LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ ssbev_2stage()

 subroutine ssbev_2stage ( character JOBZ, character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO )

SSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Download SSBEV_2STAGE + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` SSBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
a real symmetric 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 REAL array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric 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 REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.``` [out] Z ``` Z is REAL 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 REAL 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 + N 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 size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK 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).
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 202 of file ssbev_2stage.f.

204 *
205  IMPLICIT NONE
206 *
207 * -- LAPACK driver routine --
208 * -- LAPACK is a software package provided by Univ. of Tennessee, --
209 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
210 *
211 * .. Scalar Arguments ..
212  CHARACTER JOBZ, UPLO
213  INTEGER INFO, KD, LDAB, LDZ, N, LWORK
214 * ..
215 * .. Array Arguments ..
216  REAL AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
217 * ..
218 *
219 * =====================================================================
220 *
221 * .. Parameters ..
222  REAL ZERO, ONE
223  parameter( zero = 0.0e0, one = 1.0e0 )
224 * ..
225 * .. Local Scalars ..
226  LOGICAL LOWER, WANTZ, LQUERY
227  INTEGER IINFO, IMAX, INDE, INDWRK, ISCALE,
228  \$ LLWORK, LWMIN, LHTRD, LWTRD, IB, INDHOUS
229  REAL ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
230  \$ SMLNUM
231 * ..
232 * .. External Functions ..
233  LOGICAL LSAME
234  INTEGER ILAENV2STAGE
235  REAL SLAMCH, SLANSB
236  EXTERNAL lsame, slamch, slansb, ilaenv2stage
237 * ..
238 * .. External Subroutines ..
239  EXTERNAL slascl, sscal, ssteqr, ssterf, xerbla,
240  \$ ssytrd_sb2st
241 * ..
242 * .. Intrinsic Functions ..
243  INTRINSIC sqrt
244 * ..
245 * .. Executable Statements ..
246 *
247 * Test the input parameters.
248 *
249  wantz = lsame( jobz, 'V' )
250  lower = lsame( uplo, 'L' )
251  lquery = ( lwork.EQ.-1 )
252 *
253  info = 0
254  IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
255  info = -1
256  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
257  info = -2
258  ELSE IF( n.LT.0 ) THEN
259  info = -3
260  ELSE IF( kd.LT.0 ) THEN
261  info = -4
262  ELSE IF( ldab.LT.kd+1 ) THEN
263  info = -6
264  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
265  info = -9
266  END IF
267 *
268  IF( info.EQ.0 ) THEN
269  IF( n.LE.1 ) THEN
270  lwmin = 1
271  work( 1 ) = lwmin
272  ELSE
273  ib = ilaenv2stage( 2, 'SSYTRD_SB2ST', jobz,
274  \$ n, kd, -1, -1 )
275  lhtrd = ilaenv2stage( 3, 'SSYTRD_SB2ST', jobz,
276  \$ n, kd, ib, -1 )
277  lwtrd = ilaenv2stage( 4, 'SSYTRD_SB2ST', jobz,
278  \$ n, kd, ib, -1 )
279  lwmin = n + lhtrd + lwtrd
280  work( 1 ) = lwmin
281  ENDIF
282 *
283  IF( lwork.LT.lwmin .AND. .NOT.lquery )
284  \$ info = -11
285  END IF
286 *
287  IF( info.NE.0 ) THEN
288  CALL xerbla( 'SSBEV_2STAGE ', -info )
289  RETURN
290  ELSE IF( lquery ) THEN
291  RETURN
292  END IF
293 *
294 * Quick return if possible
295 *
296  IF( n.EQ.0 )
297  \$ RETURN
298 *
299  IF( n.EQ.1 ) THEN
300  IF( lower ) THEN
301  w( 1 ) = ab( 1, 1 )
302  ELSE
303  w( 1 ) = ab( kd+1, 1 )
304  END IF
305  IF( wantz )
306  \$ z( 1, 1 ) = one
307  RETURN
308  END IF
309 *
310 * Get machine constants.
311 *
312  safmin = slamch( 'Safe minimum' )
313  eps = slamch( 'Precision' )
314  smlnum = safmin / eps
315  bignum = one / smlnum
316  rmin = sqrt( smlnum )
317  rmax = sqrt( bignum )
318 *
319 * Scale matrix to allowable range, if necessary.
320 *
321  anrm = slansb( 'M', uplo, n, kd, ab, ldab, work )
322  iscale = 0
323  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
324  iscale = 1
325  sigma = rmin / anrm
326  ELSE IF( anrm.GT.rmax ) THEN
327  iscale = 1
328  sigma = rmax / anrm
329  END IF
330  IF( iscale.EQ.1 ) THEN
331  IF( lower ) THEN
332  CALL slascl( 'B', kd, kd, one, sigma, n, n, ab, ldab, info )
333  ELSE
334  CALL slascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab, info )
335  END IF
336  END IF
337 *
338 * Call SSYTRD_SB2ST to reduce symmetric band matrix to tridiagonal form.
339 *
340  inde = 1
341  indhous = inde + n
342  indwrk = indhous + lhtrd
343  llwork = lwork - indwrk + 1
344 *
345  CALL ssytrd_sb2st( "N", jobz, uplo, n, kd, ab, ldab, w,
346  \$ work( inde ), work( indhous ), lhtrd,
347  \$ work( indwrk ), llwork, iinfo )
348 *
349 * For eigenvalues only, call SSTERF. For eigenvectors, call SSTEQR.
350 *
351  IF( .NOT.wantz ) THEN
352  CALL ssterf( n, w, work( inde ), info )
353  ELSE
354  CALL ssteqr( jobz, n, w, work( inde ), z, ldz, work( indwrk ),
355  \$ info )
356  END IF
357 *
358 * If matrix was scaled, then rescale eigenvalues appropriately.
359 *
360  IF( iscale.EQ.1 ) THEN
361  IF( info.EQ.0 ) THEN
362  imax = n
363  ELSE
364  imax = info - 1
365  END IF
366  CALL sscal( imax, one / sigma, w, 1 )
367  END IF
368 *
369 * Set WORK(1) to optimal workspace size.
370 *
371  work( 1 ) = lwmin
372 *
373  RETURN
374 *
375 * End of SSBEV_2STAGE
376 *
subroutine slascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: slascl.f:143
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 ssteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SSTEQR
Definition: ssteqr.f:131
subroutine ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:86
real function slansb(NORM, UPLO, N, K, AB, LDAB, WORK)
SLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slansb.f:129
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
subroutine ssytrd_sb2st(STAGE1, VECT, UPLO, N, KD, AB, LDAB, D, E, HOUS, LHOUS, WORK, LWORK, INFO)
SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T
Definition: ssytrd_sb2st.F:230
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