 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ chegvx()

 subroutine chegvx ( integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

CHEGVX

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Purpose:
``` CHEGVX 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 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] A ``` A is COMPLEX array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] B ``` B is COMPLEX array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B. On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= 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 C to tridiagonal form, where C is the symmetric matrix of the standard symmetric problem to which the generalized problem is transformed. 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) The first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is COMPLEX array, dimension (LDZ, max(1,M)) 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**T*B*Z = I; if ITYPE = 3, Z**T*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 (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of the array WORK. LWORK >= max(1,2*N). For optimal efficiency, LWORK >= (NB+1)*N, where NB is the blocksize for CHETRD returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [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: CPOTRF or CHEEVX returned an error code: <= N: if INFO = i, CHEEVX 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 304 of file chegvx.f.

307 *
308 * -- LAPACK driver routine --
309 * -- LAPACK is a software package provided by Univ. of Tennessee, --
310 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
311 *
312 * .. Scalar Arguments ..
313  CHARACTER JOBZ, RANGE, UPLO
314  INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
315  REAL ABSTOL, VL, VU
316 * ..
317 * .. Array Arguments ..
318  INTEGER IFAIL( * ), IWORK( * )
319  REAL RWORK( * ), W( * )
320  COMPLEX A( LDA, * ), B( LDB, * ), WORK( * ),
321  \$ Z( LDZ, * )
322 * ..
323 *
324 * =====================================================================
325 *
326 * .. Parameters ..
327  COMPLEX CONE
328  parameter( cone = ( 1.0e+0, 0.0e+0 ) )
329 * ..
330 * .. Local Scalars ..
331  LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
332  CHARACTER TRANS
333  INTEGER LWKOPT, NB
334 * ..
335 * .. External Functions ..
336  LOGICAL LSAME
337  INTEGER ILAENV
338  EXTERNAL ilaenv, lsame
339 * ..
340 * .. External Subroutines ..
341  EXTERNAL cheevx, chegst, cpotrf, ctrmm, ctrsm, xerbla
342 * ..
343 * .. Intrinsic Functions ..
344  INTRINSIC max, 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  lquery = ( lwork.EQ.-1 )
356 *
357  info = 0
358  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
359  info = -1
360  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
361  info = -2
362  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
363  info = -3
364  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
365  info = -4
366  ELSE IF( n.LT.0 ) THEN
367  info = -5
368  ELSE IF( lda.LT.max( 1, n ) ) THEN
369  info = -7
370  ELSE IF( ldb.LT.max( 1, n ) ) THEN
371  info = -9
372  ELSE
373  IF( valeig ) THEN
374  IF( n.GT.0 .AND. vu.LE.vl )
375  \$ info = -11
376  ELSE IF( indeig ) THEN
377  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
378  info = -12
379  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
380  info = -13
381  END IF
382  END IF
383  END IF
384  IF (info.EQ.0) THEN
385  IF (ldz.LT.1 .OR. (wantz .AND. ldz.LT.n)) THEN
386  info = -18
387  END IF
388  END IF
389 *
390  IF( info.EQ.0 ) THEN
391  nb = ilaenv( 1, 'CHETRD', uplo, n, -1, -1, -1 )
392  lwkopt = max( 1, ( nb + 1 )*n )
393  work( 1 ) = lwkopt
394 *
395  IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
396  info = -20
397  END IF
398  END IF
399 *
400  IF( info.NE.0 ) THEN
401  CALL xerbla( 'CHEGVX', -info )
402  RETURN
403  ELSE IF( lquery ) THEN
404  RETURN
405  END IF
406 *
407 * Quick return if possible
408 *
409  m = 0
410  IF( n.EQ.0 ) THEN
411  RETURN
412  END IF
413 *
414 * Form a Cholesky factorization of B.
415 *
416  CALL cpotrf( uplo, n, b, ldb, info )
417  IF( info.NE.0 ) THEN
418  info = n + info
419  RETURN
420  END IF
421 *
422 * Transform problem to standard eigenvalue problem and solve.
423 *
424  CALL chegst( itype, uplo, n, a, lda, b, ldb, info )
425  CALL cheevx( jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
426  \$ m, w, z, ldz, work, lwork, rwork, iwork, ifail,
427  \$ info )
428 *
429  IF( wantz ) THEN
430 *
431 * Backtransform eigenvectors to the original problem.
432 *
433  IF( info.GT.0 )
434  \$ m = info - 1
435  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
436 *
437 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
438 * backtransform eigenvectors: x = inv(L)**H*y or inv(U)*y
439 *
440  IF( upper ) THEN
441  trans = 'N'
442  ELSE
443  trans = 'C'
444  END IF
445 *
446  CALL ctrsm( 'Left', uplo, trans, 'Non-unit', n, m, cone, b,
447  \$ ldb, z, ldz )
448 *
449  ELSE IF( itype.EQ.3 ) THEN
450 *
451 * For B*A*x=(lambda)*x;
452 * backtransform eigenvectors: x = L*y or U**H*y
453 *
454  IF( upper ) THEN
455  trans = 'C'
456  ELSE
457  trans = 'N'
458  END IF
459 *
460  CALL ctrmm( 'Left', uplo, trans, 'Non-unit', n, m, cone, b,
461  \$ ldb, z, ldz )
462  END IF
463  END IF
464 *
465 * Set WORK(1) to optimal complex workspace size.
466 *
467  work( 1 ) = lwkopt
468 *
469  RETURN
470 *
471 * End of CHEGVX
472 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ctrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRMM
Definition: ctrmm.f:177
subroutine ctrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRSM
Definition: ctrsm.f:180
subroutine chegst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
CHEGST
Definition: chegst.f:128
subroutine cheevx(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, INFO)
CHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
Definition: cheevx.f:259
subroutine cpotrf(UPLO, N, A, LDA, INFO)
CPOTRF
Definition: cpotrf.f:107
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