LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine sstevx ( character JOBZ, character RANGE, integer N, real, dimension( * ) D, real, dimension( * ) E, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

SSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:
``` SSTEVX computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric tridiagonal matrix A.  Eigenvalues and
eigenvectors can be selected by specifying either a range of values
or a range of indices for the desired eigenvalues.```
Parameters
 [in] JOBZ ``` JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.``` [in] RANGE ``` RANGE is CHARACTER*1 = 'A': all eigenvalues will be found. = 'V': all eigenvalues in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th eigenvalues will be found.``` [in] N ``` N is INTEGER The order of the matrix. N >= 0.``` [in,out] D ``` D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, D may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.``` [in,out] E ``` E is REAL array, dimension (max(1,N-1)) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A in elements 1 to N-1 of E. On exit, E may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.``` [in] VL ``` VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.``` [in] VU ``` VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.``` [in] IL ``` IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.``` [in] IU ``` IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.``` [in] ABSTOL ``` ABSTOL is REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.``` [out] M ``` M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.``` [out] W ``` W is REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is REAL array, dimension (LDZ, max(1,M) ) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge (INFO > 0), then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).``` [out] WORK ` WORK is REAL array, dimension (5*N)` [out] IWORK ` IWORK is INTEGER array, dimension (5*N)` [out] IFAIL ``` IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.``` [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. Their indices are stored in array IFAIL.```
Date
June 2016

Definition at line 229 of file sstevx.f.

229 *
230 * -- LAPACK driver routine (version 3.6.1) --
231 * -- LAPACK is a software package provided by Univ. of Tennessee, --
232 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
233 * June 2016
234 *
235 * .. Scalar Arguments ..
236  CHARACTER jobz, range
237  INTEGER il, info, iu, ldz, m, n
238  REAL abstol, vl, vu
239 * ..
240 * .. Array Arguments ..
241  INTEGER ifail( * ), iwork( * )
242  REAL d( * ), e( * ), w( * ), work( * ), z( ldz, * )
243 * ..
244 *
245 * =====================================================================
246 *
247 * .. Parameters ..
248  REAL zero, one
249  parameter ( zero = 0.0e0, one = 1.0e0 )
250 * ..
251 * .. Local Scalars ..
252  LOGICAL alleig, indeig, test, valeig, wantz
253  CHARACTER order
254  INTEGER i, imax, indibl, indisp, indiwo, indwrk,
255  \$ iscale, itmp1, j, jj, nsplit
256  REAL bignum, eps, rmax, rmin, safmin, sigma, smlnum,
257  \$ tmp1, tnrm, vll, vuu
258 * ..
259 * .. External Functions ..
260  LOGICAL lsame
261  REAL slamch, slanst
262  EXTERNAL lsame, slamch, slanst
263 * ..
264 * .. External Subroutines ..
265  EXTERNAL scopy, sscal, sstebz, sstein, ssteqr, ssterf,
266  \$ sswap, xerbla
267 * ..
268 * .. Intrinsic Functions ..
269  INTRINSIC max, min, sqrt
270 * ..
271 * .. Executable Statements ..
272 *
273 * Test the input parameters.
274 *
275  wantz = lsame( jobz, 'V' )
276  alleig = lsame( range, 'A' )
277  valeig = lsame( range, 'V' )
278  indeig = lsame( range, 'I' )
279 *
280  info = 0
281  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
282  info = -1
283  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
284  info = -2
285  ELSE IF( n.LT.0 ) THEN
286  info = -3
287  ELSE
288  IF( valeig ) THEN
289  IF( n.GT.0 .AND. vu.LE.vl )
290  \$ info = -7
291  ELSE IF( indeig ) THEN
292  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
293  info = -8
294  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
295  info = -9
296  END IF
297  END IF
298  END IF
299  IF( info.EQ.0 ) THEN
300  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) )
301  \$ info = -14
302  END IF
303 *
304  IF( info.NE.0 ) THEN
305  CALL xerbla( 'SSTEVX', -info )
306  RETURN
307  END IF
308 *
309 * Quick return if possible
310 *
311  m = 0
312  IF( n.EQ.0 )
313  \$ RETURN
314 *
315  IF( n.EQ.1 ) THEN
316  IF( alleig .OR. indeig ) THEN
317  m = 1
318  w( 1 ) = d( 1 )
319  ELSE
320  IF( vl.LT.d( 1 ) .AND. vu.GE.d( 1 ) ) THEN
321  m = 1
322  w( 1 ) = d( 1 )
323  END IF
324  END IF
325  IF( wantz )
326  \$ z( 1, 1 ) = one
327  RETURN
328  END IF
329 *
330 * Get machine constants.
331 *
332  safmin = slamch( 'Safe minimum' )
333  eps = slamch( 'Precision' )
334  smlnum = safmin / eps
335  bignum = one / smlnum
336  rmin = sqrt( smlnum )
337  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
338 *
339 * Scale matrix to allowable range, if necessary.
340 *
341  iscale = 0
342  IF ( valeig ) THEN
343  vll = vl
344  vuu = vu
345  ELSE
346  vll = zero
347  vuu = zero
348  ENDIF
349  tnrm = slanst( 'M', n, d, e )
350  IF( tnrm.GT.zero .AND. tnrm.LT.rmin ) THEN
351  iscale = 1
352  sigma = rmin / tnrm
353  ELSE IF( tnrm.GT.rmax ) THEN
354  iscale = 1
355  sigma = rmax / tnrm
356  END IF
357  IF( iscale.EQ.1 ) THEN
358  CALL sscal( n, sigma, d, 1 )
359  CALL sscal( n-1, sigma, e( 1 ), 1 )
360  IF( valeig ) THEN
361  vll = vl*sigma
362  vuu = vu*sigma
363  END IF
364  END IF
365 *
366 * If all eigenvalues are desired and ABSTOL is less than zero, then
367 * call SSTERF or SSTEQR. If this fails for some eigenvalue, then
368 * try SSTEBZ.
369 *
370  test = .false.
371  IF( indeig ) THEN
372  IF( il.EQ.1 .AND. iu.EQ.n ) THEN
373  test = .true.
374  END IF
375  END IF
376  IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
377  CALL scopy( n, d, 1, w, 1 )
378  CALL scopy( n-1, e( 1 ), 1, work( 1 ), 1 )
379  indwrk = n + 1
380  IF( .NOT.wantz ) THEN
381  CALL ssterf( n, w, work, info )
382  ELSE
383  CALL ssteqr( 'I', n, w, work, z, ldz, work( indwrk ), info )
384  IF( info.EQ.0 ) THEN
385  DO 10 i = 1, n
386  ifail( i ) = 0
387  10 CONTINUE
388  END IF
389  END IF
390  IF( info.EQ.0 ) THEN
391  m = n
392  GO TO 20
393  END IF
394  info = 0
395  END IF
396 *
397 * Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
398 *
399  IF( wantz ) THEN
400  order = 'B'
401  ELSE
402  order = 'E'
403  END IF
404  indwrk = 1
405  indibl = 1
406  indisp = indibl + n
407  indiwo = indisp + n
408  CALL sstebz( range, order, n, vll, vuu, il, iu, abstol, d, e, m,
409  \$ nsplit, w, iwork( indibl ), iwork( indisp ),
410  \$ work( indwrk ), iwork( indiwo ), info )
411 *
412  IF( wantz ) THEN
413  CALL sstein( n, d, e, m, w, iwork( indibl ), iwork( indisp ),
414  \$ z, ldz, work( indwrk ), iwork( indiwo ), ifail,
415  \$ info )
416  END IF
417 *
418 * If matrix was scaled, then rescale eigenvalues appropriately.
419 *
420  20 CONTINUE
421  IF( iscale.EQ.1 ) THEN
422  IF( info.EQ.0 ) THEN
423  imax = m
424  ELSE
425  imax = info - 1
426  END IF
427  CALL sscal( imax, one / sigma, w, 1 )
428  END IF
429 *
430 * If eigenvalues are not in order, then sort them, along with
431 * eigenvectors.
432 *
433  IF( wantz ) THEN
434  DO 40 j = 1, m - 1
435  i = 0
436  tmp1 = w( j )
437  DO 30 jj = j + 1, m
438  IF( w( jj ).LT.tmp1 ) THEN
439  i = jj
440  tmp1 = w( jj )
441  END IF
442  30 CONTINUE
443 *
444  IF( i.NE.0 ) THEN
445  itmp1 = iwork( indibl+i-1 )
446  w( i ) = w( j )
447  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
448  w( j ) = tmp1
449  iwork( indibl+j-1 ) = itmp1
450  CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
451  IF( info.NE.0 ) THEN
452  itmp1 = ifail( i )
453  ifail( i ) = ifail( j )
454  ifail( j ) = itmp1
455  END IF
456  END IF
457  40 CONTINUE
458  END IF
459 *
460  RETURN
461 *
462 * End of SSTEVX
463 *
subroutine sstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
SSTEBZ
Definition: sstebz.f:275
real function slanst(NORM, N, D, E)
SLANST returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.
Definition: slanst.f:102
subroutine sstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
SSTEIN
Definition: sstein.f:176
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine ssteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SSTEQR
Definition: ssteqr.f:133
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:55
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:53
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:88
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53

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