LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zhbevd_2stage()

 subroutine zhbevd_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 LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

ZHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:
``` ZHBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
a complex Hermitian band matrix A using the 2stage technique for
the reduction to tridiagonal.  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. 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 (MAX(1,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 (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.```
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).
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 257 of file zhbevd_2stage.f.

260 *
261  IMPLICIT NONE
262 *
263 * -- LAPACK driver routine --
264 * -- LAPACK is a software package provided by Univ. of Tennessee, --
265 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
266 *
267 * .. Scalar Arguments ..
268  CHARACTER JOBZ, UPLO
269  INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
270 * ..
271 * .. Array Arguments ..
272  INTEGER IWORK( * )
273  DOUBLE PRECISION RWORK( * ), W( * )
274  COMPLEX*16 AB( LDAB, * ), WORK( * ), Z( LDZ, * )
275 * ..
276 *
277 * =====================================================================
278 *
279 * .. Parameters ..
280  DOUBLE PRECISION ZERO, ONE
281  parameter( zero = 0.0d0, one = 1.0d0 )
282  COMPLEX*16 CZERO, CONE
283  parameter( czero = ( 0.0d0, 0.0d0 ),
284  \$ cone = ( 1.0d0, 0.0d0 ) )
285 * ..
286 * .. Local Scalars ..
287  LOGICAL LOWER, LQUERY, WANTZ
288  INTEGER IINFO, IMAX, INDE, INDWK2, INDRWK, ISCALE,
289  \$ LLWORK, INDWK, LHTRD, LWTRD, IB, INDHOUS,
290  \$ LIWMIN, LLRWK, LLWK2, LRWMIN, LWMIN
291  DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
292  \$ SMLNUM
293 * ..
294 * .. External Functions ..
295  LOGICAL LSAME
296  INTEGER ILAENV2STAGE
297  DOUBLE PRECISION DLAMCH, ZLANHB
298  EXTERNAL lsame, dlamch, zlanhb, ilaenv2stage
299 * ..
300 * .. External Subroutines ..
301  EXTERNAL dscal, dsterf, xerbla, zgemm, zlacpy,
303 * ..
304 * .. Intrinsic Functions ..
305  INTRINSIC dble, sqrt
306 * ..
307 * .. Executable Statements ..
308 *
309 * Test the input parameters.
310 *
311  wantz = lsame( jobz, 'V' )
312  lower = lsame( uplo, 'L' )
313  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 .OR. lrwork.EQ.-1 )
314 *
315  info = 0
316  IF( n.LE.1 ) THEN
317  lwmin = 1
318  lrwmin = 1
319  liwmin = 1
320  ELSE
321  ib = ilaenv2stage( 2, 'ZHETRD_HB2ST', jobz, n, kd, -1, -1 )
322  lhtrd = ilaenv2stage( 3, 'ZHETRD_HB2ST', jobz, n, kd, ib, -1 )
323  lwtrd = ilaenv2stage( 4, 'ZHETRD_HB2ST', jobz, n, kd, ib, -1 )
324  IF( wantz ) THEN
325  lwmin = 2*n**2
326  lrwmin = 1 + 5*n + 2*n**2
327  liwmin = 3 + 5*n
328  ELSE
329  lwmin = max( n, lhtrd + lwtrd )
330  lrwmin = n
331  liwmin = 1
332  END IF
333  END IF
334  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
335  info = -1
336  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
337  info = -2
338  ELSE IF( n.LT.0 ) THEN
339  info = -3
340  ELSE IF( kd.LT.0 ) THEN
341  info = -4
342  ELSE IF( ldab.LT.kd+1 ) THEN
343  info = -6
344  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
345  info = -9
346  END IF
347 *
348  IF( info.EQ.0 ) THEN
349  work( 1 ) = lwmin
350  rwork( 1 ) = lrwmin
351  iwork( 1 ) = liwmin
352 *
353  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
354  info = -11
355  ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
356  info = -13
357  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
358  info = -15
359  END IF
360  END IF
361 *
362  IF( info.NE.0 ) THEN
363  CALL xerbla( 'ZHBEVD_2STAGE', -info )
364  RETURN
365  ELSE IF( lquery ) THEN
366  RETURN
367  END IF
368 *
369 * Quick return if possible
370 *
371  IF( n.EQ.0 )
372  \$ RETURN
373 *
374  IF( n.EQ.1 ) THEN
375  w( 1 ) = dble( ab( 1, 1 ) )
376  IF( wantz )
377  \$ z( 1, 1 ) = cone
378  RETURN
379  END IF
380 *
381 * Get machine constants.
382 *
383  safmin = dlamch( 'Safe minimum' )
384  eps = dlamch( 'Precision' )
385  smlnum = safmin / eps
386  bignum = one / smlnum
387  rmin = sqrt( smlnum )
388  rmax = sqrt( bignum )
389 *
390 * Scale matrix to allowable range, if necessary.
391 *
392  anrm = zlanhb( 'M', uplo, n, kd, ab, ldab, rwork )
393  iscale = 0
394  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
395  iscale = 1
396  sigma = rmin / anrm
397  ELSE IF( anrm.GT.rmax ) THEN
398  iscale = 1
399  sigma = rmax / anrm
400  END IF
401  IF( iscale.EQ.1 ) THEN
402  IF( lower ) THEN
403  CALL zlascl( 'B', kd, kd, one, sigma, n, n, ab, ldab, info )
404  ELSE
405  CALL zlascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab, info )
406  END IF
407  END IF
408 *
409 * Call ZHBTRD_HB2ST to reduce Hermitian band matrix to tridiagonal form.
410 *
411  inde = 1
412  indrwk = inde + n
413  llrwk = lrwork - indrwk + 1
414  indhous = 1
415  indwk = indhous + lhtrd
416  llwork = lwork - indwk + 1
417  indwk2 = indwk + n*n
418  llwk2 = lwork - indwk2 + 1
419 *
420  CALL zhetrd_hb2st( "N", jobz, uplo, n, kd, ab, ldab, w,
421  \$ rwork( inde ), work( indhous ), lhtrd,
422  \$ work( indwk ), llwork, iinfo )
423 *
424 * For eigenvalues only, call DSTERF. For eigenvectors, call ZSTEDC.
425 *
426  IF( .NOT.wantz ) THEN
427  CALL dsterf( n, w, rwork( inde ), info )
428  ELSE
429  CALL zstedc( 'I', n, w, rwork( inde ), work, n, work( indwk2 ),
430  \$ llwk2, rwork( indrwk ), llrwk, iwork, liwork,
431  \$ info )
432  CALL zgemm( 'N', 'N', n, n, n, cone, z, ldz, work, n, czero,
433  \$ work( indwk2 ), n )
434  CALL zlacpy( 'A', n, n, work( indwk2 ), n, z, ldz )
435  END IF
436 *
437 * If matrix was scaled, then rescale eigenvalues appropriately.
438 *
439  IF( iscale.EQ.1 ) THEN
440  IF( info.EQ.0 ) THEN
441  imax = n
442  ELSE
443  imax = info - 1
444  END IF
445  CALL dscal( imax, one / sigma, w, 1 )
446  END IF
447 *
448  work( 1 ) = lwmin
449  rwork( 1 ) = lrwmin
450  iwork( 1 ) = liwmin
451  RETURN
452 *
453 * End of ZHBEVD_2STAGE
454 *
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 zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
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
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
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 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 zstedc(COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
ZSTEDC
Definition: zstedc.f:212
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
Here is the call graph for this function:
Here is the caller graph for this function: