 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ cheevd()

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

CHEEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:
``` CHEEVD computes all eigenvalues and, optionally, eigenvectors of a
complex Hermitian 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.```
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 COMPLEX array, dimension (LDA, N) On entry, the Hermitian 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 COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of the array WORK. If N <= 1, LWORK must be at least 1. If JOBZ = 'N' and N > 1, LWORK must be at least N + 1. If JOBZ = 'V' and N > 1, LWORK must be at least 2*N + N**2. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.``` [out] RWORK ``` RWORK is REAL array, dimension (LRWORK) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.``` [in] LRWORK ``` LRWORK is INTEGER The dimension of the array RWORK. If N <= 1, LRWORK must be at least 1. If JOBZ = 'N' and N > 1, LRWORK must be at least N. If JOBZ = 'V' and N > 1, LRWORK must be at least 1 + 5*N + 2*N**2. If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK 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, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK 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).```
Further Details:
Modified description of INFO. Sven, 16 Feb 05.
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 203 of file cheevd.f.

205 *
206 * -- LAPACK driver routine --
207 * -- LAPACK is a software package provided by Univ. of Tennessee, --
208 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
209 *
210 * .. Scalar Arguments ..
211  CHARACTER JOBZ, UPLO
212  INTEGER INFO, LDA, LIWORK, LRWORK, LWORK, N
213 * ..
214 * .. Array Arguments ..
215  INTEGER IWORK( * )
216  REAL RWORK( * ), W( * )
217  COMPLEX A( LDA, * ), WORK( * )
218 * ..
219 *
220 * =====================================================================
221 *
222 * .. Parameters ..
223  REAL ZERO, ONE
224  parameter( zero = 0.0e0, one = 1.0e0 )
225  COMPLEX CONE
226  parameter( cone = ( 1.0e0, 0.0e0 ) )
227 * ..
228 * .. Local Scalars ..
229  LOGICAL LOWER, LQUERY, WANTZ
230  INTEGER IINFO, IMAX, INDE, INDRWK, INDTAU, INDWK2,
231  \$ INDWRK, ISCALE, LIOPT, LIWMIN, LLRWK, LLWORK,
232  \$ LLWRK2, LOPT, LROPT, LRWMIN, LWMIN
233  REAL ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
234  \$ SMLNUM
235 * ..
236 * .. External Functions ..
237  LOGICAL LSAME
238  INTEGER ILAENV
239  REAL CLANHE, SLAMCH
240  EXTERNAL ilaenv, lsame, clanhe, slamch
241 * ..
242 * .. External Subroutines ..
243  EXTERNAL chetrd, clacpy, clascl, cstedc, cunmtr, sscal,
244  \$ ssterf, xerbla
245 * ..
246 * .. Intrinsic Functions ..
247  INTRINSIC max, sqrt
248 * ..
249 * .. Executable Statements ..
250 *
251 * Test the input parameters.
252 *
253  wantz = lsame( jobz, 'V' )
254  lower = lsame( uplo, 'L' )
255  lquery = ( lwork.EQ.-1 .OR. lrwork.EQ.-1 .OR. liwork.EQ.-1 )
256 *
257  info = 0
258  IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
259  info = -1
260  ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
261  info = -2
262  ELSE IF( n.LT.0 ) THEN
263  info = -3
264  ELSE IF( lda.LT.max( 1, n ) ) THEN
265  info = -5
266  END IF
267 *
268  IF( info.EQ.0 ) THEN
269  IF( n.LE.1 ) THEN
270  lwmin = 1
271  lrwmin = 1
272  liwmin = 1
273  lopt = lwmin
274  lropt = lrwmin
275  liopt = liwmin
276  ELSE
277  IF( wantz ) THEN
278  lwmin = 2*n + n*n
279  lrwmin = 1 + 5*n + 2*n**2
280  liwmin = 3 + 5*n
281  ELSE
282  lwmin = n + 1
283  lrwmin = n
284  liwmin = 1
285  END IF
286  lopt = max( lwmin, n +
287  \$ ilaenv( 1, 'CHETRD', uplo, n, -1, -1, -1 ) )
288  lropt = lrwmin
289  liopt = liwmin
290  END IF
291  work( 1 ) = lopt
292  rwork( 1 ) = lropt
293  iwork( 1 ) = liopt
294 *
295  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
296  info = -8
297  ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
298  info = -10
299  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
300  info = -12
301  END IF
302  END IF
303 *
304  IF( info.NE.0 ) THEN
305  CALL xerbla( 'CHEEVD', -info )
306  RETURN
307  ELSE IF( lquery ) THEN
308  RETURN
309  END IF
310 *
311 * Quick return if possible
312 *
313  IF( n.EQ.0 )
314  \$ RETURN
315 *
316  IF( n.EQ.1 ) THEN
317  w( 1 ) = real( a( 1, 1 ) )
318  IF( wantz )
319  \$ a( 1, 1 ) = cone
320  RETURN
321  END IF
322 *
323 * Get machine constants.
324 *
325  safmin = slamch( 'Safe minimum' )
326  eps = slamch( 'Precision' )
327  smlnum = safmin / eps
328  bignum = one / smlnum
329  rmin = sqrt( smlnum )
330  rmax = sqrt( bignum )
331 *
332 * Scale matrix to allowable range, if necessary.
333 *
334  anrm = clanhe( 'M', uplo, n, a, lda, rwork )
335  iscale = 0
336  IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
337  iscale = 1
338  sigma = rmin / anrm
339  ELSE IF( anrm.GT.rmax ) THEN
340  iscale = 1
341  sigma = rmax / anrm
342  END IF
343  IF( iscale.EQ.1 )
344  \$ CALL clascl( uplo, 0, 0, one, sigma, n, n, a, lda, info )
345 *
346 * Call CHETRD to reduce Hermitian matrix to tridiagonal form.
347 *
348  inde = 1
349  indtau = 1
350  indwrk = indtau + n
351  indrwk = inde + n
352  indwk2 = indwrk + n*n
353  llwork = lwork - indwrk + 1
354  llwrk2 = lwork - indwk2 + 1
355  llrwk = lrwork - indrwk + 1
356  CALL chetrd( uplo, n, a, lda, w, rwork( inde ), work( indtau ),
357  \$ work( indwrk ), llwork, iinfo )
358 *
359 * For eigenvalues only, call SSTERF. For eigenvectors, first call
360 * CSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
361 * tridiagonal matrix, then call CUNMTR to multiply it to the
362 * Householder transformations represented as Householder vectors in
363 * A.
364 *
365  IF( .NOT.wantz ) THEN
366  CALL ssterf( n, w, rwork( inde ), info )
367  ELSE
368  CALL cstedc( 'I', n, w, rwork( inde ), work( indwrk ), n,
369  \$ work( indwk2 ), llwrk2, rwork( indrwk ), llrwk,
370  \$ iwork, liwork, info )
371  CALL cunmtr( 'L', uplo, 'N', n, n, a, lda, work( indtau ),
372  \$ work( indwrk ), n, work( indwk2 ), llwrk2, iinfo )
373  CALL clacpy( 'A', n, n, work( indwrk ), n, a, lda )
374  END IF
375 *
376 * If matrix was scaled, then rescale eigenvalues appropriately.
377 *
378  IF( iscale.EQ.1 ) THEN
379  IF( info.EQ.0 ) THEN
380  imax = n
381  ELSE
382  imax = info - 1
383  END IF
384  CALL sscal( imax, one / sigma, w, 1 )
385  END IF
386 *
387  work( 1 ) = lopt
388  rwork( 1 ) = lropt
389  iwork( 1 ) = liopt
390 *
391  RETURN
392 *
393 * End of CHEEVD
394 *
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 ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:86
real function clanhe(NORM, UPLO, N, A, LDA, WORK)
CLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clanhe.f:124
subroutine chetrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
CHETRD
Definition: chetrd.f:192
subroutine clascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: clascl.f:143
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine cunmtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMTR
Definition: cunmtr.f:172
subroutine cstedc(COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
CSTEDC
Definition: cstedc.f:212
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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