LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zhpgvx()

 subroutine zhpgvx ( integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, complex*16, dimension( * ) AP, complex*16, dimension( * ) BP, 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 )

ZHPGVX

Purpose:
``` ZHPGVX 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*16 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*16 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 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) On normal exit, the first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is COMPLEX*16 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*16 array, dimension (2*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: ZPPTRF or ZHPEVX returned an error code: <= N: if INFO = i, ZHPEVX 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 minor of order i of 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 274 of file zhpgvx.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  DOUBLE PRECISION ABSTOL, VL, VU
286 * ..
287 * .. Array Arguments ..
288  INTEGER IFAIL( * ), IWORK( * )
289  DOUBLE PRECISION RWORK( * ), W( * )
290  COMPLEX*16 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 xerbla, zhpevx, zhpgst, zpptrf, ztpmv, ztpsv
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( 'ZHPGVX', -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 zpptrf( 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 zhpgst( itype, uplo, n, ap, bp, info )
371  CALL zhpevx( 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 ztpsv( 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 ztpmv( 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 ZHPGVX
417 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ztpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
ZTPSV
Definition: ztpsv.f:144
subroutine ztpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
ZTPMV
Definition: ztpmv.f:142
subroutine zhpgst(ITYPE, UPLO, N, AP, BP, INFO)
ZHPGST
Definition: zhpgst.f:113
subroutine zpptrf(UPLO, N, AP, INFO)
ZPPTRF
Definition: zpptrf.f:119
subroutine zhpevx(JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
ZHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrice...
Definition: zhpevx.f:240
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