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

 subroutine chpgvx ( integer itype, character jobz, character range, character uplo, integer n, complex, dimension( * ) ap, complex, dimension( * ) bp, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info )

CHPGVX

Purpose:
``` CHPGVX computes selected eigenvalues and, optionally, eigenvectors
of a complex generalized Hermitian-definite eigenproblem, of the form
A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
B are assumed to be Hermitian, stored in packed format, and B is also
positive definite.  Eigenvalues and eigenvectors can be selected by
specifying either a range of values or a range of indices for the
desired eigenvalues.```
Parameters
 [in] ITYPE ``` ITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*x``` [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,out] AP ``` AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian 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, the contents of AP are destroyed.``` [in,out] BP ``` BP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, in the same storage format as B.``` [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 AP 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) On normal exit, the first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'N', then Z is not referenced. 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). The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = 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. 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 COMPLEX array, dimension (2*N)` [out] RWORK ` RWORK is REAL 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: CPPTRF or CHPEVX returned an error code: <= N: if INFO = i, CHPEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL. > N: 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 274 of file chpgvx.f.

277*
278* -- LAPACK driver routine --
279* -- LAPACK is a software package provided by Univ. of Tennessee, --
280* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
281*
282* .. Scalar Arguments ..
283 CHARACTER JOBZ, RANGE, UPLO
284 INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
285 REAL ABSTOL, VL, VU
286* ..
287* .. Array Arguments ..
288 INTEGER IFAIL( * ), IWORK( * )
289 REAL RWORK( * ), W( * )
290 COMPLEX AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
291* ..
292*
293* =====================================================================
294*
295* .. Local Scalars ..
296 LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
297 CHARACTER TRANS
298 INTEGER J
299* ..
300* .. External Functions ..
301 LOGICAL LSAME
302 EXTERNAL lsame
303* ..
304* .. External Subroutines ..
305 EXTERNAL chpevx, chpgst, cpptrf, ctpmv, ctpsv, xerbla
306* ..
307* .. Intrinsic Functions ..
308 INTRINSIC min
309* ..
310* .. Executable Statements ..
311*
312* Test the input parameters.
313*
314 wantz = lsame( jobz, 'V' )
315 upper = lsame( uplo, 'U' )
316 alleig = lsame( range, 'A' )
317 valeig = lsame( range, 'V' )
318 indeig = lsame( range, 'I' )
319*
320 info = 0
321 IF( itype.LT.1 .OR. itype.GT.3 ) THEN
322 info = -1
323 ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
324 info = -2
325 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
326 info = -3
327 ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
328 info = -4
329 ELSE IF( n.LT.0 ) THEN
330 info = -5
331 ELSE
332 IF( valeig ) THEN
333 IF( n.GT.0 .AND. vu.LE.vl ) THEN
334 info = -9
335 END IF
336 ELSE IF( indeig ) THEN
337 IF( il.LT.1 ) THEN
338 info = -10
339 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
340 info = -11
341 END IF
342 END IF
343 END IF
344 IF( info.EQ.0 ) THEN
345 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
346 info = -16
347 END IF
348 END IF
349*
350 IF( info.NE.0 ) THEN
351 CALL xerbla( 'CHPGVX', -info )
352 RETURN
353 END IF
354*
355* Quick return if possible
356*
357 IF( n.EQ.0 )
358 \$ RETURN
359*
360* Form a Cholesky factorization of B.
361*
362 CALL cpptrf( uplo, n, bp, info )
363 IF( info.NE.0 ) THEN
364 info = n + info
365 RETURN
366 END IF
367*
368* Transform problem to standard eigenvalue problem and solve.
369*
370 CALL chpgst( itype, uplo, n, ap, bp, info )
371 CALL chpevx( jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m,
372 \$ w, z, ldz, work, rwork, iwork, ifail, info )
373*
374 IF( wantz ) THEN
375*
376* Backtransform eigenvectors to the original problem.
377*
378 IF( info.GT.0 )
379 \$ m = info - 1
380 IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
381*
382* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
383* backtransform eigenvectors: x = inv(L)**H*y or inv(U)*y
384*
385 IF( upper ) THEN
386 trans = 'N'
387 ELSE
388 trans = 'C'
389 END IF
390*
391 DO 10 j = 1, m
392 CALL ctpsv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
393 \$ 1 )
394 10 CONTINUE
395*
396 ELSE IF( itype.EQ.3 ) THEN
397*
398* For B*A*x=(lambda)*x;
399* backtransform eigenvectors: x = L*y or U**H*y
400*
401 IF( upper ) THEN
402 trans = 'C'
403 ELSE
404 trans = 'N'
405 END IF
406*
407 DO 20 j = 1, m
408 CALL ctpmv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
409 \$ 1 )
410 20 CONTINUE
411 END IF
412 END IF
413*
414 RETURN
415*
416* End of CHPGVX
417*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine chpevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)
CHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrice...
Definition chpevx.f:240
subroutine chpgst(itype, uplo, n, ap, bp, info)
CHPGST
Definition chpgst.f:113
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine cpptrf(uplo, n, ap, info)
CPPTRF
Definition cpptrf.f:119
subroutine ctpmv(uplo, trans, diag, n, ap, x, incx)
CTPMV
Definition ctpmv.f:142
subroutine ctpsv(uplo, trans, diag, n, ap, x, incx)
CTPSV
Definition ctpsv.f:144
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