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

 subroutine zhbgvx ( character JOBZ, character RANGE, character UPLO, integer N, integer KA, integer KB, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldbb, * ) BB, integer LDBB, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

ZHBGVX

Purpose:
``` ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors
of a complex generalized Hermitian-definite banded eigenproblem, of
the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
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 COMPLEX*16 array, dimension (LDAB, N) On entry, the upper or lower triangle of the Hermitian 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 COMPLEX*16 array, dimension (LDBB, N) On entry, the upper or lower triangle of the Hermitian 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(kb+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**H*S, as returned by ZPBSTF.``` [in] LDBB ``` LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.``` [out] Q ``` Q is COMPLEX*16 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 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').``` [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 eigenvalues in ascending order.``` [out] Z ``` Z is COMPLEX*16 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 that Z**H*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 >= N.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (N)` [out] RWORK ` RWORK 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, and i is: <= N: then i eigenvectors failed to converge. Their indices are stored in array IFAIL. > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF returned INFO = i: B is not positive definite. 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 297 of file zhbgvx.f.

300*
301* -- LAPACK driver routine --
302* -- LAPACK is a software package provided by Univ. of Tennessee, --
303* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
304*
305* .. Scalar Arguments ..
306 CHARACTER JOBZ, RANGE, UPLO
307 INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
308 \$ N
309 DOUBLE PRECISION ABSTOL, VL, VU
310* ..
311* .. Array Arguments ..
312 INTEGER IFAIL( * ), IWORK( * )
313 DOUBLE PRECISION RWORK( * ), W( * )
314 COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
315 \$ WORK( * ), Z( LDZ, * )
316* ..
317*
318* =====================================================================
319*
320* .. Parameters ..
321 DOUBLE PRECISION ZERO
322 parameter( zero = 0.0d+0 )
323 COMPLEX*16 CZERO, CONE
324 parameter( czero = ( 0.0d+0, 0.0d+0 ),
325 \$ cone = ( 1.0d+0, 0.0d+0 ) )
326* ..
327* .. Local Scalars ..
328 LOGICAL ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
329 CHARACTER ORDER, VECT
330 INTEGER I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
331 \$ INDIWK, INDRWK, INDWRK, ITMP1, J, JJ, NSPLIT
332 DOUBLE PRECISION TMP1
333* ..
334* .. External Functions ..
335 LOGICAL LSAME
336 EXTERNAL lsame
337* ..
338* .. External Subroutines ..
339 EXTERNAL dcopy, dstebz, dsterf, xerbla, zcopy, zgemv,
341 \$ zswap
342* ..
343* .. Intrinsic Functions ..
344 INTRINSIC min
345* ..
346* .. Executable Statements ..
347*
348* Test the input parameters.
349*
350 wantz = lsame( jobz, 'V' )
351 upper = lsame( uplo, 'U' )
352 alleig = lsame( range, 'A' )
353 valeig = lsame( range, 'V' )
354 indeig = lsame( range, 'I' )
355*
356 info = 0
357 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
358 info = -1
359 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
360 info = -2
361 ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
362 info = -3
363 ELSE IF( n.LT.0 ) THEN
364 info = -4
365 ELSE IF( ka.LT.0 ) THEN
366 info = -5
367 ELSE IF( kb.LT.0 .OR. kb.GT.ka ) THEN
368 info = -6
369 ELSE IF( ldab.LT.ka+1 ) THEN
370 info = -8
371 ELSE IF( ldbb.LT.kb+1 ) THEN
372 info = -10
373 ELSE IF( ldq.LT.1 .OR. ( wantz .AND. ldq.LT.n ) ) THEN
374 info = -12
375 ELSE
376 IF( valeig ) THEN
377 IF( n.GT.0 .AND. vu.LE.vl )
378 \$ info = -14
379 ELSE IF( indeig ) THEN
380 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
381 info = -15
382 ELSE IF ( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
383 info = -16
384 END IF
385 END IF
386 END IF
387 IF( info.EQ.0) THEN
388 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
389 info = -21
390 END IF
391 END IF
392*
393 IF( info.NE.0 ) THEN
394 CALL xerbla( 'ZHBGVX', -info )
395 RETURN
396 END IF
397*
398* Quick return if possible
399*
400 m = 0
401 IF( n.EQ.0 )
402 \$ RETURN
403*
404* Form a split Cholesky factorization of B.
405*
406 CALL zpbstf( uplo, n, kb, bb, ldbb, info )
407 IF( info.NE.0 ) THEN
408 info = n + info
409 RETURN
410 END IF
411*
412* Transform problem to standard eigenvalue problem.
413*
414 CALL zhbgst( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq,
415 \$ work, rwork, iinfo )
416*
417* Solve the standard eigenvalue problem.
418* Reduce Hermitian band matrix to tridiagonal form.
419*
420 indd = 1
421 inde = indd + n
422 indrwk = inde + n
423 indwrk = 1
424 IF( wantz ) THEN
425 vect = 'U'
426 ELSE
427 vect = 'N'
428 END IF
429 CALL zhbtrd( vect, uplo, n, ka, ab, ldab, rwork( indd ),
430 \$ rwork( 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 ZSTEQR. 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, rwork( indd ), 1, w, 1 )
444 indee = indrwk + 2*n
445 CALL dcopy( n-1, rwork( inde ), 1, rwork( indee ), 1 )
446 IF( .NOT.wantz ) THEN
447 CALL dsterf( n, w, rwork( indee ), info )
448 ELSE
449 CALL zlacpy( 'A', n, n, q, ldq, z, ldz )
450 CALL zsteqr( jobz, n, w, rwork( indee ), z, ldz,
451 \$ rwork( indrwk ), info )
452 IF( info.EQ.0 ) THEN
453 DO 10 i = 1, n
454 ifail( i ) = 0
455 10 CONTINUE
456 END IF
457 END IF
458 IF( info.EQ.0 ) THEN
459 m = n
460 GO TO 30
461 END IF
462 info = 0
463 END IF
464*
465* Otherwise, call DSTEBZ and, if eigenvectors are desired,
466* call ZSTEIN.
467*
468 IF( wantz ) THEN
469 order = 'B'
470 ELSE
471 order = 'E'
472 END IF
473 indibl = 1
474 indisp = indibl + n
475 indiwk = indisp + n
476 CALL dstebz( range, order, n, vl, vu, il, iu, abstol,
477 \$ rwork( indd ), rwork( inde ), m, nsplit, w,
478 \$ iwork( indibl ), iwork( indisp ), rwork( indrwk ),
479 \$ iwork( indiwk ), info )
480*
481 IF( wantz ) THEN
482 CALL zstein( n, rwork( indd ), rwork( inde ), m, w,
483 \$ iwork( indibl ), iwork( indisp ), z, ldz,
484 \$ rwork( indrwk ), iwork( indiwk ), ifail, info )
485*
486* Apply unitary matrix used in reduction to tridiagonal
487* form to eigenvectors returned by ZSTEIN.
488*
489 DO 20 j = 1, m
490 CALL zcopy( n, z( 1, j ), 1, work( 1 ), 1 )
491 CALL zgemv( 'N', n, n, cone, q, ldq, work, 1, czero,
492 \$ z( 1, j ), 1 )
493 20 CONTINUE
494 END IF
495*
496 30 CONTINUE
497*
498* If eigenvalues are not in order, then sort them, along with
499* eigenvectors.
500*
501 IF( wantz ) THEN
502 DO 50 j = 1, m - 1
503 i = 0
504 tmp1 = w( j )
505 DO 40 jj = j + 1, m
506 IF( w( jj ).LT.tmp1 ) THEN
507 i = jj
508 tmp1 = w( jj )
509 END IF
510 40 CONTINUE
511*
512 IF( i.NE.0 ) THEN
513 itmp1 = iwork( indibl+i-1 )
514 w( i ) = w( j )
515 iwork( indibl+i-1 ) = iwork( indibl+j-1 )
516 w( j ) = tmp1
517 iwork( indibl+j-1 ) = itmp1
518 CALL zswap( n, z( 1, i ), 1, z( 1, j ), 1 )
519 IF( info.NE.0 ) THEN
520 itmp1 = ifail( i )
521 ifail( i ) = ifail( j )
522 ifail( j ) = itmp1
523 END IF
524 END IF
525 50 CONTINUE
526 END IF
527*
528 RETURN
529*
530* End of ZHBGVX
531*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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 dsterf(N, D, E, INFO)
DSTERF
Definition: dsterf.f:86
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zsteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
ZSTEQR
Definition: zsteqr.f:132
subroutine zhbgst(VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, RWORK, INFO)
ZHBGST
Definition: zhbgst.f:165
subroutine zstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
ZSTEIN
Definition: zstein.f:182
subroutine zpbstf(UPLO, N, KD, AB, LDAB, INFO)
ZPBSTF
Definition: zpbstf.f:153
subroutine zhbtrd(VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
ZHBTRD
Definition: zhbtrd.f:163
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
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