 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ sspgvd()

 subroutine sspgvd ( integer ITYPE, character JOBZ, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) BP, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

SSPGVD

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Purpose:
``` SSPGVD computes all the eigenvalues, and optionally, the eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
B are assumed to be symmetric, stored in packed format, 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] ITYPE ``` ITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*x``` [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,out] AP ``` AP is REAL array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.``` [in,out] BP ``` BP is REAL array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B.``` [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. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(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 required LWORK.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= 2*N. If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2. If LWORK = -1, then a workspace query is assumed; the routine only calculates the required 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 required 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 required 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: SPPTRF or SSPEVD returned an error code: <= N: if INFO = i, SSPEVD 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 the leading minor of order i of 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 208 of file sspgvd.f.

210 *
211 * -- LAPACK driver routine --
212 * -- LAPACK is a software package provided by Univ. of Tennessee, --
213 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
214 *
215 * .. Scalar Arguments ..
216  CHARACTER JOBZ, UPLO
217  INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N
218 * ..
219 * .. Array Arguments ..
220  INTEGER IWORK( * )
221  REAL AP( * ), BP( * ), W( * ), WORK( * ),
222  \$ Z( LDZ, * )
223 * ..
224 *
225 * =====================================================================
226 *
227 * .. Local Scalars ..
228  LOGICAL LQUERY, UPPER, WANTZ
229  CHARACTER TRANS
230  INTEGER J, LIWMIN, LWMIN, NEIG
231 * ..
232 * .. External Functions ..
233  LOGICAL LSAME
234  EXTERNAL lsame
235 * ..
236 * .. External Subroutines ..
237  EXTERNAL spptrf, sspevd, sspgst, stpmv, stpsv, xerbla
238 * ..
239 * .. Intrinsic Functions ..
240  INTRINSIC max, real
241 * ..
242 * .. Executable Statements ..
243 *
244 * Test the input parameters.
245 *
246  wantz = lsame( jobz, 'V' )
247  upper = lsame( uplo, 'U' )
248  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
249 *
250  info = 0
251  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
252  info = -1
253  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
254  info = -2
255  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
256  info = -3
257  ELSE IF( n.LT.0 ) THEN
258  info = -4
259  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
260  info = -9
261  END IF
262 *
263  IF( info.EQ.0 ) THEN
264  IF( n.LE.1 ) THEN
265  liwmin = 1
266  lwmin = 1
267  ELSE
268  IF( wantz ) THEN
269  liwmin = 3 + 5*n
270  lwmin = 1 + 6*n + 2*n**2
271  ELSE
272  liwmin = 1
273  lwmin = 2*n
274  END IF
275  END IF
276  work( 1 ) = lwmin
277  iwork( 1 ) = liwmin
278  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
279  info = -11
280  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
281  info = -13
282  END IF
283  END IF
284 *
285  IF( info.NE.0 ) THEN
286  CALL xerbla( 'SSPGVD', -info )
287  RETURN
288  ELSE IF( lquery ) THEN
289  RETURN
290  END IF
291 *
292 * Quick return if possible
293 *
294  IF( n.EQ.0 )
295  \$ RETURN
296 *
297 * Form a Cholesky factorization of BP.
298 *
299  CALL spptrf( uplo, n, bp, info )
300  IF( info.NE.0 ) THEN
301  info = n + info
302  RETURN
303  END IF
304 *
305 * Transform problem to standard eigenvalue problem and solve.
306 *
307  CALL sspgst( itype, uplo, n, ap, bp, info )
308  CALL sspevd( jobz, uplo, n, ap, w, z, ldz, work, lwork, iwork,
309  \$ liwork, info )
310  lwmin = max( real( lwmin ), real( work( 1 ) ) )
311  liwmin = max( real( liwmin ), real( iwork( 1 ) ) )
312 *
313  IF( wantz ) THEN
314 *
315 * Backtransform eigenvectors to the original problem.
316 *
317  neig = n
318  IF( info.GT.0 )
319  \$ neig = info - 1
320  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
321 *
322 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
323 * backtransform eigenvectors: x = inv(L)**T *y or inv(U)*y
324 *
325  IF( upper ) THEN
326  trans = 'N'
327  ELSE
328  trans = 'T'
329  END IF
330 *
331  DO 10 j = 1, neig
332  CALL stpsv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
333  \$ 1 )
334  10 CONTINUE
335 *
336  ELSE IF( itype.EQ.3 ) THEN
337 *
338 * For B*A*x=(lambda)*x;
339 * backtransform eigenvectors: x = L*y or U**T *y
340 *
341  IF( upper ) THEN
342  trans = 'T'
343  ELSE
344  trans = 'N'
345  END IF
346 *
347  DO 20 j = 1, neig
348  CALL stpmv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
349  \$ 1 )
350  20 CONTINUE
351  END IF
352  END IF
353 *
354  work( 1 ) = lwmin
355  iwork( 1 ) = liwmin
356 *
357  RETURN
358 *
359 * End of SSPGVD
360 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sspgst(ITYPE, UPLO, N, AP, BP, INFO)
SSPGST
Definition: sspgst.f:113
subroutine spptrf(UPLO, N, AP, INFO)
SPPTRF
Definition: spptrf.f:119
subroutine sspevd(JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
SSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrice...
Definition: sspevd.f:178
subroutine stpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
STPMV
Definition: stpmv.f:142
subroutine stpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
STPSV
Definition: stpsv.f:144
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