LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine chbevx ( character JOBZ, character RANGE, character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldq, * ) Q, integer LDQ, 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 )

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

Purpose:
``` CHBEVX computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian band 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] KD ``` KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.``` [in,out] AB ``` AB is COMPLEX array, dimension (LDAB, N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form.``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.``` [out] Q ``` Q is COMPLEX array, dimension (LDQ, N) If JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.``` [in] LDQ ``` LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= 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 AB 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) The first M elements contain 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 (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.```
Date
June 2016

Definition at line 269 of file chbevx.f.

269 *
270 * -- LAPACK driver routine (version 3.6.1) --
271 * -- LAPACK is a software package provided by Univ. of Tennessee, --
272 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
273 * June 2016
274 *
275 * .. Scalar Arguments ..
276  CHARACTER jobz, range, uplo
277  INTEGER il, info, iu, kd, ldab, ldq, ldz, m, n
278  REAL abstol, vl, vu
279 * ..
280 * .. Array Arguments ..
281  INTEGER ifail( * ), iwork( * )
282  REAL rwork( * ), w( * )
283  COMPLEX ab( ldab, * ), q( ldq, * ), work( * ),
284  \$ z( ldz, * )
285 * ..
286 *
287 * =====================================================================
288 *
289 * .. Parameters ..
290  REAL zero, one
291  parameter ( zero = 0.0e0, one = 1.0e0 )
292  COMPLEX czero, cone
293  parameter ( czero = ( 0.0e0, 0.0e0 ),
294  \$ cone = ( 1.0e0, 0.0e0 ) )
295 * ..
296 * .. Local Scalars ..
297  LOGICAL alleig, indeig, lower, test, valeig, wantz
298  CHARACTER order
299  INTEGER i, iinfo, imax, indd, inde, indee, indibl,
300  \$ indisp, indiwk, indrwk, indwrk, iscale, itmp1,
301  \$ j, jj, nsplit
302  REAL abstll, anrm, bignum, eps, rmax, rmin, safmin,
303  \$ sigma, smlnum, tmp1, vll, vuu
304  COMPLEX ctmp1
305 * ..
306 * .. External Functions ..
307  LOGICAL lsame
308  REAL clanhb, slamch
309  EXTERNAL lsame, clanhb, slamch
310 * ..
311 * .. External Subroutines ..
312  EXTERNAL ccopy, cgemv, chbtrd, clacpy, clascl, cstein,
314  \$ xerbla
315 * ..
316 * .. Intrinsic Functions ..
317  INTRINSIC max, min, REAL, sqrt
318 * ..
319 * .. Executable Statements ..
320 *
321 * Test the input parameters.
322 *
323  wantz = lsame( jobz, 'V' )
324  alleig = lsame( range, 'A' )
325  valeig = lsame( range, 'V' )
326  indeig = lsame( range, 'I' )
327  lower = lsame( uplo, 'L' )
328 *
329  info = 0
330  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
331  info = -1
332  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
333  info = -2
334  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
335  info = -3
336  ELSE IF( n.LT.0 ) THEN
337  info = -4
338  ELSE IF( kd.LT.0 ) THEN
339  info = -5
340  ELSE IF( ldab.LT.kd+1 ) THEN
341  info = -7
342  ELSE IF( wantz .AND. ldq.LT.max( 1, n ) ) THEN
343  info = -9
344  ELSE
345  IF( valeig ) THEN
346  IF( n.GT.0 .AND. vu.LE.vl )
347  \$ info = -11
348  ELSE IF( indeig ) THEN
349  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
350  info = -12
351  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
352  info = -13
353  END IF
354  END IF
355  END IF
356  IF( info.EQ.0 ) THEN
357  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) )
358  \$ info = -18
359  END IF
360 *
361  IF( info.NE.0 ) THEN
362  CALL xerbla( 'CHBEVX', -info )
363  RETURN
364  END IF
365 *
366 * Quick return if possible
367 *
368  m = 0
369  IF( n.EQ.0 )
370  \$ RETURN
371 *
372  IF( n.EQ.1 ) THEN
373  m = 1
374  IF( lower ) THEN
375  ctmp1 = ab( 1, 1 )
376  ELSE
377  ctmp1 = ab( kd+1, 1 )
378  END IF
379  tmp1 = REAL( ctmp1 )
380  IF( valeig ) THEN
381  IF( .NOT.( vl.LT.tmp1 .AND. vu.GE.tmp1 ) )
382  \$ m = 0
383  END IF
384  IF( m.EQ.1 ) THEN
385  w( 1 ) = ctmp1
386  IF( wantz )
387  \$ z( 1, 1 ) = cone
388  END IF
389  RETURN
390  END IF
391 *
392 * Get machine constants.
393 *
394  safmin = slamch( 'Safe minimum' )
395  eps = slamch( 'Precision' )
396  smlnum = safmin / eps
397  bignum = one / smlnum
398  rmin = sqrt( smlnum )
399  rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
400 *
401 * Scale matrix to allowable range, if necessary.
402 *
403  iscale = 0
404  abstll = abstol
405  IF ( valeig ) THEN
406  vll = vl
407  vuu = vu
408  ELSE
409  vll = zero
410  vuu = zero
411  ENDIF
412  anrm = clanhb( 'M', uplo, n, kd, ab, ldab, rwork )
413  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
414  iscale = 1
415  sigma = rmin / anrm
416  ELSE IF( anrm.GT.rmax ) THEN
417  iscale = 1
418  sigma = rmax / anrm
419  END IF
420  IF( iscale.EQ.1 ) THEN
421  IF( lower ) THEN
422  CALL clascl( 'B', kd, kd, one, sigma, n, n, ab, ldab, info )
423  ELSE
424  CALL clascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab, info )
425  END IF
426  IF( abstol.GT.0 )
427  \$ abstll = abstol*sigma
428  IF( valeig ) THEN
429  vll = vl*sigma
430  vuu = vu*sigma
431  END IF
432  END IF
433 *
434 * Call CHBTRD to reduce Hermitian band matrix to tridiagonal form.
435 *
436  indd = 1
437  inde = indd + n
438  indrwk = inde + n
439  indwrk = 1
440  CALL chbtrd( jobz, uplo, n, kd, ab, ldab, rwork( indd ),
441  \$ rwork( inde ), q, ldq, work( indwrk ), iinfo )
442 *
443 * If all eigenvalues are desired and ABSTOL is less than or equal
444 * to zero, then call SSTERF or CSTEQR. If this fails for some
445 * eigenvalue, then try SSTEBZ.
446 *
447  test = .false.
448  IF (indeig) THEN
449  IF (il.EQ.1 .AND. iu.EQ.n) THEN
450  test = .true.
451  END IF
452  END IF
453  IF ((alleig .OR. test) .AND. (abstol.LE.zero)) THEN
454  CALL scopy( n, rwork( indd ), 1, w, 1 )
455  indee = indrwk + 2*n
456  IF( .NOT.wantz ) THEN
457  CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
458  CALL ssterf( n, w, rwork( indee ), info )
459  ELSE
460  CALL clacpy( 'A', n, n, q, ldq, z, ldz )
461  CALL scopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
462  CALL csteqr( jobz, n, w, rwork( indee ), z, ldz,
463  \$ rwork( indrwk ), info )
464  IF( info.EQ.0 ) THEN
465  DO 10 i = 1, n
466  ifail( i ) = 0
467  10 CONTINUE
468  END IF
469  END IF
470  IF( info.EQ.0 ) THEN
471  m = n
472  GO TO 30
473  END IF
474  info = 0
475  END IF
476 *
477 * Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
478 *
479  IF( wantz ) THEN
480  order = 'B'
481  ELSE
482  order = 'E'
483  END IF
484  indibl = 1
485  indisp = indibl + n
486  indiwk = indisp + n
487  CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
488  \$ rwork( indd ), rwork( inde ), m, nsplit, w,
489  \$ iwork( indibl ), iwork( indisp ), rwork( indrwk ),
490  \$ iwork( indiwk ), info )
491 *
492  IF( wantz ) THEN
493  CALL cstein( n, rwork( indd ), rwork( inde ), m, w,
494  \$ iwork( indibl ), iwork( indisp ), z, ldz,
495  \$ rwork( indrwk ), iwork( indiwk ), ifail, info )
496 *
497 * Apply unitary matrix used in reduction to tridiagonal
498 * form to eigenvectors returned by CSTEIN.
499 *
500  DO 20 j = 1, m
501  CALL ccopy( n, z( 1, j ), 1, work( 1 ), 1 )
502  CALL cgemv( 'N', n, n, cone, q, ldq, work, 1, czero,
503  \$ z( 1, j ), 1 )
504  20 CONTINUE
505  END IF
506 *
507 * If matrix was scaled, then rescale eigenvalues appropriately.
508 *
509  30 CONTINUE
510  IF( iscale.EQ.1 ) THEN
511  IF( info.EQ.0 ) THEN
512  imax = m
513  ELSE
514  imax = info - 1
515  END IF
516  CALL sscal( imax, one / sigma, w, 1 )
517  END IF
518 *
519 * If eigenvalues are not in order, then sort them, along with
520 * eigenvectors.
521 *
522  IF( wantz ) THEN
523  DO 50 j = 1, m - 1
524  i = 0
525  tmp1 = w( j )
526  DO 40 jj = j + 1, m
527  IF( w( jj ).LT.tmp1 ) THEN
528  i = jj
529  tmp1 = w( jj )
530  END IF
531  40 CONTINUE
532 *
533  IF( i.NE.0 ) THEN
534  itmp1 = iwork( indibl+i-1 )
535  w( i ) = w( j )
536  iwork( indibl+i-1 ) = iwork( indibl+j-1 )
537  w( j ) = tmp1
538  iwork( indibl+j-1 ) = itmp1
539  CALL cswap( n, z( 1, i ), 1, z( 1, j ), 1 )
540  IF( info.NE.0 ) THEN
541  itmp1 = ifail( i )
542  ifail( i ) = ifail( j )
543  ifail( j ) = itmp1
544  END IF
545  END IF
546  50 CONTINUE
547  END IF
548 *
549  RETURN
550 *
551 * End of CHBEVX
552 *
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 clascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: clascl.f:145
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine chbtrd(VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
CHBTRD
Definition: chbtrd.f:165
subroutine csteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
CSTEQR
Definition: csteqr.f:134
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:160
subroutine cstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
CSTEIN
Definition: cstein.f:184
real function clanhb(NORM, UPLO, N, K, AB, LDAB, WORK)
CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix.
Definition: clanhb.f:134
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:55
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:52
subroutine cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:52
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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