LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine ssyevx ( character JOBZ, character RANGE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, 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 LWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

SSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:
``` SSYEVX computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric 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] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] A ``` A is REAL array, dimension (LDA, N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [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 obtained by reducing A to tridiagonal form. 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) On normal exit, 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, 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 (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise 8*N. For optimal efficiency, LWORK >= (NB+3)*N, where NB is the max of the blocksize for SSYTRD and SORMTR returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [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 255 of file ssyevx.f.

255 *
256 * -- LAPACK driver routine (version 3.6.1) --
257 * -- LAPACK is a software package provided by Univ. of Tennessee, --
258 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
259 * June 2016
260 *
261 * .. Scalar Arguments ..
262  CHARACTER jobz, range, uplo
263  INTEGER il, info, iu, lda, ldz, lwork, m, n
264  REAL abstol, vl, vu
265 * ..
266 * .. Array Arguments ..
267  INTEGER ifail( * ), iwork( * )
268  REAL a( lda, * ), w( * ), work( * ), z( ldz, * )
269 * ..
270 *
271 * =====================================================================
272 *
273 * .. Parameters ..
274  REAL zero, one
275  parameter ( zero = 0.0e+0, one = 1.0e+0 )
276 * ..
277 * .. Local Scalars ..
278  LOGICAL alleig, indeig, lower, lquery, test, valeig,
279  \$ wantz
280  CHARACTER order
281  INTEGER i, iinfo, imax, indd, inde, indee, indibl,
282  \$ indisp, indiwo, indtau, indwkn, indwrk, iscale,
283  \$ itmp1, j, jj, llwork, llwrkn, lwkmin,
284  \$ lwkopt, nb, nsplit
285  REAL abstll, anrm, bignum, eps, rmax, rmin, safmin,
286  \$ sigma, smlnum, tmp1, vll, vuu
287 * ..
288 * .. External Functions ..
289  LOGICAL lsame
290  INTEGER ilaenv
291  REAL slamch, slansy
292  EXTERNAL lsame, ilaenv, slamch, slansy
293 * ..
294 * .. External Subroutines ..
295  EXTERNAL scopy, slacpy, sorgtr, sormtr, sscal, sstebz,
297 * ..
298 * .. Intrinsic Functions ..
299  INTRINSIC max, min, sqrt
300 * ..
301 * .. Executable Statements ..
302 *
303 * Test the input parameters.
304 *
305  lower = lsame( uplo, 'L' )
306  wantz = lsame( jobz, 'V' )
307  alleig = lsame( range, 'A' )
308  valeig = lsame( range, 'V' )
309  indeig = lsame( range, 'I' )
310  lquery = ( lwork.EQ.-1 )
311 *
312  info = 0
313  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
314  info = -1
315  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
316  info = -2
317  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
318  info = -3
319  ELSE IF( n.LT.0 ) THEN
320  info = -4
321  ELSE IF( lda.LT.max( 1, n ) ) THEN
322  info = -6
323  ELSE
324  IF( valeig ) THEN
325  IF( n.GT.0 .AND. vu.LE.vl )
326  \$ info = -8
327  ELSE IF( indeig ) THEN
328  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
329  info = -9
330  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
331  info = -10
332  END IF
333  END IF
334  END IF
335  IF( info.EQ.0 ) THEN
336  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
337  info = -15
338  END IF
339  END IF
340 *
341  IF( info.EQ.0 ) THEN
342  IF( n.LE.1 ) THEN
343  lwkmin = 1
344  work( 1 ) = lwkmin
345  ELSE
346  lwkmin = 8*n
347  nb = ilaenv( 1, 'SSYTRD', uplo, n, -1, -1, -1 )
348  nb = max( nb, ilaenv( 1, 'SORMTR', uplo, n, -1, -1, -1 ) )
349  lwkopt = max( lwkmin, ( nb + 3 )*n )
350  work( 1 ) = lwkopt
351  END IF
352 *
353  IF( lwork.LT.lwkmin .AND. .NOT.lquery )
354  \$ info = -17
355  END IF
356 *
357  IF( info.NE.0 ) THEN
358  CALL xerbla( 'SSYEVX', -info )
359  RETURN
360  ELSE IF( lquery ) THEN
361  RETURN
362  END IF
363 *
364 * Quick return if possible
365 *
366  m = 0
367  IF( n.EQ.0 ) THEN
368  RETURN
369  END IF
370 *
371  IF( n.EQ.1 ) THEN
372  IF( alleig .OR. indeig ) THEN
373  m = 1
374  w( 1 ) = a( 1, 1 )
375  ELSE
376  IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
377  m = 1
378  w( 1 ) = a( 1, 1 )
379  END IF
380  END IF
381  IF( wantz )
382  \$ z( 1, 1 ) = one
383  RETURN
384  END IF
385 *
386 * Get machine constants.
387 *
388  safmin = slamch( 'Safe minimum' )
389  eps = slamch( 'Precision' )
390  smlnum = safmin / eps
391  bignum = one / smlnum
392  rmin = sqrt( smlnum )
393  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
394 *
395 * Scale matrix to allowable range, if necessary.
396 *
397  iscale = 0
398  abstll = abstol
399  IF( valeig ) THEN
400  vll = vl
401  vuu = vu
402  END IF
403  anrm = slansy( 'M', uplo, n, a, lda, work )
404  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
405  iscale = 1
406  sigma = rmin / anrm
407  ELSE IF( anrm.GT.rmax ) THEN
408  iscale = 1
409  sigma = rmax / anrm
410  END IF
411  IF( iscale.EQ.1 ) THEN
412  IF( lower ) THEN
413  DO 10 j = 1, n
414  CALL sscal( n-j+1, sigma, a( j, j ), 1 )
415  10 CONTINUE
416  ELSE
417  DO 20 j = 1, n
418  CALL sscal( j, sigma, a( 1, j ), 1 )
419  20 CONTINUE
420  END IF
421  IF( abstol.GT.0 )
422  \$ abstll = abstol*sigma
423  IF( valeig ) THEN
424  vll = vl*sigma
425  vuu = vu*sigma
426  END IF
427  END IF
428 *
429 * Call SSYTRD to reduce symmetric matrix to tridiagonal form.
430 *
431  indtau = 1
432  inde = indtau + n
433  indd = inde + n
434  indwrk = indd + n
435  llwork = lwork - indwrk + 1
436  CALL ssytrd( uplo, n, a, lda, work( indd ), work( inde ),
437  \$ work( indtau ), work( indwrk ), llwork, iinfo )
438 *
439 * If all eigenvalues are desired and ABSTOL is less than or equal to
440 * zero, then call SSTERF or SORGTR and SSTEQR. If this fails for
441 * some eigenvalue, then try SSTEBZ.
442 *
443  test = .false.
444  IF( indeig ) THEN
445  IF( il.EQ.1 .AND. iu.EQ.n ) THEN
446  test = .true.
447  END IF
448  END IF
449  IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
450  CALL scopy( n, work( indd ), 1, w, 1 )
451  indee = indwrk + 2*n
452  IF( .NOT.wantz ) THEN
453  CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
454  CALL ssterf( n, w, work( indee ), info )
455  ELSE
456  CALL slacpy( 'A', n, n, a, lda, z, ldz )
457  CALL sorgtr( uplo, n, z, ldz, work( indtau ),
458  \$ work( indwrk ), llwork, iinfo )
459  CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
460  CALL ssteqr( jobz, n, w, work( indee ), z, ldz,
461  \$ work( indwrk ), info )
462  IF( info.EQ.0 ) THEN
463  DO 30 i = 1, n
464  ifail( i ) = 0
465  30 CONTINUE
466  END IF
467  END IF
468  IF( info.EQ.0 ) THEN
469  m = n
470  GO TO 40
471  END IF
472  info = 0
473  END IF
474 *
475 * Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
476 *
477  IF( wantz ) THEN
478  order = 'B'
479  ELSE
480  order = 'E'
481  END IF
482  indibl = 1
483  indisp = indibl + n
484  indiwo = indisp + n
485  CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
486  \$ work( indd ), work( inde ), m, nsplit, w,
487  \$ iwork( indibl ), iwork( indisp ), work( indwrk ),
488  \$ iwork( indiwo ), info )
489 *
490  IF( wantz ) THEN
491  CALL sstein( n, work( indd ), work( inde ), m, w,
492  \$ iwork( indibl ), iwork( indisp ), z, ldz,
493  \$ work( indwrk ), iwork( indiwo ), ifail, info )
494 *
495 * Apply orthogonal matrix used in reduction to tridiagonal
496 * form to eigenvectors returned by SSTEIN.
497 *
498  indwkn = inde
499  llwrkn = lwork - indwkn + 1
500  CALL sormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
501  \$ ldz, work( indwkn ), llwrkn, iinfo )
502  END IF
503 *
504 * If matrix was scaled, then rescale eigenvalues appropriately.
505 *
506  40 CONTINUE
507  IF( iscale.EQ.1 ) THEN
508  IF( info.EQ.0 ) THEN
509  imax = m
510  ELSE
511  imax = info - 1
512  END IF
513  CALL sscal( imax, one / sigma, w, 1 )
514  END IF
515 *
516 * If eigenvalues are not in order, then sort them, along with
517 * eigenvectors.
518 *
519  IF( wantz ) THEN
520  DO 60 j = 1, m - 1
521  i = 0
522  tmp1 = w( j )
523  DO 50 jj = j + 1, m
524  IF( w( jj ).LT.tmp1 ) THEN
525  i = jj
526  tmp1 = w( jj )
527  END IF
528  50 CONTINUE
529 *
530  IF( i.NE.0 ) THEN
531  itmp1 = iwork( indibl+i-1 )
532  w( i ) = w( j )
533  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
534  w( j ) = tmp1
535  iwork( indibl+j-1 ) = itmp1
536  CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
537  IF( info.NE.0 ) THEN
538  itmp1 = ifail( i )
539  ifail( i ) = ifail( j )
540  ifail( j ) = itmp1
541  END IF
542  END IF
543  60 CONTINUE
544  END IF
545 *
546 * Set WORK(1) to optimal workspace size.
547 *
548  work( 1 ) = lwkopt
549 *
550  RETURN
551 *
552 * End of SSYEVX
553 *
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
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 ssytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
SSYTRD
Definition: ssytrd.f:194
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 ssteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SSTEQR
Definition: ssteqr.f:133
subroutine sorgtr(UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
SORGTR
Definition: sorgtr.f:125
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
Definition: tstiee.f:83
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
subroutine sormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMTR
Definition: sormtr.f:174
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53
real function slansy(NORM, UPLO, N, A, LDA, WORK)
SLANSY 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 matrix.
Definition: slansy.f:124

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