 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dsyevd()

 subroutine dsyevd ( character JOBZ, character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) W, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

DSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:
``` DSYEVD computes all eigenvalues and, optionally, eigenvectors of a
real symmetric matrix A. 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.

Because of large use of BLAS of level 3, DSYEVD needs N**2 more
workspace than DSYEVX.```
Parameters
 [in] JOBZ ``` JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] A ``` A is DOUBLE PRECISION 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 orthonormal eigenvectors of the matrix A. If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] W ``` W is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.``` [out] WORK ``` WORK is DOUBLE PRECISION array, dimension (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 must be at least 1. If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1. If JOBZ = 'V' and N > 1, LWORK must be at least 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 must be at least 1. If JOBZ = 'N' and N > 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 1, LIWORK must be at least 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 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).```
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified description of INFO. Sven, 16 Feb 05.

Definition at line 183 of file dsyevd.f.

185 *
186 * -- LAPACK driver routine --
187 * -- LAPACK is a software package provided by Univ. of Tennessee, --
188 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189 *
190 * .. Scalar Arguments ..
191  CHARACTER JOBZ, UPLO
192  INTEGER INFO, LDA, LIWORK, LWORK, N
193 * ..
194 * .. Array Arguments ..
195  INTEGER IWORK( * )
196  DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
197 * ..
198 *
199 * =====================================================================
200 *
201 * .. Parameters ..
202  DOUBLE PRECISION ZERO, ONE
203  parameter( zero = 0.0d+0, one = 1.0d+0 )
204 * ..
205 * .. Local Scalars ..
206 *
207  LOGICAL LOWER, LQUERY, WANTZ
208  INTEGER IINFO, INDE, INDTAU, INDWK2, INDWRK, ISCALE,
209  \$ LIOPT, LIWMIN, LLWORK, LLWRK2, LOPT, LWMIN
210  DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
211  \$ SMLNUM
212 * ..
213 * .. External Functions ..
214  LOGICAL LSAME
215  INTEGER ILAENV
216  DOUBLE PRECISION DLAMCH, DLANSY
217  EXTERNAL lsame, dlamch, dlansy, ilaenv
218 * ..
219 * .. External Subroutines ..
220  EXTERNAL dlacpy, dlascl, dormtr, dscal, dstedc, dsterf,
221  \$ dsytrd, xerbla
222 * ..
223 * .. Intrinsic Functions ..
224  INTRINSIC max, sqrt
225 * ..
226 * .. Executable Statements ..
227 *
228 * Test the input parameters.
229 *
230  wantz = lsame( jobz, 'V' )
231  lower = lsame( uplo, 'L' )
232  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
233 *
234  info = 0
235  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
236  info = -1
237  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
238  info = -2
239  ELSE IF( n.LT.0 ) THEN
240  info = -3
241  ELSE IF( lda.LT.max( 1, n ) ) THEN
242  info = -5
243  END IF
244 *
245  IF( info.EQ.0 ) THEN
246  IF( n.LE.1 ) THEN
247  liwmin = 1
248  lwmin = 1
249  lopt = lwmin
250  liopt = liwmin
251  ELSE
252  IF( wantz ) THEN
253  liwmin = 3 + 5*n
254  lwmin = 1 + 6*n + 2*n**2
255  ELSE
256  liwmin = 1
257  lwmin = 2*n + 1
258  END IF
259  lopt = max( lwmin, 2*n +
260  \$ ilaenv( 1, 'DSYTRD', uplo, n, -1, -1, -1 ) )
261  liopt = liwmin
262  END IF
263  work( 1 ) = lopt
264  iwork( 1 ) = liopt
265 *
266  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
267  info = -8
268  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
269  info = -10
270  END IF
271  END IF
272 *
273  IF( info.NE.0 ) THEN
274  CALL xerbla( 'DSYEVD', -info )
275  RETURN
276  ELSE IF( lquery ) THEN
277  RETURN
278  END IF
279 *
280 * Quick return if possible
281 *
282  IF( n.EQ.0 )
283  \$ RETURN
284 *
285  IF( n.EQ.1 ) THEN
286  w( 1 ) = a( 1, 1 )
287  IF( wantz )
288  \$ a( 1, 1 ) = one
289  RETURN
290  END IF
291 *
292 * Get machine constants.
293 *
294  safmin = dlamch( 'Safe minimum' )
295  eps = dlamch( 'Precision' )
296  smlnum = safmin / eps
297  bignum = one / smlnum
298  rmin = sqrt( smlnum )
299  rmax = sqrt( bignum )
300 *
301 * Scale matrix to allowable range, if necessary.
302 *
303  anrm = dlansy( 'M', uplo, n, a, lda, work )
304  iscale = 0
305  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
306  iscale = 1
307  sigma = rmin / anrm
308  ELSE IF( anrm.GT.rmax ) THEN
309  iscale = 1
310  sigma = rmax / anrm
311  END IF
312  IF( iscale.EQ.1 )
313  \$ CALL dlascl( uplo, 0, 0, one, sigma, n, n, a, lda, info )
314 *
315 * Call DSYTRD to reduce symmetric matrix to tridiagonal form.
316 *
317  inde = 1
318  indtau = inde + n
319  indwrk = indtau + n
320  llwork = lwork - indwrk + 1
321  indwk2 = indwrk + n*n
322  llwrk2 = lwork - indwk2 + 1
323 *
324  CALL dsytrd( uplo, n, a, lda, w, work( inde ), work( indtau ),
325  \$ work( indwrk ), llwork, iinfo )
326 *
327 * For eigenvalues only, call DSTERF. For eigenvectors, first call
328 * DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
329 * tridiagonal matrix, then call DORMTR to multiply it by the
330 * Householder transformations stored in A.
331 *
332  IF( .NOT.wantz ) THEN
333  CALL dsterf( n, w, work( inde ), info )
334  ELSE
335  CALL dstedc( 'I', n, w, work( inde ), work( indwrk ), n,
336  \$ work( indwk2 ), llwrk2, iwork, liwork, info )
337  CALL dormtr( 'L', uplo, 'N', n, n, a, lda, work( indtau ),
338  \$ work( indwrk ), n, work( indwk2 ), llwrk2, iinfo )
339  CALL dlacpy( 'A', n, n, work( indwrk ), n, a, lda )
340  END IF
341 *
342 * If matrix was scaled, then rescale eigenvalues appropriately.
343 *
344  IF( iscale.EQ.1 )
345  \$ CALL dscal( n, one / sigma, w, 1 )
346 *
347  work( 1 ) = lopt
348  iwork( 1 ) = liopt
349 *
350  RETURN
351 *
352 * End of DSYEVD
353 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:143
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dstedc(COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSTEDC
Definition: dstedc.f:188
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:86
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMTR
Definition: dormtr.f:171
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansy.f:122
subroutine dsytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
DSYTRD
Definition: dsytrd.f:192
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