LAPACK 3.12.0 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

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 "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," 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.```

Definition at line 262 of file dsbevx.f.

265*
266* -- LAPACK driver routine --
267* -- LAPACK is a software package provided by Univ. of Tennessee, --
268* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
269*
270* .. Scalar Arguments ..
271 CHARACTER JOBZ, RANGE, UPLO
272 INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
273 DOUBLE PRECISION ABSTOL, VL, VU
274* ..
275* .. Array Arguments ..
276 INTEGER IFAIL( * ), IWORK( * )
277 DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
278 \$ Z( LDZ, * )
279* ..
280*
281* =====================================================================
282*
283* .. Parameters ..
284 DOUBLE PRECISION ZERO, ONE
285 parameter( zero = 0.0d0, one = 1.0d0 )
286* ..
287* .. Local Scalars ..
288 LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
289 CHARACTER ORDER
290 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
291 \$ INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ,
292 \$ NSPLIT
293 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
294 \$ SIGMA, SMLNUM, TMP1, VLL, VUU
295* ..
296* .. External Functions ..
297 LOGICAL LSAME
298 DOUBLE PRECISION DLAMCH, DLANSB
299 EXTERNAL lsame, dlamch, dlansb
300* ..
301* .. External Subroutines ..
302 EXTERNAL dcopy, dgemv, dlacpy, dlascl, dsbtrd, dscal,
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, info )
411 ELSE
412 CALL dlascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab, info )
413 END IF
414 IF( abstol.GT.0 )
415 \$ abstll = abstol*sigma
416 IF( valeig ) THEN
417 vll = vl*sigma
418 vuu = vu*sigma
419 END IF
420 END IF
421*
422* Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
423*
424 indd = 1
425 inde = indd + n
426 indwrk = inde + n
427 CALL dsbtrd( jobz, uplo, n, kd, ab, ldab, work( indd ),
428 \$ work( inde ), q, ldq, work( indwrk ), iinfo )
429*
430* If all eigenvalues are desired and ABSTOL is less than or equal
431* to zero, then call DSTERF or SSTEQR. If this fails for some
432* eigenvalue, then try DSTEBZ.
433*
434 test = .false.
435 IF (indeig) THEN
436 IF (il.EQ.1 .AND. iu.EQ.n) THEN
437 test = .true.
438 END IF
439 END IF
440 IF ((alleig .OR. test) .AND. (abstol.LE.zero)) THEN
441 CALL dcopy( n, work( indd ), 1, w, 1 )
442 indee = indwrk + 2*n
443 IF( .NOT.wantz ) THEN
444 CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
445 CALL dsterf( n, w, work( indee ), info )
446 ELSE
447 CALL dlacpy( 'A', n, n, q, ldq, z, ldz )
448 CALL dcopy( n-1, work( inde ), 1, work( indee ), 1 )
449 CALL dsteqr( jobz, n, w, work( indee ), z, ldz,
450 \$ work( indwrk ), info )
451 IF( info.EQ.0 ) THEN
452 DO 10 i = 1, n
453 ifail( i ) = 0
454 10 CONTINUE
455 END IF
456 END IF
457 IF( info.EQ.0 ) THEN
458 m = n
459 GO TO 30
460 END IF
461 info = 0
462 END IF
463*
464* Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
465*
466 IF( wantz ) THEN
467 order = 'B'
468 ELSE
469 order = 'E'
470 END IF
471 indibl = 1
472 indisp = indibl + n
473 indiwo = indisp + n
474 CALL dstebz( range, order, n, vll, vuu, il, iu, abstll,
475 \$ work( indd ), work( inde ), m, nsplit, w,
476 \$ iwork( indibl ), iwork( indisp ), work( indwrk ),
477 \$ iwork( indiwo ), info )
478*
479 IF( wantz ) THEN
480 CALL dstein( n, work( indd ), work( inde ), m, w,
481 \$ iwork( indibl ), iwork( indisp ), z, ldz,
482 \$ work( indwrk ), iwork( indiwo ), ifail, info )
483*
484* Apply orthogonal matrix used in reduction to tridiagonal
485* form to eigenvectors returned by DSTEIN.
486*
487 DO 20 j = 1, m
488 CALL dcopy( n, z( 1, j ), 1, work( 1 ), 1 )
489 CALL dgemv( 'N', n, n, one, q, ldq, work, 1, zero,
490 \$ z( 1, j ), 1 )
491 20 CONTINUE
492 END IF
493*
494* If matrix was scaled, then rescale eigenvalues appropriately.
495*
496 30 CONTINUE
497 IF( iscale.EQ.1 ) THEN
498 IF( info.EQ.0 ) THEN
499 imax = m
500 ELSE
501 imax = info - 1
502 END IF
503 CALL dscal( imax, one / sigma, w, 1 )
504 END IF
505*
506* If eigenvalues are not in order, then sort them, along with
507* eigenvectors.
508*
509 IF( wantz ) THEN
510 DO 50 j = 1, m - 1
511 i = 0
512 tmp1 = w( j )
513 DO 40 jj = j + 1, m
514 IF( w( jj ).LT.tmp1 ) THEN
515 i = jj
516 tmp1 = w( jj )
517 END IF
518 40 CONTINUE
519*
520 IF( i.NE.0 ) THEN
521 itmp1 = iwork( indibl+i-1 )
522 w( i ) = w( j )
523 iwork( indibl+i-1 ) = iwork( indibl+j-1 )
524 w( j ) = tmp1
525 iwork( indibl+j-1 ) = itmp1
526 CALL dswap( n, z( 1, i ), 1, z( 1, j ), 1 )
527 IF( info.NE.0 ) THEN
528 itmp1 = ifail( i )
529 ifail( i ) = ifail( j )
530 ifail( j ) = itmp1
531 END IF
532 END IF
533 50 CONTINUE
534 END IF
535*
536 RETURN
537*
538* End of DSBEVX
539*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dcopy(n, dx, incx, dy, incy)
DCOPY
Definition dcopy.f:82
subroutine dgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
DGEMV
Definition dgemv.f:158
subroutine dsbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
DSBTRD
Definition dsbtrd.f:163
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
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
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:129
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
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
subroutine dstebz(range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
DSTEBZ
Definition dstebz.f:273
subroutine dstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)
DSTEIN
Definition dstein.f:174
subroutine dsteqr(compz, n, d, e, z, ldz, work, info)
DSTEQR
Definition dsteqr.f:131
subroutine dsterf(n, d, e, info)
DSTERF
Definition dsterf.f:86
subroutine dswap(n, dx, incx, dy, incy)
DSWAP
Definition dswap.f:82
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