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

 subroutine ssbgvx ( character jobz, character range, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info )

SSBGVX

Purpose:
``` SSBGVX computes selected eigenvalues, and optionally, eigenvectors
of a real generalized symmetric-definite banded eigenproblem, of
the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric
and banded, and B is also positive definite.  Eigenvalues and
eigenvectors can be selected by specifying either all eigenvalues,
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 triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.``` [in] N ``` N is INTEGER The order of the matrices A and B. N >= 0.``` [in] KA ``` KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.``` [in] KB ``` KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.``` [in,out] AB ``` AB is REAL array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.``` [in,out] BB ``` BB is REAL array, dimension (LDBB, N) On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by SPBSTF.``` [in] LDBB ``` LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.``` [out] Q ``` Q is REAL array, dimension (LDQ, N) If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C 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 = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).``` [in] VL ``` VL is REAL 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 REAL 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 REAL 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 A to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').``` [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 REAL array, dimension (N) If INFO = 0, the eigenvalues in ascending order.``` [out] Z ``` Z is REAL array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = 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 REAL array, dimension (7*N)` [out] IWORK ` IWORK is INTEGER array, dimension (5*N)` [out] IFAIL ``` IFAIL is INTEGER array, dimension (M) 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 eigenvalues 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 <= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL. > N: SPBSTF returned an error code; i.e., if INFO = N + i, for 1 <= i <= N, then the leading principal minor of order i of B is not positive. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.```
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 291 of file ssbgvx.f.

294*
295* -- LAPACK driver routine --
296* -- LAPACK is a software package provided by Univ. of Tennessee, --
297* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
298*
299* .. Scalar Arguments ..
300 CHARACTER JOBZ, RANGE, UPLO
301 INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
302 \$ N
303 REAL ABSTOL, VL, VU
304* ..
305* .. Array Arguments ..
306 INTEGER IFAIL( * ), IWORK( * )
307 REAL AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
308 \$ W( * ), WORK( * ), Z( LDZ, * )
309* ..
310*
311* =====================================================================
312*
313* .. Parameters ..
314 REAL ZERO, ONE
315 parameter( zero = 0.0e+0, one = 1.0e+0 )
316* ..
317* .. Local Scalars ..
318 LOGICAL ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
319 CHARACTER ORDER, VECT
320 INTEGER I, IINFO, INDD, INDE, INDEE, INDISP,
321 \$ INDIWO, INDWRK, ITMP1, J, JJ, NSPLIT
322 REAL TMP1
323* ..
324* .. External Functions ..
325 LOGICAL LSAME
326 EXTERNAL lsame
327* ..
328* .. External Subroutines ..
329 EXTERNAL scopy, sgemv, slacpy, spbstf, ssbgst, ssbtrd,
331* ..
332* .. Intrinsic Functions ..
333 INTRINSIC min
334* ..
335* .. Executable Statements ..
336*
337* Test the input parameters.
338*
339 wantz = lsame( jobz, 'V' )
340 upper = lsame( uplo, 'U' )
341 alleig = lsame( range, 'A' )
342 valeig = lsame( range, 'V' )
343 indeig = lsame( range, 'I' )
344*
345 info = 0
346 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
347 info = -1
348 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
349 info = -2
350 ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
351 info = -3
352 ELSE IF( n.LT.0 ) THEN
353 info = -4
354 ELSE IF( ka.LT.0 ) THEN
355 info = -5
356 ELSE IF( kb.LT.0 .OR. kb.GT.ka ) THEN
357 info = -6
358 ELSE IF( ldab.LT.ka+1 ) THEN
359 info = -8
360 ELSE IF( ldbb.LT.kb+1 ) THEN
361 info = -10
362 ELSE IF( ldq.LT.1 .OR. ( wantz .AND. ldq.LT.n ) ) THEN
363 info = -12
364 ELSE
365 IF( valeig ) THEN
366 IF( n.GT.0 .AND. vu.LE.vl )
367 \$ info = -14
368 ELSE IF( indeig ) THEN
369 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
370 info = -15
371 ELSE IF ( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
372 info = -16
373 END IF
374 END IF
375 END IF
376 IF( info.EQ.0) THEN
377 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
378 info = -21
379 END IF
380 END IF
381*
382 IF( info.NE.0 ) THEN
383 CALL xerbla( 'SSBGVX', -info )
384 RETURN
385 END IF
386*
387* Quick return if possible
388*
389 m = 0
390 IF( n.EQ.0 )
391 \$ RETURN
392*
393* Form a split Cholesky factorization of B.
394*
395 CALL spbstf( uplo, n, kb, bb, ldbb, info )
396 IF( info.NE.0 ) THEN
397 info = n + info
398 RETURN
399 END IF
400*
401* Transform problem to standard eigenvalue problem.
402*
403 CALL ssbgst( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq,
404 \$ work, iinfo )
405*
406* Reduce symmetric band matrix to tridiagonal form.
407*
408 indd = 1
409 inde = indd + n
410 indwrk = inde + n
411 IF( wantz ) THEN
412 vect = 'U'
413 ELSE
414 vect = 'N'
415 END IF
416 CALL ssbtrd( vect, uplo, n, ka, ab, ldab, work( indd ),
417 \$ work( inde ), q, ldq, work( indwrk ), iinfo )
418*
419* If all eigenvalues are desired and ABSTOL is less than or equal
420* to zero, then call SSTERF or SSTEQR. If this fails for some
421* eigenvalue, then try SSTEBZ.
422*
423 test = .false.
424 IF( indeig ) THEN
425 IF( il.EQ.1 .AND. iu.EQ.n ) THEN
426 test = .true.
427 END IF
428 END IF
429 IF( ( alleig .OR. test ) .AND. ( abstol.LE.zero ) ) THEN
430 CALL scopy( n, work( indd ), 1, w, 1 )
431 indee = indwrk + 2*n
432 CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
433 IF( .NOT.wantz ) THEN
434 CALL ssterf( n, w, work( indee ), info )
435 ELSE
436 CALL slacpy( 'A', n, n, q, ldq, z, ldz )
437 CALL ssteqr( jobz, n, w, work( indee ), z, ldz,
438 \$ work( indwrk ), info )
439 IF( info.EQ.0 ) THEN
440 DO 10 i = 1, n
441 ifail( i ) = 0
442 10 CONTINUE
443 END IF
444 END IF
445 IF( info.EQ.0 ) THEN
446 m = n
447 GO TO 30
448 END IF
449 info = 0
450 END IF
451*
452* Otherwise, call SSTEBZ and, if eigenvectors are desired,
453* call SSTEIN.
454*
455 IF( wantz ) THEN
456 order = 'B'
457 ELSE
458 order = 'E'
459 END IF
460 indisp = 1 + n
461 indiwo = indisp + n
462 CALL sstebz( range, order, n, vl, vu, il, iu, abstol,
463 \$ work( indd ), work( inde ), m, nsplit, w,
464 \$ iwork( 1 ), iwork( indisp ), work( indwrk ),
465 \$ iwork( indiwo ), info )
466*
467 IF( wantz ) THEN
468 CALL sstein( n, work( indd ), work( inde ), m, w,
469 \$ iwork( 1 ), iwork( indisp ), z, ldz,
470 \$ work( indwrk ), iwork( indiwo ), ifail, info )
471*
472* Apply transformation matrix used in reduction to tridiagonal
473* form to eigenvectors returned by SSTEIN.
474*
475 DO 20 j = 1, m
476 CALL scopy( n, z( 1, j ), 1, work( 1 ), 1 )
477 CALL sgemv( 'N', n, n, one, q, ldq, work, 1, zero,
478 \$ z( 1, j ), 1 )
479 20 CONTINUE
480 END IF
481*
482 30 CONTINUE
483*
484* If eigenvalues are not in order, then sort them, along with
485* eigenvectors.
486*
487 IF( wantz ) THEN
488 DO 50 j = 1, m - 1
489 i = 0
490 tmp1 = w( j )
491 DO 40 jj = j + 1, m
492 IF( w( jj ).LT.tmp1 ) THEN
493 i = jj
494 tmp1 = w( jj )
495 END IF
496 40 CONTINUE
497*
498 IF( i.NE.0 ) THEN
499 itmp1 = iwork( 1 + i-1 )
500 w( i ) = w( j )
501 iwork( 1 + i-1 ) = iwork( 1 + j-1 )
502 w( j ) = tmp1
503 iwork( 1 + j-1 ) = itmp1
504 CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
505 IF( info.NE.0 ) THEN
506 itmp1 = ifail( i )
507 ifail( i ) = ifail( j )
508 ifail( j ) = itmp1
509 END IF
510 END IF
511 50 CONTINUE
512 END IF
513*
514 RETURN
515*
516* End of SSBGVX
517*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine sgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
SGEMV
Definition sgemv.f:158
subroutine ssbgst(vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, info)
SSBGST
Definition ssbgst.f:159
subroutine ssbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
SSBTRD
Definition ssbtrd.f:163
subroutine slacpy(uplo, m, n, a, lda, b, ldb)
SLACPY copies all or part of one two-dimensional array to another.
Definition slacpy.f:103
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine spbstf(uplo, n, kd, ab, ldab, info)
SPBSTF
Definition spbstf.f:152
subroutine sstebz(range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
SSTEBZ
Definition sstebz.f:273
subroutine sstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)
SSTEIN
Definition sstein.f:174
subroutine ssteqr(compz, n, d, e, z, ldz, work, info)
SSTEQR
Definition ssteqr.f:131
subroutine ssterf(n, d, e, info)
SSTERF
Definition ssterf.f:86
subroutine sswap(n, sx, incx, sy, incy)
SSWAP
Definition sswap.f:82
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