpose:
``` SSTEIN computes the eigenvectors of a real symmetric tridiagonal
matrix T corresponding to specified eigenvalues, using inverse
iteration.

The maximum number of iterations allowed for each eigenvector is
specified by an internal parameter MAXITS (currently set to 5).```
Parameters
 [in] N ``` N is INTEGER The order of the matrix. N >= 0.``` [in] D ``` D is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T.``` [in] E ``` E is REAL array, dimension (N-1) The (n-1) subdiagonal elements of the tridiagonal matrix T, in elements 1 to N-1.``` [in] M ``` M is INTEGER The number of eigenvectors to be found. 0 <= M <= N.``` [in] W ``` W is REAL array, dimension (N) The first M elements of W contain the eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block. ( The output array W from SSTEBZ with ORDER = 'B' is expected here. )``` [in] IBLOCK ``` IBLOCK is INTEGER array, dimension (N) The submatrix indices associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first submatrix from the top, =2 if W(i) belongs to the second submatrix, etc. ( The output array IBLOCK from SSTEBZ is expected here. )``` [in] ISPLIT ``` ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc. ( The output array ISPLIT from SSTEBZ is expected here. )``` [out] Z ``` Z is REAL array, dimension (LDZ, M) The computed eigenvectors. The eigenvector associated with the eigenvalue W(i) is stored in the i-th column of Z. Any vector which fails to converge is set to its current iterate after MAXITS iterations.``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,N).``` [out] WORK ` WORK is REAL array, dimension (5*N)` [out] IWORK ` IWORK is INTEGER array, dimension (N)` [out] IFAIL ``` IFAIL is INTEGER array, dimension (M) On normal exit, all elements of IFAIL are zero. If one or more eigenvectors fail to converge after MAXITS iterations, then their indices are stored in array IFAIL.``` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge in MAXITS iterations. Their indices are stored in array IFAIL.```
Internal Parameters:
```  MAXITS  INTEGER, default = 5
The maximum number of iterations performed.

EXTRA   INTEGER, default = 2
The number of iterations performed after norm growth
criterion is satisfied, should be at least 1.```

Definition at line 172 of file sstein.f.

174 *
175 * -- LAPACK computational routine --
176 * -- LAPACK is a software package provided by Univ. of Tennessee, --
177 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
178 *
179 * .. Scalar Arguments ..
180  INTEGER INFO, LDZ, M, N
181 * ..
182 * .. Array Arguments ..
183  INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
184  \$ IWORK( * )
185  REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
186 * ..
187 *
188 * =====================================================================
189 *
190 * .. Parameters ..
191  REAL ZERO, ONE, TEN, ODM3, ODM1
192  parameter( zero = 0.0e+0, one = 1.0e+0, ten = 1.0e+1,
193  \$ odm3 = 1.0e-3, odm1 = 1.0e-1 )
194  INTEGER MAXITS, EXTRA
195  parameter( maxits = 5, extra = 2 )
196 * ..
197 * .. Local Scalars ..
198  INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
199  \$ INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1,
200  \$ JBLK, JMAX, NBLK, NRMCHK
201  REAL CTR, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
202  \$ SCL, SEP, STPCRT, TOL, XJ, XJM
203 * ..
204 * .. Local Arrays ..
205  INTEGER ISEED( 4 )
206 * ..
207 * .. External Functions ..
208  INTEGER ISAMAX
209  REAL SDOT, SLAMCH, SNRM2
210  EXTERNAL isamax, sdot, slamch, snrm2
211 * ..
212 * .. External Subroutines ..
213  EXTERNAL saxpy, scopy, slagtf, slagts, slarnv, sscal,
214  \$ xerbla
215 * ..
216 * .. Intrinsic Functions ..
217  INTRINSIC abs, max, sqrt
218 * ..
219 * .. Executable Statements ..
220 *
221 * Test the input parameters.
222 *
223  info = 0
224  DO 10 i = 1, m
225  ifail( i ) = 0
226  10 CONTINUE
227 *
228  IF( n.LT.0 ) THEN
229  info = -1
230  ELSE IF( m.LT.0 .OR. m.GT.n ) THEN
231  info = -4
232  ELSE IF( ldz.LT.max( 1, n ) ) THEN
233  info = -9
234  ELSE
235  DO 20 j = 2, m
236  IF( iblock( j ).LT.iblock( j-1 ) ) THEN
237  info = -6
238  GO TO 30
239  END IF
240  IF( iblock( j ).EQ.iblock( j-1 ) .AND. w( j ).LT.w( j-1 ) )
241  \$ THEN
242  info = -5
243  GO TO 30
244  END IF
245  20 CONTINUE
246  30 CONTINUE
247  END IF
248 *
249  IF( info.NE.0 ) THEN
250  CALL xerbla( 'SSTEIN', -info )
251  RETURN
252  END IF
253 *
254 * Quick return if possible
255 *
256  IF( n.EQ.0 .OR. m.EQ.0 ) THEN
257  RETURN
258  ELSE IF( n.EQ.1 ) THEN
259  z( 1, 1 ) = one
260  RETURN
261  END IF
262 *
263 * Get machine constants.
264 *
265  eps = slamch( 'Precision' )
266 *
267 * Initialize seed for random number generator SLARNV.
268 *
269  DO 40 i = 1, 4
270  iseed( i ) = 1
271  40 CONTINUE
272 *
273 * Initialize pointers.
274 *
275  indrv1 = 0
276  indrv2 = indrv1 + n
277  indrv3 = indrv2 + n
278  indrv4 = indrv3 + n
279  indrv5 = indrv4 + n
280 *
281 * Compute eigenvectors of matrix blocks.
282 *
283  j1 = 1
284  DO 160 nblk = 1, iblock( m )
285 *
286 * Find starting and ending indices of block nblk.
287 *
288  IF( nblk.EQ.1 ) THEN
289  b1 = 1
290  ELSE
291  b1 = isplit( nblk-1 ) + 1
292  END IF
293  bn = isplit( nblk )
294  blksiz = bn - b1 + 1
295  IF( blksiz.EQ.1 )
296  \$ GO TO 60
297  gpind = j1
298 *
299 * Compute reorthogonalization criterion and stopping criterion.
300 *
301  onenrm = abs( d( b1 ) ) + abs( e( b1 ) )
302  onenrm = max( onenrm, abs( d( bn ) )+abs( e( bn-1 ) ) )
303  DO 50 i = b1 + 1, bn - 1
304  onenrm = max( onenrm, abs( d( i ) )+abs( e( i-1 ) )+
305  \$ abs( e( i ) ) )
306  50 CONTINUE
307  ortol = odm3*onenrm
308 *
309  stpcrt = sqrt( odm1 / blksiz )
310 *
311 * Loop through eigenvalues of block nblk.
312 *
313  60 CONTINUE
314  jblk = 0
315  DO 150 j = j1, m
316  IF( iblock( j ).NE.nblk ) THEN
317  j1 = j
318  GO TO 160
319  END IF
320  jblk = jblk + 1
321  xj = w( j )
322 *
323 * Skip all the work if the block size is one.
324 *
325  IF( blksiz.EQ.1 ) THEN
326  work( indrv1+1 ) = one
327  GO TO 120
328  END IF
329 *
330 * If eigenvalues j and j-1 are too close, add a relatively
331 * small perturbation.
332 *
333  IF( jblk.GT.1 ) THEN
334  eps1 = abs( eps*xj )
335  pertol = ten*eps1
336  sep = xj - xjm
337  IF( sep.LT.pertol )
338  \$ xj = xjm + pertol
339  END IF
340 *
341  its = 0
342  nrmchk = 0
343 *
344 * Get random starting vector.
345 *
346  CALL slarnv( 2, iseed, blksiz, work( indrv1+1 ) )
347 *
348 * Copy the matrix T so it won't be destroyed in factorization.
349 *
350  CALL scopy( blksiz, d( b1 ), 1, work( indrv4+1 ), 1 )
351  CALL scopy( blksiz-1, e( b1 ), 1, work( indrv2+2 ), 1 )
352  CALL scopy( blksiz-1, e( b1 ), 1, work( indrv3+1 ), 1 )
353 *
354 * Compute LU factors with partial pivoting ( PT = LU )
355 *
356  tol = zero
357  CALL slagtf( blksiz, work( indrv4+1 ), xj, work( indrv2+2 ),
358  \$ work( indrv3+1 ), tol, work( indrv5+1 ), iwork,
359  \$ iinfo )
360 *
361 * Update iteration count.
362 *
363  70 CONTINUE
364  its = its + 1
365  IF( its.GT.maxits )
366  \$ GO TO 100
367 *
368 * Normalize and scale the righthand side vector Pb.
369 *
370  jmax = isamax( blksiz, work( indrv1+1 ), 1 )
371  scl = blksiz*onenrm*max( eps,
372  \$ abs( work( indrv4+blksiz ) ) ) /
373  \$ abs( work( indrv1+jmax ) )
374  CALL sscal( blksiz, scl, work( indrv1+1 ), 1 )
375 *
376 * Solve the system LU = Pb.
377 *
378  CALL slagts( -1, blksiz, work( indrv4+1 ), work( indrv2+2 ),
379  \$ work( indrv3+1 ), work( indrv5+1 ), iwork,
380  \$ work( indrv1+1 ), tol, iinfo )
381 *
382 * Reorthogonalize by modified Gram-Schmidt if eigenvalues are
383 * close enough.
384 *
385  IF( jblk.EQ.1 )
386  \$ GO TO 90
387  IF( abs( xj-xjm ).GT.ortol )
388  \$ gpind = j
389  IF( gpind.NE.j ) THEN
390  DO 80 i = gpind, j - 1
391  ctr = -sdot( blksiz, work( indrv1+1 ), 1, z( b1, i ),
392  \$ 1 )
393  CALL saxpy( blksiz, ctr, z( b1, i ), 1,
394  \$ work( indrv1+1 ), 1 )
395  80 CONTINUE
396  END IF
397 *
398 * Check the infinity norm of the iterate.
399 *
400  90 CONTINUE
401  jmax = isamax( blksiz, work( indrv1+1 ), 1 )
402  nrm = abs( work( indrv1+jmax ) )
403 *
404 * Continue for additional iterations after norm reaches
405 * stopping criterion.
406 *
407  IF( nrm.LT.stpcrt )
408  \$ GO TO 70
409  nrmchk = nrmchk + 1
410  IF( nrmchk.LT.extra+1 )
411  \$ GO TO 70
412 *
413  GO TO 110
414 *
415 * If stopping criterion was not satisfied, update info and
416 * store eigenvector number in array ifail.
417 *
418  100 CONTINUE
419  info = info + 1
420  ifail( info ) = j
421 *
422 * Accept iterate as jth eigenvector.
423 *
424  110 CONTINUE
425  scl = one / snrm2( blksiz, work( indrv1+1 ), 1 )
426  jmax = isamax( blksiz, work( indrv1+1 ), 1 )
427  IF( work( indrv1+jmax ).LT.zero )
428  \$ scl = -scl
429  CALL sscal( blksiz, scl, work( indrv1+1 ), 1 )
430  120 CONTINUE
431  DO 130 i = 1, n
432  z( i, j ) = zero
433  130 CONTINUE
434  DO 140 i = 1, blksiz
435  z( b1+i-1, j ) = work( indrv1+i )
436  140 CONTINUE
437 *
438 * Save the shift to check eigenvalue spacing at next
439 * iteration.
440 *
441  xjm = xj
442 *
443  150 CONTINUE
444  160 CONTINUE
445 *
446  RETURN
447 *
448 * End of SSTEIN
449 *
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
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:71
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 scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function sdot(N, SX, INCX, SY, INCY)
SDOT
Definition: sdot.f:82
real(wp) function snrm2(n, x, incx)
SNRM2
Definition: snrm2.f90:89
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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