 LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine ssygvd ( integer ITYPE, character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

SSYGVD

Purpose:
``` SSYGVD 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 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] A ``` A is REAL array, dimension (LDA, N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, if JOBZ = 'V', then if INFO = 0, A 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 on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of A, including the diagonal, is destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] B ``` B is REAL array, dimension (LDB, N) On entry, the symmetric matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B. On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] W ``` W is REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.``` [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 >= 2*N+1. 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 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 N <= 1, LIWORK >= 1. If JOBZ = 'N' and 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: SPOTRF or SSYEVD returned an error code: <= N: if INFO = i and JOBZ = 'N', then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = 'V', then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1); > 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
November 2015
Further Details:
```  Modified so that no backsubstitution is performed if SSYEVD fails to
converge (NEIG in old code could be greater than N causing out of
bounds reference to A - reported by Ralf Meyer).  Also corrected the
description of INFO and the test on ITYPE. Sven, 16 Feb 05.```
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 229 of file ssygvd.f.

229 *
230 * -- LAPACK driver routine (version 3.6.0) --
231 * -- LAPACK is a software package provided by Univ. of Tennessee, --
232 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
233 * November 2015
234 *
235 * .. Scalar Arguments ..
236  CHARACTER jobz, uplo
237  INTEGER info, itype, lda, ldb, liwork, lwork, n
238 * ..
239 * .. Array Arguments ..
240  INTEGER iwork( * )
241  REAL a( lda, * ), b( ldb, * ), w( * ), work( * )
242 * ..
243 *
244 * =====================================================================
245 *
246 * .. Parameters ..
247  REAL one
248  parameter ( one = 1.0e+0 )
249 * ..
250 * .. Local Scalars ..
251  LOGICAL lquery, upper, wantz
252  CHARACTER trans
253  INTEGER liopt, liwmin, lopt, lwmin
254 * ..
255 * .. External Functions ..
256  LOGICAL lsame
257  EXTERNAL lsame
258 * ..
259 * .. External Subroutines ..
260  EXTERNAL spotrf, ssyevd, ssygst, strmm, strsm, xerbla
261 * ..
262 * .. Intrinsic Functions ..
263  INTRINSIC max, real
264 * ..
265 * .. Executable Statements ..
266 *
267 * Test the input parameters.
268 *
269  wantz = lsame( jobz, 'V' )
270  upper = lsame( uplo, 'U' )
271  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
272 *
273  info = 0
274  IF( n.LE.1 ) THEN
275  liwmin = 1
276  lwmin = 1
277  ELSE IF( wantz ) THEN
278  liwmin = 3 + 5*n
279  lwmin = 1 + 6*n + 2*n**2
280  ELSE
281  liwmin = 1
282  lwmin = 2*n + 1
283  END IF
284  lopt = lwmin
285  liopt = liwmin
286  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
287  info = -1
288  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
289  info = -2
290  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
291  info = -3
292  ELSE IF( n.LT.0 ) THEN
293  info = -4
294  ELSE IF( lda.LT.max( 1, n ) ) THEN
295  info = -6
296  ELSE IF( ldb.LT.max( 1, n ) ) THEN
297  info = -8
298  END IF
299 *
300  IF( info.EQ.0 ) THEN
301  work( 1 ) = lopt
302  iwork( 1 ) = liopt
303 *
304  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
305  info = -11
306  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
307  info = -13
308  END IF
309  END IF
310 *
311  IF( info.NE.0 ) THEN
312  CALL xerbla( 'SSYGVD', -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 Cholesky factorization of B.
324 *
325  CALL spotrf( uplo, n, b, ldb, info )
326  IF( info.NE.0 ) THEN
327  info = n + info
328  RETURN
329  END IF
330 *
331 * Transform problem to standard eigenvalue problem and solve.
332 *
333  CALL ssygst( itype, uplo, n, a, lda, b, ldb, info )
334  CALL ssyevd( jobz, uplo, n, a, lda, w, work, lwork, iwork, liwork,
335  \$ info )
336  lopt = max( REAL( LOPT ), REAL( WORK( 1 ) ) )
337  liopt = max( REAL( LIOPT ), REAL( IWORK( 1 ) ) )
338 *
339  IF( wantz .AND. info.EQ.0 ) THEN
340 *
341 * Backtransform eigenvectors to the original problem.
342 *
343  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
344 *
345 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
346 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
347 *
348  IF( upper ) THEN
349  trans = 'N'
350  ELSE
351  trans = 'T'
352  END IF
353 *
354  CALL strsm( 'Left', uplo, trans, 'Non-unit', n, n, one,
355  \$ b, ldb, a, lda )
356 *
357  ELSE IF( itype.EQ.3 ) THEN
358 *
359 * For B*A*x=(lambda)*x;
360 * backtransform eigenvectors: x = L*y or U**T*y
361 *
362  IF( upper ) THEN
363  trans = 'T'
364  ELSE
365  trans = 'N'
366  END IF
367 *
368  CALL strmm( 'Left', uplo, trans, 'Non-unit', n, n, one,
369  \$ b, ldb, a, lda )
370  END IF
371  END IF
372 *
373  work( 1 ) = lopt
374  iwork( 1 ) = liopt
375 *
376  RETURN
377 *
378 * End of SSYGVD
379 *
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:183
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine spotrf(UPLO, N, A, LDA, INFO)
SPOTRF
Definition: spotrf.f:109
subroutine strmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRMM
Definition: strmm.f:179
subroutine ssyevd(JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO)
SSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices ...
Definition: ssyevd.f:185
subroutine ssygst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
SSYGST
Definition: ssygst.f:129
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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