LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dspevd()

 subroutine dspevd ( character jobz, character uplo, integer n, double precision, dimension( * ) ap, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info )

DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:
``` DSPEVD computes all the eigenvalues and, optionally, eigenvectors
of a real symmetric matrix A in packed storage. If eigenvectors are
desired, it uses a divide and conquer algorithm.```
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] 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, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.``` [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 orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(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 must be at least 1. If JOBZ = 'N' and N > 1, LWORK must be at least 2*N. If JOBZ = 'V' and N > 1, LWORK must be at least 1 + 6*N + 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 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 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: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.```

Definition at line 170 of file dspevd.f.

172*
173* -- LAPACK driver routine --
174* -- LAPACK is a software package provided by Univ. of Tennessee, --
175* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
176*
177* .. Scalar Arguments ..
178 CHARACTER JOBZ, UPLO
179 INTEGER INFO, LDZ, LIWORK, LWORK, N
180* ..
181* .. Array Arguments ..
182 INTEGER IWORK( * )
183 DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
184* ..
185*
186* =====================================================================
187*
188* .. Parameters ..
189 DOUBLE PRECISION ZERO, ONE
190 parameter( zero = 0.0d+0, one = 1.0d+0 )
191* ..
192* .. Local Scalars ..
193 LOGICAL LQUERY, WANTZ
194 INTEGER IINFO, INDE, INDTAU, INDWRK, ISCALE, LIWMIN,
195 \$ LLWORK, LWMIN
196 DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
197 \$ SMLNUM
198* ..
199* .. External Functions ..
200 LOGICAL LSAME
201 DOUBLE PRECISION DLAMCH, DLANSP
202 EXTERNAL lsame, dlamch, dlansp
203* ..
204* .. External Subroutines ..
205 EXTERNAL dopmtr, dscal, dsptrd, dstedc, dsterf, xerbla
206* ..
207* .. Intrinsic Functions ..
208 INTRINSIC sqrt
209* ..
210* .. Executable Statements ..
211*
212* Test the input parameters.
213*
214 wantz = lsame( jobz, 'V' )
215 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
216*
217 info = 0
218 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
219 info = -1
220 ELSE IF( .NOT.( lsame( uplo, 'U' ) .OR. lsame( uplo, 'L' ) ) )
221 \$ THEN
222 info = -2
223 ELSE IF( n.LT.0 ) THEN
224 info = -3
225 ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
226 info = -7
227 END IF
228*
229 IF( info.EQ.0 ) THEN
230 IF( n.LE.1 ) THEN
231 liwmin = 1
232 lwmin = 1
233 ELSE
234 IF( wantz ) THEN
235 liwmin = 3 + 5*n
236 lwmin = 1 + 6*n + n**2
237 ELSE
238 liwmin = 1
239 lwmin = 2*n
240 END IF
241 END IF
242 iwork( 1 ) = liwmin
243 work( 1 ) = lwmin
244*
245 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
246 info = -9
247 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
248 info = -11
249 END IF
250 END IF
251*
252 IF( info.NE.0 ) THEN
253 CALL xerbla( 'DSPEVD', -info )
254 RETURN
255 ELSE IF( lquery ) THEN
256 RETURN
257 END IF
258*
259* Quick return if possible
260*
261 IF( n.EQ.0 )
262 \$ RETURN
263*
264 IF( n.EQ.1 ) THEN
265 w( 1 ) = ap( 1 )
266 IF( wantz )
267 \$ z( 1, 1 ) = one
268 RETURN
269 END IF
270*
271* Get machine constants.
272*
273 safmin = dlamch( 'Safe minimum' )
274 eps = dlamch( 'Precision' )
275 smlnum = safmin / eps
276 bignum = one / smlnum
277 rmin = sqrt( smlnum )
278 rmax = sqrt( bignum )
279*
280* Scale matrix to allowable range, if necessary.
281*
282 anrm = dlansp( 'M', uplo, n, ap, work )
283 iscale = 0
284 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
285 iscale = 1
286 sigma = rmin / anrm
287 ELSE IF( anrm.GT.rmax ) THEN
288 iscale = 1
289 sigma = rmax / anrm
290 END IF
291 IF( iscale.EQ.1 ) THEN
292 CALL dscal( ( n*( n+1 ) ) / 2, sigma, ap, 1 )
293 END IF
294*
295* Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
296*
297 inde = 1
298 indtau = inde + n
299 CALL dsptrd( uplo, n, ap, w, work( inde ), work( indtau ), iinfo )
300*
301* For eigenvalues only, call DSTERF. For eigenvectors, first call
302* DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
303* tridiagonal matrix, then call DOPMTR to multiply it by the
304* Householder transformations represented in AP.
305*
306 IF( .NOT.wantz ) THEN
307 CALL dsterf( n, w, work( inde ), info )
308 ELSE
309 indwrk = indtau + n
310 llwork = lwork - indwrk + 1
311 CALL dstedc( 'I', n, w, work( inde ), z, ldz, work( indwrk ),
312 \$ llwork, iwork, liwork, info )
313 CALL dopmtr( 'L', uplo, 'N', n, n, ap, work( indtau ), z, ldz,
314 \$ work( indwrk ), iinfo )
315 END IF
316*
317* If matrix was scaled, then rescale eigenvalues appropriately.
318*
319 IF( iscale.EQ.1 )
320 \$ CALL dscal( n, one / sigma, w, 1 )
321*
322 work( 1 ) = lwmin
323 iwork( 1 ) = liwmin
324 RETURN
325*
326* End of DSPEVD
327*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dsptrd(uplo, n, ap, d, e, tau, info)
DSPTRD
Definition dsptrd.f:150
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
double precision function dlansp(norm, uplo, n, ap, work)
DLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition dlansp.f:114
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
subroutine dstedc(compz, n, d, e, z, ldz, work, lwork, iwork, liwork, info)
DSTEDC
Definition dstedc.f:182
subroutine dsterf(n, d, e, info)
DSTERF
Definition dsterf.f:86
subroutine dopmtr(side, uplo, trans, m, n, ap, tau, c, ldc, work, info)
DOPMTR
Definition dopmtr.f:150
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