LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 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

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.```
Date
June 2016
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 254 of file zhbgvd.f.

254 *
255 * -- LAPACK driver routine (version 3.6.1) --
256 * -- LAPACK is a software package provided by Univ. of Tennessee, --
257 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
258 * June 2016
259 *
260 * .. Scalar Arguments ..
261  CHARACTER jobz, uplo
262  INTEGER info, ka, kb, ldab, ldbb, ldz, liwork, lrwork,
263  \$ lwork, n
264 * ..
265 * .. Array Arguments ..
266  INTEGER iwork( * )
267  DOUBLE PRECISION rwork( * ), w( * )
268  COMPLEX*16 ab( ldab, * ), bb( ldbb, * ), work( * ),
269  \$ z( ldz, * )
270 * ..
271 *
272 * =====================================================================
273 *
274 * .. Parameters ..
275  COMPLEX*16 cone, czero
276  parameter ( cone = ( 1.0d+0, 0.0d+0 ),
277  \$ czero = ( 0.0d+0, 0.0d+0 ) )
278 * ..
279 * .. Local Scalars ..
280  LOGICAL lquery, upper, wantz
281  CHARACTER vect
282  INTEGER iinfo, inde, indwk2, indwrk, liwmin, llrwk,
283  \$ llwk2, lrwmin, lwmin
284 * ..
285 * .. External Functions ..
286  LOGICAL lsame
287  EXTERNAL lsame
288 * ..
289 * .. External Subroutines ..
290  EXTERNAL dsterf, xerbla, zgemm, zhbgst, zhbtrd, zlacpy,
291  \$ zpbstf, zstedc
292 * ..
293 * .. Executable Statements ..
294 *
295 * Test the input parameters.
296 *
297  wantz = lsame( jobz, 'V' )
298  upper = lsame( uplo, 'U' )
299  lquery = ( lwork.EQ.-1 .OR. lrwork.EQ.-1 .OR. liwork.EQ.-1 )
300 *
301  info = 0
302  IF( n.LE.1 ) THEN
303  lwmin = 1+n
304  lrwmin = 1+n
305  liwmin = 1
306  ELSE IF( wantz ) THEN
307  lwmin = 2*n**2
308  lrwmin = 1 + 5*n + 2*n**2
309  liwmin = 3 + 5*n
310  ELSE
311  lwmin = n
312  lrwmin = n
313  liwmin = 1
314  END IF
315  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
316  info = -1
317  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
318  info = -2
319  ELSE IF( n.LT.0 ) THEN
320  info = -3
321  ELSE IF( ka.LT.0 ) THEN
322  info = -4
323  ELSE IF( kb.LT.0 .OR. kb.GT.ka ) THEN
324  info = -5
325  ELSE IF( ldab.LT.ka+1 ) THEN
326  info = -7
327  ELSE IF( ldbb.LT.kb+1 ) THEN
328  info = -9
329  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
330  info = -12
331  END IF
332 *
333  IF( info.EQ.0 ) THEN
334  work( 1 ) = lwmin
335  rwork( 1 ) = lrwmin
336  iwork( 1 ) = liwmin
337 *
338  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
339  info = -14
340  ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
341  info = -16
342  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
343  info = -18
344  END IF
345  END IF
346 *
347  IF( info.NE.0 ) THEN
348  CALL xerbla( 'ZHBGVD', -info )
349  RETURN
350  ELSE IF( lquery ) THEN
351  RETURN
352  END IF
353 *
354 * Quick return if possible
355 *
356  IF( n.EQ.0 )
357  \$ RETURN
358 *
359 * Form a split Cholesky factorization of B.
360 *
361  CALL zpbstf( uplo, n, kb, bb, ldbb, info )
362  IF( info.NE.0 ) THEN
363  info = n + info
364  RETURN
365  END IF
366 *
367 * Transform problem to standard eigenvalue problem.
368 *
369  inde = 1
370  indwrk = inde + n
371  indwk2 = 1 + n*n
372  llwk2 = lwork - indwk2 + 2
373  llrwk = lrwork - indwrk + 2
374  CALL zhbgst( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, z, ldz,
375  \$ work, rwork, iinfo )
376 *
377 * Reduce Hermitian band matrix to tridiagonal form.
378 *
379  IF( wantz ) THEN
380  vect = 'U'
381  ELSE
382  vect = 'N'
383  END IF
384  CALL zhbtrd( vect, uplo, n, ka, ab, ldab, w, rwork( inde ), z,
385  \$ ldz, work, iinfo )
386 *
387 * For eigenvalues only, call DSTERF. For eigenvectors, call ZSTEDC.
388 *
389  IF( .NOT.wantz ) THEN
390  CALL dsterf( n, w, rwork( inde ), info )
391  ELSE
392  CALL zstedc( 'I', n, w, rwork( inde ), work, n, work( indwk2 ),
393  \$ llwk2, rwork( indwrk ), llrwk, iwork, liwork,
394  \$ info )
395  CALL zgemm( 'N', 'N', n, n, n, cone, z, ldz, work, n, czero,
396  \$ work( indwk2 ), n )
397  CALL zlacpy( 'A', n, n, work( indwk2 ), n, z, ldz )
398  END IF
399 *
400  work( 1 ) = lwmin
401  rwork( 1 ) = lrwmin
402  iwork( 1 ) = liwmin
403  RETURN
404 *
405 * End of ZHBGVD
406 *
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:88
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:189
subroutine zhbgst(VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, RWORK, INFO)
ZHBGST
Definition: zhbgst.f:167
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zstedc(COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
ZSTEDC
Definition: zstedc.f:215
subroutine zpbstf(UPLO, N, KD, AB, LDAB, INFO)
ZPBSTF
Definition: zpbstf.f:155
subroutine zhbtrd(VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
ZHBTRD
Definition: zhbtrd.f:165
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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