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

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

SSBGVD

Purpose:
``` SSBGVD computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite banded eigenproblem, of the
form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric 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 REAL array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric 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 REAL array, dimension (LDBB, N) On entry, the upper or lower triangle of the symmetric 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(ka+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**T*S, as returned by SPBSTF.``` [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 REAL 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 Z**T*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 >= max(1,N).``` [out] WORK ``` WORK is REAL 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 >= 3*N. If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2. 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 LIWORK > 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 >= 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 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, 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 SPBSTF 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 225 of file ssbgvd.f.

227*
228* -- LAPACK driver routine --
229* -- LAPACK is a software package provided by Univ. of Tennessee, --
230* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
231*
232* .. Scalar Arguments ..
233 CHARACTER JOBZ, UPLO
234 INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N
235* ..
236* .. Array Arguments ..
237 INTEGER IWORK( * )
238 REAL AB( LDAB, * ), BB( LDBB, * ), W( * ),
239 \$ WORK( * ), Z( LDZ, * )
240* ..
241*
242* =====================================================================
243*
244* .. Parameters ..
245 REAL ONE, ZERO
246 parameter( one = 1.0e+0, zero = 0.0e+0 )
247* ..
248* .. Local Scalars ..
249 LOGICAL LQUERY, UPPER, WANTZ
250 CHARACTER VECT
251 INTEGER IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLWRK2,
252 \$ LWMIN
253* ..
254* .. External Functions ..
255 LOGICAL LSAME
256 EXTERNAL lsame
257* ..
258* .. External Subroutines ..
259 EXTERNAL sgemm, slacpy, spbstf, ssbgst, ssbtrd, sstedc,
260 \$ ssterf, xerbla
261* ..
262* .. Executable Statements ..
263*
264* Test the input parameters.
265*
266 wantz = lsame( jobz, 'V' )
267 upper = lsame( uplo, 'U' )
268 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
269*
270 info = 0
271 IF( n.LE.1 ) THEN
272 liwmin = 1
273 lwmin = 1
274 ELSE IF( wantz ) THEN
275 liwmin = 3 + 5*n
276 lwmin = 1 + 5*n + 2*n**2
277 ELSE
278 liwmin = 1
279 lwmin = 2*n
280 END IF
281*
282 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
283 info = -1
284 ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
285 info = -2
286 ELSE IF( n.LT.0 ) THEN
287 info = -3
288 ELSE IF( ka.LT.0 ) THEN
289 info = -4
290 ELSE IF( kb.LT.0 .OR. kb.GT.ka ) THEN
291 info = -5
292 ELSE IF( ldab.LT.ka+1 ) THEN
293 info = -7
294 ELSE IF( ldbb.LT.kb+1 ) THEN
295 info = -9
296 ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
297 info = -12
298 END IF
299*
300 IF( info.EQ.0 ) THEN
301 work( 1 ) = lwmin
302 iwork( 1 ) = liwmin
303*
304 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
305 info = -14
306 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
307 info = -16
308 END IF
309 END IF
310*
311 IF( info.NE.0 ) THEN
312 CALL xerbla( 'SSBGVD', -info )
313 RETURN
314 ELSE IF( lquery ) THEN
315 RETURN
316 END IF
317*
318* Quick return if possible
319*
320 IF( n.EQ.0 )
321 \$ RETURN
322*
323* Form a split Cholesky factorization of B.
324*
325 CALL spbstf( uplo, n, kb, bb, ldbb, info )
326 IF( info.NE.0 ) THEN
327 info = n + info
328 RETURN
329 END IF
330*
331* Transform problem to standard eigenvalue problem.
332*
333 inde = 1
334 indwrk = inde + n
335 indwk2 = indwrk + n*n
336 llwrk2 = lwork - indwk2 + 1
337 CALL ssbgst( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, z, ldz,
338 \$ work, iinfo )
339*
340* Reduce to tridiagonal form.
341*
342 IF( wantz ) THEN
343 vect = 'U'
344 ELSE
345 vect = 'N'
346 END IF
347 CALL ssbtrd( vect, uplo, n, ka, ab, ldab, w, work( inde ), z, ldz,
348 \$ work( indwrk ), iinfo )
349*
350* For eigenvalues only, call SSTERF. For eigenvectors, call SSTEDC.
351*
352 IF( .NOT.wantz ) THEN
353 CALL ssterf( n, w, work( inde ), info )
354 ELSE
355 CALL sstedc( 'I', n, w, work( inde ), work( indwrk ), n,
356 \$ work( indwk2 ), llwrk2, iwork, liwork, info )
357 CALL sgemm( 'N', 'N', n, n, n, one, z, ldz, work( indwrk ), n,
358 \$ zero, work( indwk2 ), n )
359 CALL slacpy( 'A', n, n, work( indwk2 ), n, z, ldz )
360 END IF
361*
362 work( 1 ) = lwmin
363 iwork( 1 ) = liwmin
364*
365 RETURN
366*
367* End of SSBGVD
368*
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
subroutine spbstf(UPLO, N, KD, AB, LDAB, INFO)
SPBSTF
Definition: spbstf.f:152
subroutine ssbtrd(VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
SSBTRD
Definition: ssbtrd.f:163
subroutine ssbgst(VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, INFO)
SSBGST
Definition: ssbgst.f:159
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
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