LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ dspgvd()

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

DSPGVD

Purpose:
``` DSPGVD 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.``` [out] Z ``` Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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: DPPTRF or DSPEVD returned an error code: <= N: if INFO = i, DSPEVD 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.```
Date
December 2016
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 212 of file dspgvd.f.

212 *
213 * -- LAPACK driver routine (version 3.7.0) --
214 * -- LAPACK is a software package provided by Univ. of Tennessee, --
215 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216 * December 2016
217 *
218 * .. Scalar Arguments ..
219  CHARACTER jobz, uplo
220  INTEGER info, itype, ldz, liwork, lwork, n
221 * ..
222 * .. Array Arguments ..
223  INTEGER iwork( * )
224  DOUBLE PRECISION ap( * ), bp( * ), w( * ), work( * ),
225  \$ z( ldz, * )
226 * ..
227 *
228 * =====================================================================
229 *
230 * .. Local Scalars ..
231  LOGICAL lquery, upper, wantz
232  CHARACTER trans
233  INTEGER j, liwmin, lwmin, neig
234 * ..
235 * .. External Functions ..
236  LOGICAL lsame
237  EXTERNAL lsame
238 * ..
239 * .. External Subroutines ..
240  EXTERNAL dpptrf, dspevd, dspgst, dtpmv, dtpsv, xerbla
241 * ..
242 * .. Intrinsic Functions ..
243  INTRINSIC dble, max
244 * ..
245 * .. Executable Statements ..
246 *
247 * Test the input parameters.
248 *
249  wantz = lsame( jobz, 'V' )
250  upper = lsame( uplo, 'U' )
251  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
252 *
253  info = 0
254  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
255  info = -1
256  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
257  info = -2
258  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
259  info = -3
260  ELSE IF( n.LT.0 ) THEN
261  info = -4
262  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
263  info = -9
264  END IF
265 *
266  IF( info.EQ.0 ) THEN
267  IF( n.LE.1 ) THEN
268  liwmin = 1
269  lwmin = 1
270  ELSE
271  IF( wantz ) THEN
272  liwmin = 3 + 5*n
273  lwmin = 1 + 6*n + 2*n**2
274  ELSE
275  liwmin = 1
276  lwmin = 2*n
277  END IF
278  END IF
279  work( 1 ) = lwmin
280  iwork( 1 ) = liwmin
281  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
282  info = -11
283  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
284  info = -13
285  END IF
286  END IF
287 *
288  IF( info.NE.0 ) THEN
289  CALL xerbla( 'DSPGVD', -info )
290  RETURN
291  ELSE IF( lquery ) THEN
292  RETURN
293  END IF
294 *
295 * Quick return if possible
296 *
297  IF( n.EQ.0 )
298  \$ RETURN
299 *
300 * Form a Cholesky factorization of BP.
301 *
302  CALL dpptrf( uplo, n, bp, info )
303  IF( info.NE.0 ) THEN
304  info = n + info
305  RETURN
306  END IF
307 *
308 * Transform problem to standard eigenvalue problem and solve.
309 *
310  CALL dspgst( itype, uplo, n, ap, bp, info )
311  CALL dspevd( jobz, uplo, n, ap, w, z, ldz, work, lwork, iwork,
312  \$ liwork, info )
313  lwmin = max( dble( lwmin ), dble( work( 1 ) ) )
314  liwmin = max( dble( liwmin ), dble( iwork( 1 ) ) )
315 *
316  IF( wantz ) THEN
317 *
318 * Backtransform eigenvectors to the original problem.
319 *
320  neig = n
321  IF( info.GT.0 )
322  \$ neig = info - 1
323  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
324 *
325 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
326 * backtransform eigenvectors: x = inv(L)**T *y or inv(U)*y
327 *
328  IF( upper ) THEN
329  trans = 'N'
330  ELSE
331  trans = 'T'
332  END IF
333 *
334  DO 10 j = 1, neig
335  CALL dtpsv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
336  \$ 1 )
337  10 CONTINUE
338 *
339  ELSE IF( itype.EQ.3 ) THEN
340 *
341 * For B*A*x=(lambda)*x;
342 * backtransform eigenvectors: x = L*y or U**T *y
343 *
344  IF( upper ) THEN
345  trans = 'T'
346  ELSE
347  trans = 'N'
348  END IF
349 *
350  DO 20 j = 1, neig
351  CALL dtpmv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
352  \$ 1 )
353  20 CONTINUE
354  END IF
355  END IF
356 *
357  work( 1 ) = lwmin
358  iwork( 1 ) = liwmin
359 *
360  RETURN
361 *
362 * End of DSPGVD
363 *
subroutine dspgst(ITYPE, UPLO, N, AP, BP, INFO)
DSPGST
Definition: dspgst.f:115
subroutine dtpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
DTPSV
Definition: dtpsv.f:146
subroutine dpptrf(UPLO, N, AP, INFO)
DPPTRF
Definition: dpptrf.f:121
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dtpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
DTPMV
Definition: dtpmv.f:144
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dspevd(JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matric...
Definition: dspevd.f:180
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