 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ chbgvd()

 subroutine chbgvd ( character JOBZ, character UPLO, integer N, integer KA, integer KB, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldbb, * ) BB, integer LDBB, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

CHBGVD

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Purpose:
``` CHBGVD 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 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 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 CPBSTF.``` [in] LDBB ``` LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.``` [out] W ``` W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.``` [out] Z ``` Z is COMPLEX 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 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 REAL 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 CPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.```
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 249 of file chbgvd.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  REAL RWORK( * ), W( * )
265  COMPLEX AB( LDAB, * ), BB( LDBB, * ), WORK( * ),
266  \$ Z( LDZ, * )
267 * ..
268 *
269 * =====================================================================
270 *
271 * .. Parameters ..
272  COMPLEX CONE, CZERO
273  parameter( cone = ( 1.0e+0, 0.0e+0 ),
274  \$ czero = ( 0.0e+0, 0.0e+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 ssterf, xerbla, cgemm, chbgst, chbtrd, clacpy,
288  \$ cpbstf, cstedc
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( 'CHBGVD', -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 cpbstf( 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 chbgst( 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 chbtrd( vect, uplo, n, ka, ab, ldab, w, rwork( inde ), z,
382  \$ ldz, work, iinfo )
383 *
384 * For eigenvalues only, call SSTERF. For eigenvectors, call CSTEDC.
385 *
386  IF( .NOT.wantz ) THEN
387  CALL ssterf( n, w, rwork( inde ), info )
388  ELSE
389  CALL cstedc( 'I', n, w, rwork( inde ), work, n, work( indwk2 ),
390  \$ llwk2, rwork( indwrk ), llrwk, iwork, liwork,
391  \$ info )
392  CALL cgemm( 'N', 'N', n, n, n, cone, z, ldz, work, n, czero,
393  \$ work( indwk2 ), n )
394  CALL clacpy( '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 CHBGVD
403 *
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 cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine cpbstf(UPLO, N, KD, AB, LDAB, INFO)
CPBSTF
Definition: cpbstf.f:153
subroutine chbtrd(VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
CHBTRD
Definition: chbtrd.f:163
subroutine chbgst(VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, RWORK, INFO)
CHBGST
Definition: chbgst.f:165
subroutine cstedc(COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
CSTEDC
Definition: cstedc.f:212
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