 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ ssyevd()

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

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

Purpose:
``` SSYEVD 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, SSYEVD needs N**2 more
workspace than SSYEVX.```
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 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 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 REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.``` [out] WORK ``` WORK is REAL 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 181 of file ssyevd.f.

183 *
184 * -- LAPACK driver routine --
185 * -- LAPACK is a software package provided by Univ. of Tennessee, --
186 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
187 *
188 * .. Scalar Arguments ..
189  CHARACTER JOBZ, UPLO
190  INTEGER INFO, LDA, LIWORK, LWORK, N
191 * ..
192 * .. Array Arguments ..
193  INTEGER IWORK( * )
194  REAL A( LDA, * ), W( * ), WORK( * )
195 * ..
196 *
197 * =====================================================================
198 *
199 * .. Parameters ..
200  REAL ZERO, ONE
201  parameter( zero = 0.0e+0, one = 1.0e+0 )
202 * ..
203 * .. Local Scalars ..
204 *
205  LOGICAL LOWER, LQUERY, WANTZ
206  INTEGER IINFO, INDE, INDTAU, INDWK2, INDWRK, ISCALE,
207  \$ LIOPT, LIWMIN, LLWORK, LLWRK2, LOPT, LWMIN
208  REAL ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
209  \$ SMLNUM
210 * ..
211 * .. External Functions ..
212  LOGICAL LSAME
213  INTEGER ILAENV
214  REAL SLAMCH, SLANSY
215  EXTERNAL ilaenv, lsame, slamch, slansy
216 * ..
217 * .. External Subroutines ..
218  EXTERNAL slacpy, slascl, sormtr, sscal, sstedc, ssterf,
219  \$ ssytrd, xerbla
220 * ..
221 * .. Intrinsic Functions ..
222  INTRINSIC max, sqrt
223 * ..
224 * .. Executable Statements ..
225 *
226 * Test the input parameters.
227 *
228  wantz = lsame( jobz, 'V' )
229  lower = lsame( uplo, 'L' )
230  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
231 *
232  info = 0
233  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
234  info = -1
235  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
236  info = -2
237  ELSE IF( n.LT.0 ) THEN
238  info = -3
239  ELSE IF( lda.LT.max( 1, n ) ) THEN
240  info = -5
241  END IF
242 *
243  IF( info.EQ.0 ) THEN
244  IF( n.LE.1 ) THEN
245  liwmin = 1
246  lwmin = 1
247  lopt = lwmin
248  liopt = liwmin
249  ELSE
250  IF( wantz ) THEN
251  liwmin = 3 + 5*n
252  lwmin = 1 + 6*n + 2*n**2
253  ELSE
254  liwmin = 1
255  lwmin = 2*n + 1
256  END IF
257  lopt = max( lwmin, 2*n +
258  \$ ilaenv( 1, 'SSYTRD', uplo, n, -1, -1, -1 ) )
259  liopt = liwmin
260  END IF
261  work( 1 ) = lopt
262  iwork( 1 ) = liopt
263 *
264  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
265  info = -8
266  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
267  info = -10
268  END IF
269  END IF
270 *
271  IF( info.NE.0 ) THEN
272  CALL xerbla( 'SSYEVD', -info )
273  RETURN
274  ELSE IF( lquery ) THEN
275  RETURN
276  END IF
277 *
278 * Quick return if possible
279 *
280  IF( n.EQ.0 )
281  \$ RETURN
282 *
283  IF( n.EQ.1 ) THEN
284  w( 1 ) = a( 1, 1 )
285  IF( wantz )
286  \$ a( 1, 1 ) = one
287  RETURN
288  END IF
289 *
290 * Get machine constants.
291 *
292  safmin = slamch( 'Safe minimum' )
293  eps = slamch( 'Precision' )
294  smlnum = safmin / eps
295  bignum = one / smlnum
296  rmin = sqrt( smlnum )
297  rmax = sqrt( bignum )
298 *
299 * Scale matrix to allowable range, if necessary.
300 *
301  anrm = slansy( 'M', uplo, n, a, lda, work )
302  iscale = 0
303  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
304  iscale = 1
305  sigma = rmin / anrm
306  ELSE IF( anrm.GT.rmax ) THEN
307  iscale = 1
308  sigma = rmax / anrm
309  END IF
310  IF( iscale.EQ.1 )
311  \$ CALL slascl( uplo, 0, 0, one, sigma, n, n, a, lda, info )
312 *
313 * Call SSYTRD to reduce symmetric matrix to tridiagonal form.
314 *
315  inde = 1
316  indtau = inde + n
317  indwrk = indtau + n
318  llwork = lwork - indwrk + 1
319  indwk2 = indwrk + n*n
320  llwrk2 = lwork - indwk2 + 1
321 *
322  CALL ssytrd( uplo, n, a, lda, w, work( inde ), work( indtau ),
323  \$ work( indwrk ), llwork, iinfo )
324 *
325 * For eigenvalues only, call SSTERF. For eigenvectors, first call
326 * SSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
327 * tridiagonal matrix, then call SORMTR to multiply it by the
328 * Householder transformations stored in A.
329 *
330  IF( .NOT.wantz ) THEN
331  CALL ssterf( n, w, work( inde ), info )
332  ELSE
333  CALL sstedc( 'I', n, w, work( inde ), work( indwrk ), n,
334  \$ work( indwk2 ), llwrk2, iwork, liwork, info )
335  CALL sormtr( 'L', uplo, 'N', n, n, a, lda, work( indtau ),
336  \$ work( indwrk ), n, work( indwk2 ), llwrk2, iinfo )
337  CALL slacpy( 'A', n, n, work( indwrk ), n, a, lda )
338  END IF
339 *
340 * If matrix was scaled, then rescale eigenvalues appropriately.
341 *
342  IF( iscale.EQ.1 )
343  \$ CALL sscal( n, one / sigma, w, 1 )
344 *
345  work( 1 ) = lopt
346  iwork( 1 ) = liopt
347 *
348  RETURN
349 *
350 * End of SSYEVD
351 *
subroutine slascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: slascl.f:143
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
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 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 sormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMTR
Definition: sormtr.f:172
real function slansy(NORM, UPLO, N, A, LDA, WORK)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slansy.f:122
subroutine ssytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
SSYTRD
Definition: ssytrd.f:192
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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