 LAPACK  3.8.0 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.```
Date
June 2016
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 309 of file chegvx.f.

309 *
310 * -- LAPACK driver routine (version 3.7.0) --
311 * -- LAPACK is a software package provided by Univ. of Tennessee, --
312 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
313 * June 2016
314 *
315 * .. Scalar Arguments ..
316  CHARACTER jobz, range, uplo
317  INTEGER il, info, itype, iu, lda, ldb, ldz, lwork, m, n
318  REAL abstol, vl, vu
319 * ..
320 * .. Array Arguments ..
321  INTEGER ifail( * ), iwork( * )
322  REAL rwork( * ), w( * )
323  COMPLEX a( lda, * ), b( ldb, * ), work( * ),
324  \$ z( ldz, * )
325 * ..
326 *
327 * =====================================================================
328 *
329 * .. Parameters ..
330  COMPLEX cone
331  parameter( cone = ( 1.0e+0, 0.0e+0 ) )
332 * ..
333 * .. Local Scalars ..
334  LOGICAL alleig, indeig, lquery, upper, valeig, wantz
335  CHARACTER trans
336  INTEGER lwkopt, nb
337 * ..
338 * .. External Functions ..
339  LOGICAL lsame
340  INTEGER ilaenv
341  EXTERNAL ilaenv, lsame
342 * ..
343 * .. External Subroutines ..
344  EXTERNAL cheevx, chegst, cpotrf, ctrmm, ctrsm, xerbla
345 * ..
346 * .. Intrinsic Functions ..
347  INTRINSIC max, min
348 * ..
349 * .. Executable Statements ..
350 *
351 * Test the input parameters.
352 *
353  wantz = lsame( jobz, 'V' )
354  upper = lsame( uplo, 'U' )
355  alleig = lsame( range, 'A' )
356  valeig = lsame( range, 'V' )
357  indeig = lsame( range, 'I' )
358  lquery = ( lwork.EQ.-1 )
359 *
360  info = 0
361  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
362  info = -1
363  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
364  info = -2
365  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
366  info = -3
367  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
368  info = -4
369  ELSE IF( n.LT.0 ) THEN
370  info = -5
371  ELSE IF( lda.LT.max( 1, n ) ) THEN
372  info = -7
373  ELSE IF( ldb.LT.max( 1, n ) ) THEN
374  info = -9
375  ELSE
376  IF( valeig ) THEN
377  IF( n.GT.0 .AND. vu.LE.vl )
378  \$ info = -11
379  ELSE IF( indeig ) THEN
380  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
381  info = -12
382  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
383  info = -13
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 = -18
390  END IF
391  END IF
392 *
393  IF( info.EQ.0 ) THEN
394  nb = ilaenv( 1, 'CHETRD', uplo, n, -1, -1, -1 )
395  lwkopt = max( 1, ( nb + 1 )*n )
396  work( 1 ) = lwkopt
397 *
398  IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
399  info = -20
400  END IF
401  END IF
402 *
403  IF( info.NE.0 ) THEN
404  CALL xerbla( 'CHEGVX', -info )
405  RETURN
406  ELSE IF( lquery ) THEN
407  RETURN
408  END IF
409 *
410 * Quick return if possible
411 *
412  m = 0
413  IF( n.EQ.0 ) THEN
414  RETURN
415  END IF
416 *
417 * Form a Cholesky factorization of B.
418 *
419  CALL cpotrf( uplo, n, b, ldb, info )
420  IF( info.NE.0 ) THEN
421  info = n + info
422  RETURN
423  END IF
424 *
425 * Transform problem to standard eigenvalue problem and solve.
426 *
427  CALL chegst( itype, uplo, n, a, lda, b, ldb, info )
428  CALL cheevx( jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
429  \$ m, w, z, ldz, work, lwork, rwork, iwork, ifail,
430  \$ info )
431 *
432  IF( wantz ) THEN
433 *
434 * Backtransform eigenvectors to the original problem.
435 *
436  IF( info.GT.0 )
437  \$ m = info - 1
438  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
439 *
440 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
441 * backtransform eigenvectors: x = inv(L)**H*y or inv(U)*y
442 *
443  IF( upper ) THEN
444  trans = 'N'
445  ELSE
446  trans = 'C'
447  END IF
448 *
449  CALL ctrsm( 'Left', uplo, trans, 'Non-unit', n, m, cone, b,
450  \$ ldb, z, ldz )
451 *
452  ELSE IF( itype.EQ.3 ) THEN
453 *
454 * For B*A*x=(lambda)*x;
455 * backtransform eigenvectors: x = L*y or U**H*y
456 *
457  IF( upper ) THEN
458  trans = 'C'
459  ELSE
460  trans = 'N'
461  END IF
462 *
463  CALL ctrmm( 'Left', uplo, trans, 'Non-unit', n, m, cone, b,
464  \$ ldb, z, ldz )
465  END IF
466  END IF
467 *
468 * Set WORK(1) to optimal complex workspace size.
469 *
470  work( 1 ) = lwkopt
471 *
472  RETURN
473 *
474 * End of CHEGVX
475 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: tstiee.f:83
subroutine ctrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRSM
Definition: ctrsm.f:182
subroutine chegst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
CHEGST
Definition: chegst.f:129
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine cpotrf(UPLO, N, A, LDA, INFO)
CPOTRF
Definition: cpotrf.f:109
subroutine ctrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRMM
Definition: ctrmm.f:179
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:261
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