LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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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