LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ zhbgvd()

subroutine zhbgvd ( character  JOBZ,
character  UPLO,
integer  N,
integer  KA,
integer  KB,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
complex*16, dimension( ldbb, * )  BB,
integer  LDBB,
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 
)

ZHBGVD

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Purpose:
 ZHBGVD computes all the eigenvalues, and optionally, the eigenvectors
 of a complex generalized Hermitian-definite banded eigenproblem, of
 the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
 and banded, and B is also positive definite.  If eigenvectors are
 desired, it uses a divide and conquer algorithm.

 The divide and conquer algorithm makes very mild assumptions about
 floating point arithmetic. It will work on machines with a guard
 digit in add/subtract, or on those binary machines without guard
 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 Cray-2. It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.
Parameters
[in]JOBZ
          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.
[in]UPLO
          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.
[in]N
          N is INTEGER
          The order of the matrices A and B.  N >= 0.
[in]KA
          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'. KA >= 0.
[in]KB
          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'. KB >= 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 ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.
[in,out]BB
          BB is COMPLEX*16 array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the Hermitian band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**H*S, as returned by ZPBSTF.
[in]LDBB
          LDBB is INTEGER
          The leading dimension of the array BB.  LDBB >= KB+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 matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i). The eigenvectors are
          normalized so that Z**H*B*Z = 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 >= 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 >= 1.
          If JOBZ = 'N' and N > 1, LWORK >= N.
          If JOBZ = 'V' and N > 1, LWORK >= 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 (MAX(1,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 >= 1.
          If JOBZ = 'N' and N > 1, LRWORK >= N.
          If JOBZ = 'V' and N > 1, LRWORK >= 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 >= 1.
          If JOBZ = 'V' and N > 1, LIWORK >= 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, and i is:
             <= N:  the algorithm failed to converge:
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then ZPBSTF
                    returned INFO = i: B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 249 of file zhbgvd.f.

252 *
253 * -- LAPACK driver routine --
254 * -- LAPACK is a software package provided by Univ. of Tennessee, --
255 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
256 *
257 * .. Scalar Arguments ..
258  CHARACTER JOBZ, UPLO
259  INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LRWORK,
260  $ LWORK, N
261 * ..
262 * .. Array Arguments ..
263  INTEGER IWORK( * )
264  DOUBLE PRECISION RWORK( * ), W( * )
265  COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), WORK( * ),
266  $ Z( LDZ, * )
267 * ..
268 *
269 * =====================================================================
270 *
271 * .. Parameters ..
272  COMPLEX*16 CONE, CZERO
273  parameter( cone = ( 1.0d+0, 0.0d+0 ),
274  $ czero = ( 0.0d+0, 0.0d+0 ) )
275 * ..
276 * .. Local Scalars ..
277  LOGICAL LQUERY, UPPER, WANTZ
278  CHARACTER VECT
279  INTEGER IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLRWK,
280  $ LLWK2, LRWMIN, LWMIN
281 * ..
282 * .. External Functions ..
283  LOGICAL LSAME
284  EXTERNAL lsame
285 * ..
286 * .. External Subroutines ..
287  EXTERNAL dsterf, xerbla, zgemm, zhbgst, zhbtrd, zlacpy,
288  $ zpbstf, zstedc
289 * ..
290 * .. Executable Statements ..
291 *
292 * Test the input parameters.
293 *
294  wantz = lsame( jobz, 'V' )
295  upper = lsame( uplo, 'U' )
296  lquery = ( lwork.EQ.-1 .OR. lrwork.EQ.-1 .OR. liwork.EQ.-1 )
297 *
298  info = 0
299  IF( n.LE.1 ) THEN
300  lwmin = 1+n
301  lrwmin = 1+n
302  liwmin = 1
303  ELSE IF( wantz ) THEN
304  lwmin = 2*n**2
305  lrwmin = 1 + 5*n + 2*n**2
306  liwmin = 3 + 5*n
307  ELSE
308  lwmin = n
309  lrwmin = n
310  liwmin = 1
311  END IF
312  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
313  info = -1
314  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
315  info = -2
316  ELSE IF( n.LT.0 ) THEN
317  info = -3
318  ELSE IF( ka.LT.0 ) THEN
319  info = -4
320  ELSE IF( kb.LT.0 .OR. kb.GT.ka ) THEN
321  info = -5
322  ELSE IF( ldab.LT.ka+1 ) THEN
323  info = -7
324  ELSE IF( ldbb.LT.kb+1 ) THEN
325  info = -9
326  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
327  info = -12
328  END IF
329 *
330  IF( info.EQ.0 ) THEN
331  work( 1 ) = lwmin
332  rwork( 1 ) = lrwmin
333  iwork( 1 ) = liwmin
334 *
335  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
336  info = -14
337  ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
338  info = -16
339  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
340  info = -18
341  END IF
342  END IF
343 *
344  IF( info.NE.0 ) THEN
345  CALL xerbla( 'ZHBGVD', -info )
346  RETURN
347  ELSE IF( lquery ) THEN
348  RETURN
349  END IF
350 *
351 * Quick return if possible
352 *
353  IF( n.EQ.0 )
354  $ RETURN
355 *
356 * Form a split Cholesky factorization of B.
357 *
358  CALL zpbstf( uplo, n, kb, bb, ldbb, info )
359  IF( info.NE.0 ) THEN
360  info = n + info
361  RETURN
362  END IF
363 *
364 * Transform problem to standard eigenvalue problem.
365 *
366  inde = 1
367  indwrk = inde + n
368  indwk2 = 1 + n*n
369  llwk2 = lwork - indwk2 + 2
370  llrwk = lrwork - indwrk + 2
371  CALL zhbgst( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, z, ldz,
372  $ work, rwork, iinfo )
373 *
374 * Reduce Hermitian band matrix to tridiagonal form.
375 *
376  IF( wantz ) THEN
377  vect = 'U'
378  ELSE
379  vect = 'N'
380  END IF
381  CALL zhbtrd( vect, uplo, n, ka, ab, ldab, w, rwork( inde ), z,
382  $ ldz, work, iinfo )
383 *
384 * For eigenvalues only, call DSTERF. For eigenvectors, call ZSTEDC.
385 *
386  IF( .NOT.wantz ) THEN
387  CALL dsterf( n, w, rwork( inde ), info )
388  ELSE
389  CALL zstedc( 'I', n, w, rwork( inde ), work, n, work( indwk2 ),
390  $ llwk2, rwork( indwrk ), llrwk, iwork, liwork,
391  $ info )
392  CALL zgemm( 'N', 'N', n, n, n, cone, z, ldz, work, n, czero,
393  $ work( indwk2 ), n )
394  CALL zlacpy( 'A', n, n, work( indwk2 ), n, z, ldz )
395  END IF
396 *
397  work( 1 ) = lwmin
398  rwork( 1 ) = lrwmin
399  iwork( 1 ) = liwmin
400  RETURN
401 *
402 * End of ZHBGVD
403 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:86
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zhbgst(VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, RWORK, INFO)
ZHBGST
Definition: zhbgst.f:165
subroutine zpbstf(UPLO, N, KD, AB, LDAB, INFO)
ZPBSTF
Definition: zpbstf.f:153
subroutine zstedc(COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
ZSTEDC
Definition: zstedc.f:212
subroutine zhbtrd(VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
ZHBTRD
Definition: zhbtrd.f:163
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