LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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◆ dspevx()

subroutine dspevx ( character jobz,
character range,
character uplo,
integer n,
double precision, dimension( * ) ap,
double precision vl,
double precision vu,
integer il,
integer iu,
double precision abstol,
integer m,
double precision, dimension( * ) w,
double precision, dimension( ldz, * ) z,
integer ldz,
double precision, dimension( * ) work,
integer, dimension( * ) iwork,
integer, dimension( * ) ifail,
integer info )

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

Download DSPEVX + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> DSPEVX computes selected eigenvalues and, optionally, eigenvectors
!> of a real symmetric matrix A in packed storage.  Eigenvalues/vectors
!> can be selected by specifying either a range of values or a range of
!> indices for the desired eigenvalues.
!>
Parameters
[in]JOBZ
!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors.
!>
[in]RANGE
!>          RANGE is CHARACTER*1
!>          = 'A': all eigenvalues will be found;
!>          = 'V': all eigenvalues in the half-open interval (VL,VU]
!>                 will be found;
!>          = 'I': the IL-th through IU-th eigenvalues will be found.
!>
[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.
!>
[in]VL
!>          VL is DOUBLE PRECISION
!>          If RANGE='V', the lower bound of the interval to
!>          be searched for eigenvalues. VL < VU.
!>          Not referenced if RANGE = 'A' or 'I'.
!>
[in]VU
!>          VU is DOUBLE PRECISION
!>          If RANGE='V', the upper bound of the interval to
!>          be searched for eigenvalues. VL < VU.
!>          Not referenced if RANGE = 'A' or 'I'.
!>
[in]IL
!>          IL is INTEGER
!>          If RANGE='I', the index of the
!>          smallest eigenvalue to be returned.
!>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
!>          Not referenced if RANGE = 'A' or 'V'.
!>
[in]IU
!>          IU is INTEGER
!>          If RANGE='I', the index of the
!>          largest eigenvalue to be returned.
!>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
!>          Not referenced if RANGE = 'A' or 'V'.
!>
[in]ABSTOL
!>          ABSTOL is DOUBLE PRECISION
!>          The absolute error tolerance for the eigenvalues.
!>          An approximate eigenvalue is accepted as converged
!>          when it is determined to lie in an interval [a,b]
!>          of width less than or equal to
!>
!>                  ABSTOL + EPS *   max( |a|,|b| ) ,
!>
!>          where EPS is the machine precision.  If ABSTOL is less than
!>          or equal to zero, then  EPS*|T|  will be used in its place,
!>          where |T| is the 1-norm of the tridiagonal matrix obtained
!>          by reducing AP to tridiagonal form.
!>
!>          Eigenvalues will be computed most accurately when ABSTOL is
!>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
!>          If this routine returns with INFO>0, indicating that some
!>          eigenvectors did not converge, try setting ABSTOL to
!>          2*DLAMCH('S').
!>
!>          See  by Demmel and
!>          Kahan, LAPACK Working Note #3.
!>
[out]M
!>          M is INTEGER
!>          The total number of eigenvalues found.  0 <= M <= N.
!>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
!>
[out]W
!>          W is DOUBLE PRECISION array, dimension (N)
!>          If INFO = 0, the selected eigenvalues in ascending order.
!>
[out]Z
!>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
!>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
!>          contain the orthonormal eigenvectors of the matrix A
!>          corresponding to the selected eigenvalues, with the i-th
!>          column of Z holding the eigenvector associated with W(i).
!>          If an eigenvector fails to converge, then that column of Z
!>          contains the latest approximation to the eigenvector, and the
!>          index of the eigenvector is returned in IFAIL.
!>          If JOBZ = 'N', then Z is not referenced.
!>          Note: the user must ensure that at least max(1,M) columns are
!>          supplied in the array Z; if RANGE = 'V', the exact value of M
!>          is not known in advance and an upper bound must be used.
!>
[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 (8*N)
!>
[out]IWORK
!>          IWORK is INTEGER array, dimension (5*N)
!>
[out]IFAIL
!>          IFAIL is INTEGER array, dimension (N)
!>          If JOBZ = 'V', then if INFO = 0, the first M elements of
!>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
!>          indices of the eigenvectors that failed to converge.
!>          If JOBZ = 'N', then IFAIL is not referenced.
!>
[out]INFO
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  if INFO = i, then i eigenvectors failed to converge.
!>                Their indices are stored in array IFAIL.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 229 of file dspevx.f.

232*
233* -- LAPACK driver routine --
234* -- LAPACK is a software package provided by Univ. of Tennessee, --
235* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
236*
237* .. Scalar Arguments ..
238 CHARACTER JOBZ, RANGE, UPLO
239 INTEGER IL, INFO, IU, LDZ, M, N
240 DOUBLE PRECISION ABSTOL, VL, VU
241* ..
242* .. Array Arguments ..
243 INTEGER IFAIL( * ), IWORK( * )
244 DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
245* ..
246*
247* =====================================================================
248*
249* .. Parameters ..
250 DOUBLE PRECISION ZERO, ONE
251 parameter( zero = 0.0d0, one = 1.0d0 )
252* ..
253* .. Local Scalars ..
254 LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
255 CHARACTER ORDER
256 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE,
257 $ INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1,
258 $ J, JJ, NSPLIT
259 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
260 $ SIGMA, SMLNUM, TMP1, VLL, VUU
261* ..
262* .. External Functions ..
263 LOGICAL LSAME
264 DOUBLE PRECISION DLAMCH, DLANSP
265 EXTERNAL lsame, dlamch, dlansp
266* ..
267* .. External Subroutines ..
268 EXTERNAL dcopy, dopgtr, dopmtr, dscal, dsptrd,
269 $ dstebz,
270 $ dstein, dsteqr, dsterf, dswap, xerbla
271* ..
272* .. Intrinsic Functions ..
273 INTRINSIC max, min, sqrt
274* ..
275* .. Executable Statements ..
276*
277* Test the input parameters.
278*
279 wantz = lsame( jobz, 'V' )
280 alleig = lsame( range, 'A' )
281 valeig = lsame( range, 'V' )
282 indeig = lsame( range, 'I' )
283*
284 info = 0
285 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
286 info = -1
287 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
288 info = -2
289 ELSE IF( .NOT.( lsame( uplo, 'L' ) .OR.
290 $ lsame( uplo, 'U' ) ) )
291 $ THEN
292 info = -3
293 ELSE IF( n.LT.0 ) THEN
294 info = -4
295 ELSE
296 IF( valeig ) THEN
297 IF( n.GT.0 .AND. vu.LE.vl )
298 $ info = -7
299 ELSE IF( indeig ) THEN
300 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
301 info = -8
302 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
303 info = -9
304 END IF
305 END IF
306 END IF
307 IF( info.EQ.0 ) THEN
308 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) )
309 $ info = -14
310 END IF
311*
312 IF( info.NE.0 ) THEN
313 CALL xerbla( 'DSPEVX', -info )
314 RETURN
315 END IF
316*
317* Quick return if possible
318*
319 m = 0
320 IF( n.EQ.0 )
321 $ RETURN
322*
323 IF( n.EQ.1 ) THEN
324 IF( alleig .OR. indeig ) THEN
325 m = 1
326 w( 1 ) = ap( 1 )
327 ELSE
328 IF( vl.LT.ap( 1 ) .AND. vu.GE.ap( 1 ) ) THEN
329 m = 1
330 w( 1 ) = ap( 1 )
331 END IF
332 END IF
333 IF( wantz )
334 $ z( 1, 1 ) = one
335 RETURN
336 END IF
337*
338* Get machine constants.
339*
340 safmin = dlamch( 'Safe minimum' )
341 eps = dlamch( 'Precision' )
342 smlnum = safmin / eps
343 bignum = one / smlnum
344 rmin = sqrt( smlnum )
345 rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
346*
347* Scale matrix to allowable range, if necessary.
348*
349 iscale = 0
350 abstll = abstol
351 IF( valeig ) THEN
352 vll = vl
353 vuu = vu
354 ELSE
355 vll = zero
356 vuu = zero
357 END IF
358 anrm = dlansp( 'M', uplo, n, ap, work )
359 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
360 iscale = 1
361 sigma = rmin / anrm
362 ELSE IF( anrm.GT.rmax ) THEN
363 iscale = 1
364 sigma = rmax / anrm
365 END IF
366 IF( iscale.EQ.1 ) THEN
367 CALL dscal( ( n*( n+1 ) ) / 2, sigma, ap, 1 )
368 IF( abstol.GT.0 )
369 $ abstll = abstol*sigma
370 IF( valeig ) THEN
371 vll = vl*sigma
372 vuu = vu*sigma
373 END IF
374 END IF
375*
376* Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
377*
378 indtau = 1
379 inde = indtau + n
380 indd = inde + n
381 indwrk = indd + n
382 CALL dsptrd( uplo, n, ap, work( indd ), work( inde ),
383 $ work( indtau ), iinfo )
384*
385* If all eigenvalues are desired and ABSTOL is less than or equal
386* to zero, then call DSTERF or DOPGTR and SSTEQR. If this fails
387* for some eigenvalue, then try DSTEBZ.
388*
389 test = .false.
390 IF (indeig) THEN
391 IF (il.EQ.1 .AND. iu.EQ.n) THEN
392 test = .true.
393 END IF
394 END IF
395 IF ((alleig .OR. test) .AND. (abstol.LE.zero)) THEN
396 CALL dcopy( n, work( indd ), 1, w, 1 )
397 indee = indwrk + 2*n
398 IF( .NOT.wantz ) THEN
399 CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
400 CALL dsterf( n, w, work( indee ), info )
401 ELSE
402 CALL dopgtr( uplo, n, ap, work( indtau ), z, ldz,
403 $ work( indwrk ), iinfo )
404 CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
405 CALL dsteqr( jobz, n, w, work( indee ), z, ldz,
406 $ work( indwrk ), info )
407 IF( info.EQ.0 ) THEN
408 DO 10 i = 1, n
409 ifail( i ) = 0
410 10 CONTINUE
411 END IF
412 END IF
413 IF( info.EQ.0 ) THEN
414 m = n
415 GO TO 20
416 END IF
417 info = 0
418 END IF
419*
420* Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
421*
422 IF( wantz ) THEN
423 order = 'B'
424 ELSE
425 order = 'E'
426 END IF
427 indisp = 1 + n
428 indiwo = indisp + n
429 CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
430 $ work( indd ), work( inde ), m, nsplit, w,
431 $ iwork( 1 ), iwork( indisp ), work( indwrk ),
432 $ iwork( indiwo ), info )
433*
434 IF( wantz ) THEN
435 CALL dstein( n, work( indd ), work( inde ), m, w,
436 $ iwork( 1 ), iwork( indisp ), z, ldz,
437 $ work( indwrk ), iwork( indiwo ), ifail, info )
438*
439* Apply orthogonal matrix used in reduction to tridiagonal
440* form to eigenvectors returned by DSTEIN.
441*
442 CALL dopmtr( 'L', uplo, 'N', n, m, ap, work( indtau ), z,
443 $ ldz,
444 $ work( indwrk ), iinfo )
445 END IF
446*
447* If matrix was scaled, then rescale eigenvalues appropriately.
448*
449 20 CONTINUE
450 IF( iscale.EQ.1 ) THEN
451 IF( info.EQ.0 ) THEN
452 imax = m
453 ELSE
454 imax = info - 1
455 END IF
456 CALL dscal( imax, one / sigma, w, 1 )
457 END IF
458*
459* If eigenvalues are not in order, then sort them, along with
460* eigenvectors.
461*
462 IF( wantz ) THEN
463 DO 40 j = 1, m - 1
464 i = 0
465 tmp1 = w( j )
466 DO 30 jj = j + 1, m
467 IF( w( jj ).LT.tmp1 ) THEN
468 i = jj
469 tmp1 = w( jj )
470 END IF
471 30 CONTINUE
472*
473 IF( i.NE.0 ) THEN
474 itmp1 = iwork( 1 + i-1 )
475 w( i ) = w( j )
476 iwork( 1 + i-1 ) = iwork( 1 + j-1 )
477 w( j ) = tmp1
478 iwork( 1 + j-1 ) = itmp1
479 CALL dswap( n, z( 1, i ), 1, z( 1, j ), 1 )
480 IF( info.NE.0 ) THEN
481 itmp1 = ifail( i )
482 ifail( i ) = ifail( j )
483 ifail( j ) = itmp1
484 END IF
485 END IF
486 40 CONTINUE
487 END IF
488*
489 RETURN
490*
491* End of DSPEVX
492*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
dcopy
subroutine dcopy(n, dx, incx, dy, incy)
DCOPY
Definition dcopy.f:82
dsptrd
subroutine dsptrd(uplo, n, ap, d, e, tau, info)
DSPTRD
Definition dsptrd.f:148
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
dlansp
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:112
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
dscal
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
dstebz
subroutine dstebz(range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
DSTEBZ
Definition dstebz.f:272
dstein
subroutine dstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)
DSTEIN
Definition dstein.f:172
dsteqr
subroutine dsteqr(compz, n, d, e, z, ldz, work, info)
DSTEQR
Definition dsteqr.f:129
dsterf
subroutine dsterf(n, d, e, info)
DSTERF
Definition dsterf.f:84
dswap
subroutine dswap(n, dx, incx, dy, incy)
DSWAP
Definition dswap.f:82
dopgtr
subroutine dopgtr(uplo, n, ap, tau, q, ldq, work, info)
DOPGTR
Definition dopgtr.f:112
dopmtr
subroutine dopmtr(side, uplo, trans, m, n, ap, tau, c, ldc, work, info)
DOPMTR
Definition dopmtr.f:149
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