 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dstedc()

 subroutine dstedc ( character COMPZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

DSTEDC

Purpose:
``` DSTEDC computes all eigenvalues and, optionally, eigenvectors of a
symmetric tridiagonal matrix using the divide and conquer method.
The eigenvectors of a full or band real symmetric matrix can also be
found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this
matrix to tridiagonal form.

This code 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.  See DLAED3 for details.```
Parameters
 [in] COMPZ ``` COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'I': Compute eigenvectors of tridiagonal matrix also. = 'V': Compute eigenvectors of original dense symmetric matrix also. On entry, Z contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.``` [in] N ``` N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.``` [in,out] D ``` D is DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order.``` [in,out] E ``` E is DOUBLE PRECISION array, dimension (N-1) On entry, the subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.``` [in,out] Z ``` Z is DOUBLE PRECISION array, dimension (LDZ,N) On entry, if COMPZ = 'V', then Z contains the orthogonal matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original symmetric matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If eigenvectors are desired, then LDZ >= max(1,N).``` [out] WORK ``` WORK is DOUBLE PRECISION 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 COMPZ = 'N' or N <= 1 then LWORK must be at least 1. If COMPZ = 'V' and N > 1 then LWORK must be at least ( 1 + 3*N + 2*N*lg N + 4*N**2 ), where lg( N ) = smallest integer k such that 2**k >= N. If COMPZ = 'I' and N > 1 then LWORK must be at least ( 1 + 4*N + N**2 ). Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LWORK need only be max(1,2*(N-1)). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK 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 COMPZ = 'N' or N <= 1 then LIWORK must be at least 1. If COMPZ = 'V' and N > 1 then LIWORK must be at least ( 6 + 6*N + 5*N*lg N ). If COMPZ = 'I' and N > 1 then LIWORK must be at least ( 3 + 5*N ). Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LIWORK need only be 1. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to 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: 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

Definition at line 186 of file dstedc.f.

188*
189* -- LAPACK computational routine --
190* -- LAPACK is a software package provided by Univ. of Tennessee, --
191* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
192*
193* .. Scalar Arguments ..
194 CHARACTER COMPZ
195 INTEGER INFO, LDZ, LIWORK, LWORK, N
196* ..
197* .. Array Arguments ..
198 INTEGER IWORK( * )
199 DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * )
200* ..
201*
202* =====================================================================
203*
204* .. Parameters ..
205 DOUBLE PRECISION ZERO, ONE, TWO
206 parameter( zero = 0.0d0, one = 1.0d0, two = 2.0d0 )
207* ..
208* .. Local Scalars ..
209 LOGICAL LQUERY
210 INTEGER FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN,
211 \$ LWMIN, M, SMLSIZ, START, STOREZ, STRTRW
212 DOUBLE PRECISION EPS, ORGNRM, P, TINY
213* ..
214* .. External Functions ..
215 LOGICAL LSAME
216 INTEGER ILAENV
217 DOUBLE PRECISION DLAMCH, DLANST
218 EXTERNAL lsame, ilaenv, dlamch, dlanst
219* ..
220* .. External Subroutines ..
221 EXTERNAL dgemm, dlacpy, dlaed0, dlascl, dlaset, dlasrt,
223* ..
224* .. Intrinsic Functions ..
225 INTRINSIC abs, dble, int, log, max, mod, sqrt
226* ..
227* .. Executable Statements ..
228*
229* Test the input parameters.
230*
231 info = 0
232 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
233*
234 IF( lsame( compz, 'N' ) ) THEN
235 icompz = 0
236 ELSE IF( lsame( compz, 'V' ) ) THEN
237 icompz = 1
238 ELSE IF( lsame( compz, 'I' ) ) THEN
239 icompz = 2
240 ELSE
241 icompz = -1
242 END IF
243 IF( icompz.LT.0 ) THEN
244 info = -1
245 ELSE IF( n.LT.0 ) THEN
246 info = -2
247 ELSE IF( ( ldz.LT.1 ) .OR.
248 \$ ( icompz.GT.0 .AND. ldz.LT.max( 1, n ) ) ) THEN
249 info = -6
250 END IF
251*
252 IF( info.EQ.0 ) THEN
253*
254* Compute the workspace requirements
255*
256 smlsiz = ilaenv( 9, 'DSTEDC', ' ', 0, 0, 0, 0 )
257 IF( n.LE.1 .OR. icompz.EQ.0 ) THEN
258 liwmin = 1
259 lwmin = 1
260 ELSE IF( n.LE.smlsiz ) THEN
261 liwmin = 1
262 lwmin = 2*( n - 1 )
263 ELSE
264 lgn = int( log( dble( n ) )/log( two ) )
265 IF( 2**lgn.LT.n )
266 \$ lgn = lgn + 1
267 IF( 2**lgn.LT.n )
268 \$ lgn = lgn + 1
269 IF( icompz.EQ.1 ) THEN
270 lwmin = 1 + 3*n + 2*n*lgn + 4*n**2
271 liwmin = 6 + 6*n + 5*n*lgn
272 ELSE IF( icompz.EQ.2 ) THEN
273 lwmin = 1 + 4*n + n**2
274 liwmin = 3 + 5*n
275 END IF
276 END IF
277 work( 1 ) = lwmin
278 iwork( 1 ) = liwmin
279*
280 IF( lwork.LT.lwmin .AND. .NOT. lquery ) THEN
281 info = -8
282 ELSE IF( liwork.LT.liwmin .AND. .NOT. lquery ) THEN
283 info = -10
284 END IF
285 END IF
286*
287 IF( info.NE.0 ) THEN
288 CALL xerbla( 'DSTEDC', -info )
289 RETURN
290 ELSE IF (lquery) THEN
291 RETURN
292 END IF
293*
294* Quick return if possible
295*
296 IF( n.EQ.0 )
297 \$ RETURN
298 IF( n.EQ.1 ) THEN
299 IF( icompz.NE.0 )
300 \$ z( 1, 1 ) = one
301 RETURN
302 END IF
303*
304* If the following conditional clause is removed, then the routine
305* will use the Divide and Conquer routine to compute only the
306* eigenvalues, which requires (3N + 3N**2) real workspace and
307* (2 + 5N + 2N lg(N)) integer workspace.
308* Since on many architectures DSTERF is much faster than any other
309* algorithm for finding eigenvalues only, it is used here
310* as the default. If the conditional clause is removed, then
311* information on the size of workspace needs to be changed.
312*
313* If COMPZ = 'N', use DSTERF to compute the eigenvalues.
314*
315 IF( icompz.EQ.0 ) THEN
316 CALL dsterf( n, d, e, info )
317 GO TO 50
318 END IF
319*
320* If N is smaller than the minimum divide size (SMLSIZ+1), then
321* solve the problem with another solver.
322*
323 IF( n.LE.smlsiz ) THEN
324*
325 CALL dsteqr( compz, n, d, e, z, ldz, work, info )
326*
327 ELSE
328*
329* If COMPZ = 'V', the Z matrix must be stored elsewhere for later
330* use.
331*
332 IF( icompz.EQ.1 ) THEN
333 storez = 1 + n*n
334 ELSE
335 storez = 1
336 END IF
337*
338 IF( icompz.EQ.2 ) THEN
339 CALL dlaset( 'Full', n, n, zero, one, z, ldz )
340 END IF
341*
342* Scale.
343*
344 orgnrm = dlanst( 'M', n, d, e )
345 IF( orgnrm.EQ.zero )
346 \$ GO TO 50
347*
348 eps = dlamch( 'Epsilon' )
349*
350 start = 1
351*
352* while ( START <= N )
353*
354 10 CONTINUE
355 IF( start.LE.n ) THEN
356*
357* Let FINISH be the position of the next subdiagonal entry
358* such that E( FINISH ) <= TINY or FINISH = N if no such
359* subdiagonal exists. The matrix identified by the elements
360* between START and FINISH constitutes an independent
361* sub-problem.
362*
363 finish = start
364 20 CONTINUE
365 IF( finish.LT.n ) THEN
366 tiny = eps*sqrt( abs( d( finish ) ) )*
367 \$ sqrt( abs( d( finish+1 ) ) )
368 IF( abs( e( finish ) ).GT.tiny ) THEN
369 finish = finish + 1
370 GO TO 20
371 END IF
372 END IF
373*
374* (Sub) Problem determined. Compute its size and solve it.
375*
376 m = finish - start + 1
377 IF( m.EQ.1 ) THEN
378 start = finish + 1
379 GO TO 10
380 END IF
381 IF( m.GT.smlsiz ) THEN
382*
383* Scale.
384*
385 orgnrm = dlanst( 'M', m, d( start ), e( start ) )
386 CALL dlascl( 'G', 0, 0, orgnrm, one, m, 1, d( start ), m,
387 \$ info )
388 CALL dlascl( 'G', 0, 0, orgnrm, one, m-1, 1, e( start ),
389 \$ m-1, info )
390*
391 IF( icompz.EQ.1 ) THEN
392 strtrw = 1
393 ELSE
394 strtrw = start
395 END IF
396 CALL dlaed0( icompz, n, m, d( start ), e( start ),
397 \$ z( strtrw, start ), ldz, work( 1 ), n,
398 \$ work( storez ), iwork, info )
399 IF( info.NE.0 ) THEN
400 info = ( info / ( m+1 )+start-1 )*( n+1 ) +
401 \$ mod( info, ( m+1 ) ) + start - 1
402 GO TO 50
403 END IF
404*
405* Scale back.
406*
407 CALL dlascl( 'G', 0, 0, one, orgnrm, m, 1, d( start ), m,
408 \$ info )
409*
410 ELSE
411 IF( icompz.EQ.1 ) THEN
412*
413* Since QR won't update a Z matrix which is larger than
414* the length of D, we must solve the sub-problem in a
415* workspace and then multiply back into Z.
416*
417 CALL dsteqr( 'I', m, d( start ), e( start ), work, m,
418 \$ work( m*m+1 ), info )
419 CALL dlacpy( 'A', n, m, z( 1, start ), ldz,
420 \$ work( storez ), n )
421 CALL dgemm( 'N', 'N', n, m, m, one,
422 \$ work( storez ), n, work, m, zero,
423 \$ z( 1, start ), ldz )
424 ELSE IF( icompz.EQ.2 ) THEN
425 CALL dsteqr( 'I', m, d( start ), e( start ),
426 \$ z( start, start ), ldz, work, info )
427 ELSE
428 CALL dsterf( m, d( start ), e( start ), info )
429 END IF
430 IF( info.NE.0 ) THEN
431 info = start*( n+1 ) + finish
432 GO TO 50
433 END IF
434 END IF
435*
436 start = finish + 1
437 GO TO 10
438 END IF
439*
440* endwhile
441*
442 IF( icompz.EQ.0 ) THEN
443*
444* Use Quick Sort
445*
446 CALL dlasrt( 'I', n, d, info )
447*
448 ELSE
449*
450* Use Selection Sort to minimize swaps of eigenvectors
451*
452 DO 40 ii = 2, n
453 i = ii - 1
454 k = i
455 p = d( i )
456 DO 30 j = ii, n
457 IF( d( j ).LT.p ) THEN
458 k = j
459 p = d( j )
460 END IF
461 30 CONTINUE
462 IF( k.NE.i ) THEN
463 d( k ) = d( i )
464 d( i ) = p
465 CALL dswap( n, z( 1, i ), 1, z( 1, k ), 1 )
466 END IF
467 40 CONTINUE
468 END IF
469 END IF
470*
471 50 CONTINUE
472 work( 1 ) = lwmin
473 iwork( 1 ) = liwmin
474*
475 RETURN
476*
477* End of DSTEDC
478*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
double precision function dlanst(NORM, N, D, E)
DLANST returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlanst.f:100
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
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
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 dsteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
DSTEQR
Definition: dsteqr.f:131
subroutine dlasrt(ID, N, D, INFO)
DLASRT sorts numbers in increasing or decreasing order.
Definition: dlasrt.f:88
subroutine dlaed0(ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS, WORK, IWORK, INFO)
DLAED0 used by DSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmet...
Definition: dlaed0.f:172
subroutine dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:86
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:82
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
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