LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ ssbevd_2stage()

 subroutine ssbevd_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, dimension( * ) IWORK, integer LIWORK, integer INFO )

SSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:
``` SSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
a real symmetric 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 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 sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK 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 the array IWORK. If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 2, 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 and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK 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 232 of file ssbevd_2stage.f.

234*
235 IMPLICIT NONE
236*
237* -- LAPACK driver routine --
238* -- LAPACK is a software package provided by Univ. of Tennessee, --
239* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
240*
241* .. Scalar Arguments ..
242 CHARACTER JOBZ, UPLO
243 INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
244* ..
245* .. Array Arguments ..
246 INTEGER IWORK( * )
247 REAL AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
248* ..
249*
250* =====================================================================
251*
252* .. Parameters ..
253 REAL ZERO, ONE
254 parameter( zero = 0.0e+0, one = 1.0e+0 )
255* ..
256* .. Local Scalars ..
257 LOGICAL LOWER, LQUERY, WANTZ
258 INTEGER IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN,
259 \$ LLWORK, LWMIN, LHTRD, LWTRD, IB, INDHOUS,
260 \$ LLWRK2
261 REAL ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
262 \$ SMLNUM
263* ..
264* .. External Functions ..
265 LOGICAL LSAME
266 INTEGER ILAENV2STAGE
267 REAL SLAMCH, SLANSB
268 EXTERNAL lsame, slamch, slansb, ilaenv2stage
269* ..
270* .. External Subroutines ..
271 EXTERNAL sgemm, slacpy, slascl, sscal, sstedc,
273* ..
274* .. Intrinsic Functions ..
275 INTRINSIC sqrt
276* ..
277* .. Executable Statements ..
278*
279* Test the input parameters.
280*
281 wantz = lsame( jobz, 'V' )
282 lower = lsame( uplo, 'L' )
283 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
284*
285 info = 0
286 IF( n.LE.1 ) THEN
287 liwmin = 1
288 lwmin = 1
289 ELSE
290 ib = ilaenv2stage( 2, 'SSYTRD_SB2ST', jobz, n, kd, -1, -1 )
291 lhtrd = ilaenv2stage( 3, 'SSYTRD_SB2ST', jobz, n, kd, ib, -1 )
292 lwtrd = ilaenv2stage( 4, 'SSYTRD_SB2ST', jobz, n, kd, ib, -1 )
293 IF( wantz ) THEN
294 liwmin = 3 + 5*n
295 lwmin = 1 + 5*n + 2*n**2
296 ELSE
297 liwmin = 1
298 lwmin = max( 2*n, n+lhtrd+lwtrd )
299 END IF
300 END IF
301 IF( .NOT.( lsame( jobz, 'N' ) ) ) THEN
302 info = -1
303 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
304 info = -2
305 ELSE IF( n.LT.0 ) THEN
306 info = -3
307 ELSE IF( kd.LT.0 ) THEN
308 info = -4
309 ELSE IF( ldab.LT.kd+1 ) THEN
310 info = -6
311 ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
312 info = -9
313 END IF
314*
315 IF( info.EQ.0 ) THEN
316 work( 1 ) = lwmin
317 iwork( 1 ) = liwmin
318*
319 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
320 info = -11
321 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
322 info = -13
323 END IF
324 END IF
325*
326 IF( info.NE.0 ) THEN
327 CALL xerbla( 'SSBEVD_2STAGE', -info )
328 RETURN
329 ELSE IF( lquery ) THEN
330 RETURN
331 END IF
332*
333* Quick return if possible
334*
335 IF( n.EQ.0 )
336 \$ RETURN
337*
338 IF( n.EQ.1 ) THEN
339 w( 1 ) = ab( 1, 1 )
340 IF( wantz )
341 \$ z( 1, 1 ) = one
342 RETURN
343 END IF
344*
345* Get machine constants.
346*
347 safmin = slamch( 'Safe minimum' )
348 eps = slamch( 'Precision' )
349 smlnum = safmin / eps
350 bignum = one / smlnum
351 rmin = sqrt( smlnum )
352 rmax = sqrt( bignum )
353*
354* Scale matrix to allowable range, if necessary.
355*
356 anrm = slansb( 'M', uplo, n, kd, ab, ldab, work )
357 iscale = 0
358 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
359 iscale = 1
360 sigma = rmin / anrm
361 ELSE IF( anrm.GT.rmax ) THEN
362 iscale = 1
363 sigma = rmax / anrm
364 END IF
365 IF( iscale.EQ.1 ) THEN
366 IF( lower ) THEN
367 CALL slascl( 'B', kd, kd, one, sigma, n, n, ab, ldab, info )
368 ELSE
369 CALL slascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab, info )
370 END IF
371 END IF
372*
373* Call SSYTRD_SB2ST to reduce band symmetric matrix to tridiagonal form.
374*
375 inde = 1
376 indhous = inde + n
377 indwrk = indhous + lhtrd
378 llwork = lwork - indwrk + 1
379 indwk2 = indwrk + n*n
380 llwrk2 = lwork - indwk2 + 1
381*
382 CALL ssytrd_sb2st( "N", jobz, uplo, n, kd, ab, ldab, w,
383 \$ work( inde ), work( indhous ), lhtrd,
384 \$ work( indwrk ), llwork, iinfo )
385*
386* For eigenvalues only, call SSTERF. For eigenvectors, call SSTEDC.
387*
388 IF( .NOT.wantz ) THEN
389 CALL ssterf( n, w, work( inde ), info )
390 ELSE
391 CALL sstedc( 'I', n, w, work( inde ), work( indwrk ), n,
392 \$ work( indwk2 ), llwrk2, iwork, liwork, info )
393 CALL sgemm( 'N', 'N', n, n, n, one, z, ldz, work( indwrk ), n,
394 \$ zero, work( indwk2 ), n )
395 CALL slacpy( 'A', n, n, work( indwk2 ), n, z, ldz )
396 END IF
397*
398* If matrix was scaled, then rescale eigenvalues appropriately.
399*
400 IF( iscale.EQ.1 )
401 \$ CALL sscal( n, one / sigma, w, 1 )
402*
403 work( 1 ) = lwmin
404 iwork( 1 ) = liwmin
405 RETURN
406*
407* End of SSBEVD_2STAGE
408*
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 slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sstedc(COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
SSTEDC
Definition: sstedc.f:188
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
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
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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