LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ chpevx()

 subroutine chpevx ( character JOBZ, character RANGE, character UPLO, integer N, complex, dimension( * ) AP, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

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

Purpose:
CHPEVX computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian matrix A in packed storage.
Eigenvalues/vectors 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] AP AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A. [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 AP 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) If INFO = 0, the selected eigenvalues in ascending order. [out] Z Z is COMPLEX 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 COMPLEX array, dimension (2*N) [out] RWORK RWORK is REAL array, dimension (7*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.

Definition at line 237 of file chpevx.f.

240 *
241 * -- LAPACK driver routine --
242 * -- LAPACK is a software package provided by Univ. of Tennessee, --
243 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
244 *
245 * .. Scalar Arguments ..
246  CHARACTER JOBZ, RANGE, UPLO
247  INTEGER IL, INFO, IU, LDZ, M, N
248  REAL ABSTOL, VL, VU
249 * ..
250 * .. Array Arguments ..
251  INTEGER IFAIL( * ), IWORK( * )
252  REAL RWORK( * ), W( * )
253  COMPLEX AP( * ), WORK( * ), Z( LDZ, * )
254 * ..
255 *
256 * =====================================================================
257 *
258 * .. Parameters ..
259  REAL ZERO, ONE
260  parameter( zero = 0.0e0, one = 1.0e0 )
261  COMPLEX CONE
262  parameter( cone = ( 1.0e0, 0.0e0 ) )
263 * ..
264 * .. Local Scalars ..
265  LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
266  CHARACTER ORDER
267  INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
268  \$ INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
269  \$ ITMP1, J, JJ, NSPLIT
270  REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
271  \$ SIGMA, SMLNUM, TMP1, VLL, VUU
272 * ..
273 * .. External Functions ..
274  LOGICAL LSAME
275  REAL CLANHP, SLAMCH
276  EXTERNAL lsame, clanhp, slamch
277 * ..
278 * .. External Subroutines ..
279  EXTERNAL chptrd, csscal, cstein, csteqr, cswap, cupgtr,
281 * ..
282 * .. Intrinsic Functions ..
283  INTRINSIC max, min, real, sqrt
284 * ..
285 * .. Executable Statements ..
286 *
287 * Test the input parameters.
288 *
289  wantz = lsame( jobz, 'V' )
290  alleig = lsame( range, 'A' )
291  valeig = lsame( range, 'V' )
292  indeig = lsame( range, 'I' )
293 *
294  info = 0
295  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
296  info = -1
297  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
298  info = -2
299  ELSE IF( .NOT.( lsame( uplo, 'L' ) .OR. lsame( uplo, 'U' ) ) )
300  \$ THEN
301  info = -3
302  ELSE IF( n.LT.0 ) THEN
303  info = -4
304  ELSE
305  IF( valeig ) THEN
306  IF( n.GT.0 .AND. vu.LE.vl )
307  \$ info = -7
308  ELSE IF( indeig ) THEN
309  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
310  info = -8
311  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
312  info = -9
313  END IF
314  END IF
315  END IF
316  IF( info.EQ.0 ) THEN
317  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) )
318  \$ info = -14
319  END IF
320 *
321  IF( info.NE.0 ) THEN
322  CALL xerbla( 'CHPEVX', -info )
323  RETURN
324  END IF
325 *
326 * Quick return if possible
327 *
328  m = 0
329  IF( n.EQ.0 )
330  \$ RETURN
331 *
332  IF( n.EQ.1 ) THEN
333  IF( alleig .OR. indeig ) THEN
334  m = 1
335  w( 1 ) = real( ap( 1 ) )
336  ELSE
337  IF( vl.LT.real( ap( 1 ) ) .AND. vu.GE.real( ap( 1 ) ) ) THEN
338  m = 1
339  w( 1 ) = real( ap( 1 ) )
340  END IF
341  END IF
342  IF( wantz )
343  \$ z( 1, 1 ) = cone
344  RETURN
345  END IF
346 *
347 * Get machine constants.
348 *
349  safmin = slamch( 'Safe minimum' )
350  eps = slamch( 'Precision' )
351  smlnum = safmin / eps
352  bignum = one / smlnum
353  rmin = sqrt( smlnum )
354  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
355 *
356 * Scale matrix to allowable range, if necessary.
357 *
358  iscale = 0
359  abstll = abstol
360  IF ( valeig ) THEN
361  vll = vl
362  vuu = vu
363  ELSE
364  vll = zero
365  vuu = zero
366  ENDIF
367  anrm = clanhp( 'M', uplo, n, ap, rwork )
368  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
369  iscale = 1
370  sigma = rmin / anrm
371  ELSE IF( anrm.GT.rmax ) THEN
372  iscale = 1
373  sigma = rmax / anrm
374  END IF
375  IF( iscale.EQ.1 ) THEN
376  CALL csscal( ( n*( n+1 ) ) / 2, sigma, ap, 1 )
377  IF( abstol.GT.0 )
378  \$ abstll = abstol*sigma
379  IF( valeig ) THEN
380  vll = vl*sigma
381  vuu = vu*sigma
382  END IF
383  END IF
384 *
385 * Call CHPTRD to reduce Hermitian packed matrix to tridiagonal form.
386 *
387  indd = 1
388  inde = indd + n
389  indrwk = inde + n
390  indtau = 1
391  indwrk = indtau + n
392  CALL chptrd( uplo, n, ap, rwork( indd ), rwork( inde ),
393  \$ work( indtau ), iinfo )
394 *
395 * If all eigenvalues are desired and ABSTOL is less than or equal
396 * to zero, then call SSTERF or CUPGTR and CSTEQR. If this fails
397 * for some eigenvalue, then try SSTEBZ.
398 *
399  test = .false.
400  IF (indeig) THEN
401  IF (il.EQ.1 .AND. iu.EQ.n) THEN
402  test = .true.
403  END IF
404  END IF
405  IF ((alleig .OR. test) .AND. (abstol.LE.zero)) THEN
406  CALL scopy( n, rwork( indd ), 1, w, 1 )
407  indee = indrwk + 2*n
408  IF( .NOT.wantz ) THEN
409  CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
410  CALL ssterf( n, w, rwork( indee ), info )
411  ELSE
412  CALL cupgtr( uplo, n, ap, work( indtau ), z, ldz,
413  \$ work( indwrk ), iinfo )
414  CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
415  CALL csteqr( jobz, n, w, rwork( indee ), z, ldz,
416  \$ rwork( indrwk ), info )
417  IF( info.EQ.0 ) THEN
418  DO 10 i = 1, n
419  ifail( i ) = 0
420  10 CONTINUE
421  END IF
422  END IF
423  IF( info.EQ.0 ) THEN
424  m = n
425  GO TO 20
426  END IF
427  info = 0
428  END IF
429 *
430 * Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
431 *
432  IF( wantz ) THEN
433  order = 'B'
434  ELSE
435  order = 'E'
436  END IF
437  indibl = 1
438  indisp = indibl + n
439  indiwk = indisp + n
440  CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
441  \$ rwork( indd ), rwork( inde ), m, nsplit, w,
442  \$ iwork( indibl ), iwork( indisp ), rwork( indrwk ),
443  \$ iwork( indiwk ), info )
444 *
445  IF( wantz ) THEN
446  CALL cstein( n, rwork( indd ), rwork( inde ), m, w,
447  \$ iwork( indibl ), iwork( indisp ), z, ldz,
448  \$ rwork( indrwk ), iwork( indiwk ), ifail, info )
449 *
450 * Apply unitary matrix used in reduction to tridiagonal
451 * form to eigenvectors returned by CSTEIN.
452 *
453  indwrk = indtau + n
454  CALL cupmtr( 'L', uplo, 'N', n, m, ap, work( indtau ), z, ldz,
455  \$ work( indwrk ), iinfo )
456  END IF
457 *
458 * If matrix was scaled, then rescale eigenvalues appropriately.
459 *
460  20 CONTINUE
461  IF( iscale.EQ.1 ) THEN
462  IF( info.EQ.0 ) THEN
463  imax = m
464  ELSE
465  imax = info - 1
466  END IF
467  CALL sscal( imax, one / sigma, w, 1 )
468  END IF
469 *
470 * If eigenvalues are not in order, then sort them, along with
471 * eigenvectors.
472 *
473  IF( wantz ) THEN
474  DO 40 j = 1, m - 1
475  i = 0
476  tmp1 = w( j )
477  DO 30 jj = j + 1, m
478  IF( w( jj ).LT.tmp1 ) THEN
479  i = jj
480  tmp1 = w( jj )
481  END IF
482  30 CONTINUE
483 *
484  IF( i.NE.0 ) THEN
485  itmp1 = iwork( indibl+i-1 )
486  w( i ) = w( j )
487  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
488  w( j ) = tmp1
489  iwork( indibl+j-1 ) = itmp1
490  CALL cswap( n, z( 1, i ), 1, z( 1, j ), 1 )
491  IF( info.NE.0 ) THEN
492  itmp1 = ifail( i )
493  ifail( i ) = ifail( j )
494  ifail( j ) = itmp1
495  END IF
496  END IF
497  40 CONTINUE
498  END IF
499 *
500  RETURN
501 *
502 * End of CHPEVX
503 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:86
subroutine sstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
SSTEBZ
Definition: sstebz.f:273
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:78
subroutine cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:81
real function clanhp(NORM, UPLO, N, AP, WORK)
CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clanhp.f:117
subroutine cupgtr(UPLO, N, AP, TAU, Q, LDQ, WORK, INFO)
CUPGTR
Definition: cupgtr.f:114
subroutine chptrd(UPLO, N, AP, D, E, TAU, INFO)
CHPTRD
Definition: chptrd.f:151
subroutine cstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
CSTEIN
Definition: cstein.f:182
subroutine cupmtr(SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)
CUPMTR
Definition: cupmtr.f:150
subroutine csteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
CSTEQR
Definition: csteqr.f:132
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 slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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