LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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◆ zhbevd()

subroutine zhbevd ( character jobz,
character uplo,
integer n,
integer kd,
complex*16, dimension( ldab, * ) ab,
integer ldab,
double precision, dimension( * ) w,
complex*16, dimension( ldz, * ) z,
integer ldz,
complex*16, dimension( * ) work,
integer lwork,
double precision, dimension( * ) rwork,
integer lrwork,
integer, dimension( * ) iwork,
integer liwork,
integer info )

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

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

Purpose:
!>
!> ZHBEVD computes all the eigenvalues and, optionally, eigenvectors of
!> a complex Hermitian band matrix A.  If eigenvectors are desired, it
!> uses a divide and conquer algorithm.
!>
!>
Parameters
[in]JOBZ
!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors.
!>
[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*16 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.  If UPLO = 'U', the first
!>          superdiagonal and the diagonal of the tridiagonal matrix T
!>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
!>          the diagonal and first subdiagonal of T are returned in the
!>          first two rows of AB.
!>
[in]LDAB
!>          LDAB is INTEGER
!>          The leading dimension of the array AB.  LDAB >= KD + 1.
!>
[out]W
!>          W is DOUBLE PRECISION array, dimension (N)
!>          If INFO = 0, the eigenvalues in ascending order.
!>
[out]Z
!>          Z is COMPLEX*16 array, dimension (LDZ, N)
!>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
!>          eigenvectors of the matrix A, with the i-th column of Z
!>          holding the eigenvector associated with W(i).
!>          If JOBZ = 'N', then Z is not referenced.
!>
[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*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
[in]LWORK
!>          LWORK is INTEGER
!>          The dimension of the array WORK.
!>          If N <= 1,               LWORK must be at least 1.
!>          If JOBZ = 'N' and N > 1, LWORK must be at least N.
!>          If JOBZ = 'V' and N > 1, LWORK must be at least 2*N**2.
!>
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal sizes of the WORK, RWORK and
!>          IWORK arrays, returns these values as the first entries of
!>          the WORK, RWORK and IWORK arrays, and no error message
!>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
!>
[out]RWORK
!>          RWORK is DOUBLE PRECISION array,
!>                                         dimension (LRWORK)
!>          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
!>
[in]LRWORK
!>          LRWORK is INTEGER
!>          The dimension of array RWORK.
!>          If N <= 1,               LRWORK must be at least 1.
!>          If JOBZ = 'N' and N > 1, LRWORK must be at least N.
!>          If JOBZ = 'V' and N > 1, LRWORK must be at least
!>                        1 + 5*N + 2*N**2.
!>
!>          If LRWORK = -1, then a workspace query is assumed; the
!>          routine only calculates the optimal sizes of the WORK, RWORK
!>          and IWORK arrays, returns these values as the first entries
!>          of the WORK, RWORK and IWORK arrays, and no error message
!>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
!>
[out]IWORK
!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
!>
[in]LIWORK
!>          LIWORK is INTEGER
!>          The dimension of array IWORK.
!>          If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
!>          If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .
!>
!>          If LIWORK = -1, then a workspace query is assumed; the
!>          routine only calculates the optimal sizes of the WORK, RWORK
!>          and IWORK arrays, returns these values as the first entries
!>          of the WORK, RWORK and IWORK arrays, and no error message
!>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
!>
[out]INFO
!>          INFO is INTEGER
!>          = 0:  successful exit.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value.
!>          > 0:  if INFO = i, the algorithm failed to converge; i
!>                off-diagonal elements of an intermediate tridiagonal
!>                form did not converge to zero.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 205 of file zhbevd.f.

208*
209* -- LAPACK driver routine --
210* -- LAPACK is a software package provided by Univ. of Tennessee, --
211* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
212*
213* .. Scalar Arguments ..
214 CHARACTER JOBZ, UPLO
215 INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
216* ..
217* .. Array Arguments ..
218 INTEGER IWORK( * )
219 DOUBLE PRECISION RWORK( * ), W( * )
220 COMPLEX*16 AB( LDAB, * ), WORK( * ), Z( LDZ, * )
221* ..
222*
223* =====================================================================
224*
225* .. Parameters ..
226 DOUBLE PRECISION ZERO, ONE
227 parameter( zero = 0.0d0, one = 1.0d0 )
228 COMPLEX*16 CZERO, CONE
229 parameter( czero = ( 0.0d0, 0.0d0 ),
230 $ cone = ( 1.0d0, 0.0d0 ) )
231* ..
232* .. Local Scalars ..
233 LOGICAL LOWER, LQUERY, WANTZ
234 INTEGER IINFO, IMAX, INDE, INDWK2, INDWRK, ISCALE,
235 $ LIWMIN, LLRWK, LLWK2, LRWMIN, LWMIN
236 DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
237 $ SMLNUM
238* ..
239* .. External Functions ..
240 LOGICAL LSAME
241 DOUBLE PRECISION DLAMCH, ZLANHB
242 EXTERNAL lsame, dlamch, zlanhb
243* ..
244* .. External Subroutines ..
245 EXTERNAL dscal, dsterf, xerbla, zgemm, zhbtrd,
246 $ zlacpy,
247 $ zlascl, zstedc
248* ..
249* .. Intrinsic Functions ..
250 INTRINSIC sqrt
251* ..
252* .. Executable Statements ..
253*
254* Test the input parameters.
255*
256 wantz = lsame( jobz, 'V' )
257 lower = lsame( uplo, 'L' )
258 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 .OR. lrwork.EQ.-1 )
259*
260 info = 0
261 IF( n.LE.1 ) THEN
262 lwmin = 1
263 lrwmin = 1
264 liwmin = 1
265 ELSE
266 IF( wantz ) THEN
267 lwmin = 2*n**2
268 lrwmin = 1 + 5*n + 2*n**2
269 liwmin = 3 + 5*n
270 ELSE
271 lwmin = n
272 lrwmin = n
273 liwmin = 1
274 END IF
275 END IF
276 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
277 info = -1
278 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
279 info = -2
280 ELSE IF( n.LT.0 ) THEN
281 info = -3
282 ELSE IF( kd.LT.0 ) THEN
283 info = -4
284 ELSE IF( ldab.LT.kd+1 ) THEN
285 info = -6
286 ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
287 info = -9
288 END IF
289*
290 IF( info.EQ.0 ) THEN
291 work( 1 ) = lwmin
292 rwork( 1 ) = real( lrwmin )
293 iwork( 1 ) = liwmin
294*
295 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
296 info = -11
297 ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
298 info = -13
299 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
300 info = -15
301 END IF
302 END IF
303*
304 IF( info.NE.0 ) THEN
305 CALL xerbla( 'ZHBEVD', -info )
306 RETURN
307 ELSE IF( lquery ) THEN
308 RETURN
309 END IF
310*
311* Quick return if possible
312*
313 IF( n.EQ.0 )
314 $ RETURN
315*
316 IF( n.EQ.1 ) THEN
317 w( 1 ) = dble( ab( 1, 1 ) )
318 IF( wantz )
319 $ z( 1, 1 ) = cone
320 RETURN
321 END IF
322*
323* Get machine constants.
324*
325 safmin = dlamch( 'Safe minimum' )
326 eps = dlamch( 'Precision' )
327 smlnum = safmin / eps
328 bignum = one / smlnum
329 rmin = sqrt( smlnum )
330 rmax = sqrt( bignum )
331*
332* Scale matrix to allowable range, if necessary.
333*
334 anrm = zlanhb( 'M', uplo, n, kd, ab, ldab, rwork )
335 iscale = 0
336 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
337 iscale = 1
338 sigma = rmin / anrm
339 ELSE IF( anrm.GT.rmax ) THEN
340 iscale = 1
341 sigma = rmax / anrm
342 END IF
343 IF( iscale.EQ.1 ) THEN
344 IF( lower ) THEN
345 CALL zlascl( 'B', kd, kd, one, sigma, n, n, ab, ldab,
346 $ info )
347 ELSE
348 CALL zlascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab,
349 $ info )
350 END IF
351 END IF
352*
353* Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form.
354*
355 inde = 1
356 indwrk = inde + n
357 indwk2 = 1 + n*n
358 llwk2 = lwork - indwk2 + 1
359 llrwk = lrwork - indwrk + 1
360 CALL zhbtrd( jobz, uplo, n, kd, ab, ldab, w, rwork( inde ), z,
361 $ ldz, work, iinfo )
362*
363* For eigenvalues only, call DSTERF. For eigenvectors, call ZSTEDC.
364*
365 IF( .NOT.wantz ) THEN
366 CALL dsterf( n, w, rwork( inde ), info )
367 ELSE
368 CALL zstedc( 'I', n, w, rwork( inde ), work, n,
369 $ work( indwk2 ),
370 $ llwk2, rwork( indwrk ), llrwk, iwork, liwork,
371 $ info )
372 CALL zgemm( 'N', 'N', n, n, n, cone, z, ldz, work, n, czero,
373 $ work( indwk2 ), n )
374 CALL zlacpy( 'A', n, n, work( indwk2 ), n, z, ldz )
375 END IF
376*
377* If matrix was scaled, then rescale eigenvalues appropriately.
378*
379 IF( iscale.EQ.1 ) THEN
380 IF( info.EQ.0 ) THEN
381 imax = n
382 ELSE
383 imax = info - 1
384 END IF
385 CALL dscal( imax, one / sigma, w, 1 )
386 END IF
387*
388 work( 1 ) = lwmin
389 rwork( 1 ) = real( lrwmin )
390 iwork( 1 ) = liwmin
391 RETURN
392*
393* End of ZHBEVD
394*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
zgemm
subroutine zgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
ZGEMM
Definition zgemm.f:188
zhbtrd
subroutine zhbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
ZHBTRD
Definition zhbtrd.f:161
zlacpy
subroutine zlacpy(uplo, m, n, a, lda, b, ldb)
ZLACPY copies all or part of one two-dimensional array to another.
Definition zlacpy.f:101
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
zlanhb
double precision function zlanhb(norm, uplo, n, k, ab, ldab, work)
ZLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition zlanhb.f:130
zlascl
subroutine zlascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition zlascl.f:142
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
dscal
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
zstedc
subroutine zstedc(compz, n, d, e, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)
ZSTEDC
Definition zstedc.f:204
dsterf
subroutine dsterf(n, d, e, info)
DSTERF
Definition dsterf.f:84
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