LAPACK  3.5.0
LAPACK: Linear Algebra PACKage
dsyevx.f File Reference

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Functions/Subroutines

subroutine dsyevx (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
  DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices More...
 

Function/Subroutine Documentation

subroutine dsyevx ( character  JOBZ,
character  RANGE,
character  UPLO,
integer  N,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision  VL,
double precision  VU,
integer  IL,
integer  IU,
double precision  ABSTOL,
integer  M,
double precision, dimension( * )  W,
double precision, dimension( ldz, * )  Z,
integer  LDZ,
double precision, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer, dimension( * )  IFAIL,
integer  INFO 
)

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

Download DSYEVX + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DSYEVX 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 DOUBLE PRECISION 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 DOUBLE PRECISION
[in]VU
          VU is DOUBLE PRECISION
          If RANGE='V', the lower and upper bounds of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.
[in]IL
          IL is INTEGER
[in]IU
          IU is INTEGER
          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues 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 DOUBLE PRECISION
          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*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('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 DOUBLE PRECISION array, dimension (N)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.
[out]Z
          Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DSYTRD and DORMTR
          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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
November 2011

Definition at line 248 of file dsyevx.f.

248 *
249 * -- LAPACK driver routine (version 3.4.0) --
250 * -- LAPACK is a software package provided by Univ. of Tennessee, --
251 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
252 * November 2011
253 *
254 * .. Scalar Arguments ..
255  CHARACTER jobz, range, uplo
256  INTEGER il, info, iu, lda, ldz, lwork, m, n
257  DOUBLE PRECISION abstol, vl, vu
258 * ..
259 * .. Array Arguments ..
260  INTEGER ifail( * ), iwork( * )
261  DOUBLE PRECISION a( lda, * ), w( * ), work( * ), z( ldz, * )
262 * ..
263 *
264 * =====================================================================
265 *
266 * .. Parameters ..
267  DOUBLE PRECISION zero, one
268  parameter( zero = 0.0d+0, one = 1.0d+0 )
269 * ..
270 * .. Local Scalars ..
271  LOGICAL alleig, indeig, lower, lquery, test, valeig,
272  $ wantz
273  CHARACTER order
274  INTEGER i, iinfo, imax, indd, inde, indee, indibl,
275  $ indisp, indiwo, indtau, indwkn, indwrk, iscale,
276  $ itmp1, j, jj, llwork, llwrkn, lwkmin,
277  $ lwkopt, nb, nsplit
278  DOUBLE PRECISION abstll, anrm, bignum, eps, rmax, rmin, safmin,
279  $ sigma, smlnum, tmp1, vll, vuu
280 * ..
281 * .. External Functions ..
282  LOGICAL lsame
283  INTEGER ilaenv
284  DOUBLE PRECISION dlamch, dlansy
285  EXTERNAL lsame, ilaenv, dlamch, dlansy
286 * ..
287 * .. External Subroutines ..
288  EXTERNAL dcopy, dlacpy, dorgtr, dormtr, dscal, dstebz,
290 * ..
291 * .. Intrinsic Functions ..
292  INTRINSIC max, min, sqrt
293 * ..
294 * .. Executable Statements ..
295 *
296 * Test the input parameters.
297 *
298  lower = lsame( uplo, 'L' )
299  wantz = lsame( jobz, 'V' )
300  alleig = lsame( range, 'A' )
301  valeig = lsame( range, 'V' )
302  indeig = lsame( range, 'I' )
303  lquery = ( lwork.EQ.-1 )
304 *
305  info = 0
306  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
307  info = -1
308  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
309  info = -2
310  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
311  info = -3
312  ELSE IF( n.LT.0 ) THEN
313  info = -4
314  ELSE IF( lda.LT.max( 1, n ) ) THEN
315  info = -6
316  ELSE
317  IF( valeig ) THEN
318  IF( n.GT.0 .AND. vu.LE.vl )
319  $ info = -8
320  ELSE IF( indeig ) THEN
321  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
322  info = -9
323  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
324  info = -10
325  END IF
326  END IF
327  END IF
328  IF( info.EQ.0 ) THEN
329  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
330  info = -15
331  END IF
332  END IF
333 *
334  IF( info.EQ.0 ) THEN
335  IF( n.LE.1 ) THEN
336  lwkmin = 1
337  work( 1 ) = lwkmin
338  ELSE
339  lwkmin = 8*n
340  nb = ilaenv( 1, 'DSYTRD', uplo, n, -1, -1, -1 )
341  nb = max( nb, ilaenv( 1, 'DORMTR', uplo, n, -1, -1, -1 ) )
342  lwkopt = max( lwkmin, ( nb + 3 )*n )
343  work( 1 ) = lwkopt
344  END IF
345 *
346  IF( lwork.LT.lwkmin .AND. .NOT.lquery )
347  $ info = -17
348  END IF
349 *
350  IF( info.NE.0 ) THEN
351  CALL xerbla( 'DSYEVX', -info )
352  RETURN
353  ELSE IF( lquery ) THEN
354  RETURN
355  END IF
356 *
357 * Quick return if possible
358 *
359  m = 0
360  IF( n.EQ.0 ) THEN
361  RETURN
362  END IF
363 *
364  IF( n.EQ.1 ) THEN
365  IF( alleig .OR. indeig ) THEN
366  m = 1
367  w( 1 ) = a( 1, 1 )
368  ELSE
369  IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
370  m = 1
371  w( 1 ) = a( 1, 1 )
372  END IF
373  END IF
374  IF( wantz )
375  $ z( 1, 1 ) = one
376  RETURN
377  END IF
378 *
379 * Get machine constants.
380 *
381  safmin = dlamch( 'Safe minimum' )
382  eps = dlamch( 'Precision' )
383  smlnum = safmin / eps
384  bignum = one / smlnum
385  rmin = sqrt( smlnum )
386  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
387 *
388 * Scale matrix to allowable range, if necessary.
389 *
390  iscale = 0
391  abstll = abstol
392  IF( valeig ) THEN
393  vll = vl
394  vuu = vu
395  END IF
396  anrm = dlansy( 'M', uplo, n, a, lda, work )
397  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
398  iscale = 1
399  sigma = rmin / anrm
400  ELSE IF( anrm.GT.rmax ) THEN
401  iscale = 1
402  sigma = rmax / anrm
403  END IF
404  IF( iscale.EQ.1 ) THEN
405  IF( lower ) THEN
406  DO 10 j = 1, n
407  CALL dscal( n-j+1, sigma, a( j, j ), 1 )
408  10 CONTINUE
409  ELSE
410  DO 20 j = 1, n
411  CALL dscal( j, sigma, a( 1, j ), 1 )
412  20 CONTINUE
413  END IF
414  IF( abstol.GT.0 )
415  $ abstll = abstol*sigma
416  IF( valeig ) THEN
417  vll = vl*sigma
418  vuu = vu*sigma
419  END IF
420  END IF
421 *
422 * Call DSYTRD to reduce symmetric matrix to tridiagonal form.
423 *
424  indtau = 1
425  inde = indtau + n
426  indd = inde + n
427  indwrk = indd + n
428  llwork = lwork - indwrk + 1
429  CALL dsytrd( uplo, n, a, lda, work( indd ), work( inde ),
430  $ work( indtau ), work( indwrk ), llwork, iinfo )
431 *
432 * If all eigenvalues are desired and ABSTOL is less than or equal to
433 * zero, then call DSTERF or DORGTR and SSTEQR. If this fails for
434 * some eigenvalue, then try DSTEBZ.
435 *
436  test = .false.
437  IF( indeig ) THEN
438  IF( il.EQ.1 .AND. iu.EQ.n ) THEN
439  test = .true.
440  END IF
441  END IF
442  IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
443  CALL dcopy( n, work( indd ), 1, w, 1 )
444  indee = indwrk + 2*n
445  IF( .NOT.wantz ) THEN
446  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
447  CALL dsterf( n, w, work( indee ), info )
448  ELSE
449  CALL dlacpy( 'A', n, n, a, lda, z, ldz )
450  CALL dorgtr( uplo, n, z, ldz, work( indtau ),
451  $ work( indwrk ), llwork, iinfo )
452  CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
453  CALL dsteqr( jobz, n, w, work( indee ), z, ldz,
454  $ work( indwrk ), info )
455  IF( info.EQ.0 ) THEN
456  DO 30 i = 1, n
457  ifail( i ) = 0
458  30 CONTINUE
459  END IF
460  END IF
461  IF( info.EQ.0 ) THEN
462  m = n
463  GO TO 40
464  END IF
465  info = 0
466  END IF
467 *
468 * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
469 *
470  IF( wantz ) THEN
471  order = 'B'
472  ELSE
473  order = 'E'
474  END IF
475  indibl = 1
476  indisp = indibl + n
477  indiwo = indisp + n
478  CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
479  $ work( indd ), work( inde ), m, nsplit, w,
480  $ iwork( indibl ), iwork( indisp ), work( indwrk ),
481  $ iwork( indiwo ), info )
482 *
483  IF( wantz ) THEN
484  CALL dstein( n, work( indd ), work( inde ), m, w,
485  $ iwork( indibl ), iwork( indisp ), z, ldz,
486  $ work( indwrk ), iwork( indiwo ), ifail, info )
487 *
488 * Apply orthogonal matrix used in reduction to tridiagonal
489 * form to eigenvectors returned by DSTEIN.
490 *
491  indwkn = inde
492  llwrkn = lwork - indwkn + 1
493  CALL dormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
494  $ ldz, work( indwkn ), llwrkn, iinfo )
495  END IF
496 *
497 * If matrix was scaled, then rescale eigenvalues appropriately.
498 *
499  40 CONTINUE
500  IF( iscale.EQ.1 ) THEN
501  IF( info.EQ.0 ) THEN
502  imax = m
503  ELSE
504  imax = info - 1
505  END IF
506  CALL dscal( imax, one / sigma, w, 1 )
507  END IF
508 *
509 * If eigenvalues are not in order, then sort them, along with
510 * eigenvectors.
511 *
512  IF( wantz ) THEN
513  DO 60 j = 1, m - 1
514  i = 0
515  tmp1 = w( j )
516  DO 50 jj = j + 1, m
517  IF( w( jj ).LT.tmp1 ) THEN
518  i = jj
519  tmp1 = w( jj )
520  END IF
521  50 CONTINUE
522 *
523  IF( i.NE.0 ) THEN
524  itmp1 = iwork( indibl+i-1 )
525  w( i ) = w( j )
526  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
527  w( j ) = tmp1
528  iwork( indibl+j-1 ) = itmp1
529  CALL dswap( n, z( 1, i ), 1, z( 1, j ), 1 )
530  IF( info.NE.0 ) THEN
531  itmp1 = ifail( i )
532  ifail( i ) = ifail( j )
533  ifail( j ) = itmp1
534  END IF
535  END IF
536  60 CONTINUE
537  END IF
538 *
539 * Set WORK(1) to optimal workspace size.
540 *
541  work( 1 ) = lwkopt
542 *
543  RETURN
544 *
545 * End of DSYEVX
546 *
subroutine dsteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
DSTEQR
Definition: dsteqr.f:133
subroutine dstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSTEIN
Definition: dstein.f:176
subroutine dormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMTR
Definition: dormtr.f:173
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:88
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:53
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:55
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY 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: dlansy.f:124
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:53
subroutine dsytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
DSYTRD
Definition: dsytrd.f:194
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
Definition: tstiee.f:83
subroutine dorgtr(UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
DORGTR
Definition: dorgtr.f:125
subroutine dstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
DSTEBZ
Definition: dstebz.f:265

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