LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zhbev_2stage()

 subroutine zhbev_2stage ( 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 INFO )

ZHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Download ZHBEV_2STAGE + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` ZHBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
a complex Hermitian band matrix A using the 2stage technique for
the reduction to tridiagonal.```
Parameters
 [in] JOBZ ``` JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.``` [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 LWORK On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS where KD is the size of the band. NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available. 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(1,3*N-2))` [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.```
Further Details:
```  All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Denver, Colorado, USA, 2013.
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196 ```

Definition at line 209 of file zhbev_2stage.f.

211 *
212  IMPLICIT NONE
213 *
214 * -- LAPACK driver routine --
215 * -- LAPACK is a software package provided by Univ. of Tennessee, --
216 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
217 *
218 * .. Scalar Arguments ..
219  CHARACTER JOBZ, UPLO
220  INTEGER INFO, KD, LDAB, LDZ, N, LWORK
221 * ..
222 * .. Array Arguments ..
223  DOUBLE PRECISION RWORK( * ), W( * )
224  COMPLEX*16 AB( LDAB, * ), WORK( * ), Z( LDZ, * )
225 * ..
226 *
227 * =====================================================================
228 *
229 * .. Parameters ..
230  DOUBLE PRECISION ZERO, ONE
231  parameter( zero = 0.0d0, one = 1.0d0 )
232 * ..
233 * .. Local Scalars ..
234  LOGICAL LOWER, WANTZ, LQUERY
235  INTEGER IINFO, IMAX, INDE, INDWRK, INDRWK, ISCALE,
236  \$ LLWORK, LWMIN, LHTRD, LWTRD, IB, INDHOUS
237  DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
238  \$ SMLNUM
239 * ..
240 * .. External Functions ..
241  LOGICAL LSAME
242  INTEGER ILAENV2STAGE
243  DOUBLE PRECISION DLAMCH, ZLANHB
244  EXTERNAL lsame, dlamch, zlanhb, ilaenv2stage
245 * ..
246 * .. External Subroutines ..
247  EXTERNAL dscal, dsterf, xerbla, zlascl, zsteqr,
249 * ..
250 * .. Intrinsic Functions ..
251  INTRINSIC dble, sqrt
252 * ..
253 * .. Executable Statements ..
254 *
255 * Test the input parameters.
256 *
257  wantz = lsame( jobz, 'V' )
258  lower = lsame( uplo, 'L' )
259  lquery = ( lwork.EQ.-1 )
260 *
261  info = 0
262  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
263  info = -1
264  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
265  info = -2
266  ELSE IF( n.LT.0 ) THEN
267  info = -3
268  ELSE IF( kd.LT.0 ) THEN
269  info = -4
270  ELSE IF( ldab.LT.kd+1 ) THEN
271  info = -6
272  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
273  info = -9
274  END IF
275 *
276  IF( info.EQ.0 ) THEN
277  IF( n.LE.1 ) THEN
278  lwmin = 1
279  work( 1 ) = lwmin
280  ELSE
281  ib = ilaenv2stage( 2, 'ZHETRD_HB2ST', jobz,
282  \$ n, kd, -1, -1 )
283  lhtrd = ilaenv2stage( 3, 'ZHETRD_HB2ST', jobz,
284  \$ n, kd, ib, -1 )
285  lwtrd = ilaenv2stage( 4, 'ZHETRD_HB2ST', jobz,
286  \$ n, kd, ib, -1 )
287  lwmin = lhtrd + lwtrd
288  work( 1 ) = lwmin
289  ENDIF
290 *
291  IF( lwork.LT.lwmin .AND. .NOT.lquery )
292  \$ info = -11
293  END IF
294 *
295  IF( info.NE.0 ) THEN
296  CALL xerbla( 'ZHBEV_2STAGE ', -info )
297  RETURN
298  ELSE IF( lquery ) THEN
299  RETURN
300  END IF
301 *
302 * Quick return if possible
303 *
304  IF( n.EQ.0 )
305  \$ RETURN
306 *
307  IF( n.EQ.1 ) THEN
308  IF( lower ) THEN
309  w( 1 ) = dble( ab( 1, 1 ) )
310  ELSE
311  w( 1 ) = dble( ab( kd+1, 1 ) )
312  END IF
313  IF( wantz )
314  \$ z( 1, 1 ) = one
315  RETURN
316  END IF
317 *
318 * Get machine constants.
319 *
320  safmin = dlamch( 'Safe minimum' )
321  eps = dlamch( 'Precision' )
322  smlnum = safmin / eps
323  bignum = one / smlnum
324  rmin = sqrt( smlnum )
325  rmax = sqrt( bignum )
326 *
327 * Scale matrix to allowable range, if necessary.
328 *
329  anrm = zlanhb( 'M', uplo, n, kd, ab, ldab, rwork )
330  iscale = 0
331  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
332  iscale = 1
333  sigma = rmin / anrm
334  ELSE IF( anrm.GT.rmax ) THEN
335  iscale = 1
336  sigma = rmax / anrm
337  END IF
338  IF( iscale.EQ.1 ) THEN
339  IF( lower ) THEN
340  CALL zlascl( 'B', kd, kd, one, sigma, n, n, ab, ldab, info )
341  ELSE
342  CALL zlascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab, info )
343  END IF
344  END IF
345 *
346 * Call ZHBTRD_HB2ST to reduce Hermitian band matrix to tridiagonal form.
347 *
348  inde = 1
349  indhous = 1
350  indwrk = indhous + lhtrd
351  llwork = lwork - indwrk + 1
352 *
353  CALL zhetrd_hb2st( "N", jobz, uplo, n, kd, ab, ldab, w,
354  \$ rwork( inde ), work( indhous ), lhtrd,
355  \$ work( indwrk ), llwork, iinfo )
356 *
357 * For eigenvalues only, call DSTERF. For eigenvectors, call ZSTEQR.
358 *
359  IF( .NOT.wantz ) THEN
360  CALL dsterf( n, w, rwork( inde ), info )
361  ELSE
362  indrwk = inde + n
363  CALL zsteqr( jobz, n, w, rwork( inde ), z, ldz,
364  \$ rwork( indrwk ), info )
365  END IF
366 *
367 * If matrix was scaled, then rescale eigenvalues appropriately.
368 *
369  IF( iscale.EQ.1 ) THEN
370  IF( info.EQ.0 ) THEN
371  imax = n
372  ELSE
373  imax = info - 1
374  END IF
375  CALL dscal( imax, one / sigma, w, 1 )
376  END IF
377 *
378 * Set WORK(1) to optimal workspace size.
379 *
380  work( 1 ) = lwmin
381 *
382  RETURN
383 *
384 * End of ZHBEV_2STAGE
385 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function ilaenv2stage(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV2STAGE
Definition: ilaenv2stage.f:149
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 zhetrd_2stage(VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
ZHETRD_2STAGE
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:143
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:132
subroutine zsteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
ZSTEQR
Definition: zsteqr.f:132
subroutine zhetrd_hb2st(STAGE1, VECT, UPLO, N, KD, AB, LDAB, D, E, HOUS, LHOUS, WORK, LWORK, INFO)
ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T
Definition: zhetrd_hb2st.F:230
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
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