LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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◆ dsbevx()

subroutine dsbevx ( character jobz,
character range,
character uplo,
integer n,
integer kd,
double precision, dimension( ldab, * ) ab,
integer ldab,
double precision, dimension( ldq, * ) q,
integer ldq,
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 )

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

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

Purpose:
!>
!> DSBEVX computes selected eigenvalues and, optionally, eigenvectors
!> of a real symmetric band matrix A.  Eigenvalues and eigenvectors 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]KD
!>          KD is INTEGER
!>          The number of superdiagonals of the matrix A if UPLO = 'U',
!>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
!>
[in,out]AB
!>          AB is DOUBLE PRECISION array, dimension (LDAB, N)
!>          On entry, the upper or lower triangle of the symmetric band
!>          matrix A, stored in the first KD+1 rows of the array.  The
!>          j-th column of A is stored in the j-th column of the array AB
!>          as follows:
!>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
!>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
!>
!>          On exit, AB is overwritten by values generated during the
!>          reduction to tridiagonal form.  If UPLO = 'U', the first
!>          superdiagonal and the diagonal of the tridiagonal matrix T
!>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
!>          the diagonal and first subdiagonal of T are returned in the
!>          first two rows of AB.
!>
[in]LDAB
!>          LDAB is INTEGER
!>          The leading dimension of the array AB.  LDAB >= KD + 1.
!>
[out]Q
!>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
!>          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
!>                         reduction to tridiagonal form.
!>          If JOBZ = 'N', the array Q is not referenced.
!>
[in]LDQ
!>          LDQ is INTEGER
!>          The leading dimension of the array Q.  If JOBZ = 'V', then
!>          LDQ >= max(1,N).
!>
[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 AB 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)
!>          The first M elements contain 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 (7*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 260 of file dsbevx.f.

264*
265* -- LAPACK driver routine --
266* -- LAPACK is a software package provided by Univ. of Tennessee, --
267* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
268*
269* .. Scalar Arguments ..
270 CHARACTER JOBZ, RANGE, UPLO
271 INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
272 DOUBLE PRECISION ABSTOL, VL, VU
273* ..
274* .. Array Arguments ..
275 INTEGER IFAIL( * ), IWORK( * )
276 DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
277 $ Z( LDZ, * )
278* ..
279*
280* =====================================================================
281*
282* .. Parameters ..
283 DOUBLE PRECISION ZERO, ONE
284 parameter( zero = 0.0d0, one = 1.0d0 )
285* ..
286* .. Local Scalars ..
287 LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
288 CHARACTER ORDER
289 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
290 $ INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ,
291 $ NSPLIT
292 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
293 $ SIGMA, SMLNUM, TMP1, VLL, VUU
294* ..
295* .. External Functions ..
296 LOGICAL LSAME
297 DOUBLE PRECISION DLAMCH, DLANSB
298 EXTERNAL lsame, dlamch, dlansb
299* ..
300* .. External Subroutines ..
301 EXTERNAL dcopy, dgemv, dlacpy, dlascl, dsbtrd,
302 $ dscal,
303 $ dstebz, dstein, dsteqr, dsterf, dswap, xerbla
304* ..
305* .. Intrinsic Functions ..
306 INTRINSIC max, min, sqrt
307* ..
308* .. Executable Statements ..
309*
310* Test the input parameters.
311*
312 wantz = lsame( jobz, 'V' )
313 alleig = lsame( range, 'A' )
314 valeig = lsame( range, 'V' )
315 indeig = lsame( range, 'I' )
316 lower = lsame( uplo, 'L' )
317*
318 info = 0
319 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
320 info = -1
321 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
322 info = -2
323 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
324 info = -3
325 ELSE IF( n.LT.0 ) THEN
326 info = -4
327 ELSE IF( kd.LT.0 ) THEN
328 info = -5
329 ELSE IF( ldab.LT.kd+1 ) THEN
330 info = -7
331 ELSE IF( wantz .AND. ldq.LT.max( 1, n ) ) THEN
332 info = -9
333 ELSE
334 IF( valeig ) THEN
335 IF( n.GT.0 .AND. vu.LE.vl )
336 $ info = -11
337 ELSE IF( indeig ) THEN
338 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
339 info = -12
340 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
341 info = -13
342 END IF
343 END IF
344 END IF
345 IF( info.EQ.0 ) THEN
346 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) )
347 $ info = -18
348 END IF
349*
350 IF( info.NE.0 ) THEN
351 CALL xerbla( 'DSBEVX', -info )
352 RETURN
353 END IF
354*
355* Quick return if possible
356*
357 m = 0
358 IF( n.EQ.0 )
359 $ RETURN
360*
361 IF( n.EQ.1 ) THEN
362 m = 1
363 IF( lower ) THEN
364 tmp1 = ab( 1, 1 )
365 ELSE
366 tmp1 = ab( kd+1, 1 )
367 END IF
368 IF( valeig ) THEN
369 IF( .NOT.( vl.LT.tmp1 .AND. vu.GE.tmp1 ) )
370 $ m = 0
371 END IF
372 IF( m.EQ.1 ) THEN
373 w( 1 ) = tmp1
374 IF( wantz )
375 $ z( 1, 1 ) = one
376 END IF
377 RETURN
378 END IF
379*
380* Get machine constants.
381*
382 safmin = dlamch( 'Safe minimum' )
383 eps = dlamch( 'Precision' )
384 smlnum = safmin / eps
385 bignum = one / smlnum
386 rmin = sqrt( smlnum )
387 rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
388*
389* Scale matrix to allowable range, if necessary.
390*
391 iscale = 0
392 abstll = abstol
393 IF( valeig ) THEN
394 vll = vl
395 vuu = vu
396 ELSE
397 vll = zero
398 vuu = zero
399 END IF
400 anrm = dlansb( 'M', uplo, n, kd, ab, ldab, work )
401 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
402 iscale = 1
403 sigma = rmin / anrm
404 ELSE IF( anrm.GT.rmax ) THEN
405 iscale = 1
406 sigma = rmax / anrm
407 END IF
408 IF( iscale.EQ.1 ) THEN
409 IF( lower ) THEN
410 CALL dlascl( 'B', kd, kd, one, sigma, n, n, ab, ldab,
411 $ info )
412 ELSE
413 CALL dlascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab,
414 $ info )
415 END IF
416 IF( abstol.GT.0 )
417 $ abstll = abstol*sigma
418 IF( valeig ) THEN
419 vll = vl*sigma
420 vuu = vu*sigma
421 END IF
422 END IF
423*
424* Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
425*
426 indd = 1
427 inde = indd + n
428 indwrk = inde + n
429 CALL dsbtrd( jobz, uplo, n, kd, ab, ldab, work( indd ),
430 $ work( inde ), q, ldq, work( indwrk ), iinfo )
431*
432* If all eigenvalues are desired and ABSTOL is less than or equal
433* to zero, then call DSTERF or SSTEQR. If this fails for some
434* eigenvalue, then try DSTEBZ.
435*
436 test = .false.
437 IF (indeig) THEN
438 IF (il.EQ.1 .AND. iu.EQ.n) THEN
439 test = .true.
440 END IF
441 END IF
442 IF ((alleig .OR. test) .AND. (abstol.LE.zero)) THEN
443 CALL dcopy( n, work( indd ), 1, w, 1 )
444 indee = indwrk + 2*n
445 IF( .NOT.wantz ) THEN
446 CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
447 CALL dsterf( n, w, work( indee ), info )
448 ELSE
449 CALL dlacpy( 'A', n, n, q, ldq, z, ldz )
450 CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
451 CALL dsteqr( jobz, n, w, work( indee ), z, ldz,
452 $ work( indwrk ), info )
453 IF( info.EQ.0 ) THEN
454 DO 10 i = 1, n
455 ifail( i ) = 0
456 10 CONTINUE
457 END IF
458 END IF
459 IF( info.EQ.0 ) THEN
460 m = n
461 GO TO 30
462 END IF
463 info = 0
464 END IF
465*
466* Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
467*
468 IF( wantz ) THEN
469 order = 'B'
470 ELSE
471 order = 'E'
472 END IF
473 indibl = 1
474 indisp = indibl + n
475 indiwo = indisp + n
476 CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
477 $ work( indd ), work( inde ), m, nsplit, w,
478 $ iwork( indibl ), iwork( indisp ), work( indwrk ),
479 $ iwork( indiwo ), info )
480*
481 IF( wantz ) THEN
482 CALL dstein( n, work( indd ), work( inde ), m, w,
483 $ iwork( indibl ), iwork( indisp ), z, ldz,
484 $ work( indwrk ), iwork( indiwo ), ifail, info )
485*
486* Apply orthogonal matrix used in reduction to tridiagonal
487* form to eigenvectors returned by DSTEIN.
488*
489 DO 20 j = 1, m
490 CALL dcopy( n, z( 1, j ), 1, work( 1 ), 1 )
491 CALL dgemv( 'N', n, n, one, q, ldq, work, 1, zero,
492 $ z( 1, j ), 1 )
493 20 CONTINUE
494 END IF
495*
496* If matrix was scaled, then rescale eigenvalues appropriately.
497*
498 30 CONTINUE
499 IF( iscale.EQ.1 ) THEN
500 IF( info.EQ.0 ) THEN
501 imax = m
502 ELSE
503 imax = info - 1
504 END IF
505 CALL dscal( imax, one / sigma, w, 1 )
506 END IF
507*
508* If eigenvalues are not in order, then sort them, along with
509* eigenvectors.
510*
511 IF( wantz ) THEN
512 DO 50 j = 1, m - 1
513 i = 0
514 tmp1 = w( j )
515 DO 40 jj = j + 1, m
516 IF( w( jj ).LT.tmp1 ) THEN
517 i = jj
518 tmp1 = w( jj )
519 END IF
520 40 CONTINUE
521*
522 IF( i.NE.0 ) THEN
523 itmp1 = iwork( indibl+i-1 )
524 w( i ) = w( j )
525 iwork( indibl+i-1 ) = iwork( indibl+j-1 )
526 w( j ) = tmp1
527 iwork( indibl+j-1 ) = itmp1
528 CALL dswap( n, z( 1, i ), 1, z( 1, j ), 1 )
529 IF( info.NE.0 ) THEN
530 itmp1 = ifail( i )
531 ifail( i ) = ifail( j )
532 ifail( j ) = itmp1
533 END IF
534 END IF
535 50 CONTINUE
536 END IF
537*
538 RETURN
539*
540* End of DSBEVX
541*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
dcopy
subroutine dcopy(n, dx, incx, dy, incy)
DCOPY
Definition dcopy.f:82
dgemv
subroutine dgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
DGEMV
Definition dgemv.f:158
dsbtrd
subroutine dsbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
DSBTRD
Definition dsbtrd.f:161
dlacpy
subroutine dlacpy(uplo, m, n, a, lda, b, ldb)
DLACPY copies all or part of one two-dimensional array to another.
Definition dlacpy.f:101
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
dlansb
double precision function dlansb(norm, uplo, n, k, ab, ldab, work)
DLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition dlansb.f:127
dlascl
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:142
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
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