LAPACK  3.10.0
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

Download CHEGVX + dependencies [TGZ] [ZIP] [TXT]

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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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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