zhegv()

subroutine zhegv	(	integer	itype,
		character	jobz,
		character	uplo,
		integer	n,
		complex*16, dimension( lda, * )	a,
		integer	lda,
		complex*16, dimension( ldb, * )	b,
		integer	ldb,
		double precision, dimension( * )	w,
		complex*16, dimension( * )	work,
		integer	lwork,
		double precision, dimension( * )	rwork,
		integer	info
	)

ZHEGV

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Purpose:

 ZHEGV computes all the eigenvalues, and optionally, the 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.

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]	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*16 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, if JOBZ = 'V', then if INFO = 0, A contains the matrix Z of eigenvectors. 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 JOBZ = 'N', then on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') 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*16 array, dimension (LDB, N) On entry, the Hermitian positive definite 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).
[out]	W	W is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.
[out]	WORK	WORK is COMPLEX*16 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-1). For optimal efficiency, LWORK >= (NB+1)*N, where NB is the blocksize for ZHETRD 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 DOUBLE PRECISION array, dimension (max(1, 3*N-2))
[out]	INFO	INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: ZPOTRF or ZHEEV returned an error code: <= N: if INFO = i, ZHEEV failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 179 of file zhegv.f.

*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, ITYPE, LDA, LDB, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   RWORK( * ), W( * )
      COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         ONE
      parameter( one = ( 1.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER, WANTZ
      CHARACTER          TRANS
      INTEGER            LWKOPT, NB, NEIG
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           lsame, ilaenv
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zheev, zhegst, zpotrf, ztrmm, ztrsm
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      wantz = lsame( jobz, 'V' )
      upper = lsame( uplo, 'U' )
      lquery = ( lwork.EQ.-1 )
*
      info = 0
      IF( itype.LT.1 .OR. itype.GT.3 ) THEN
         info = -1
      ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
         info = -2
      ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
         info = -3
      ELSE IF( n.LT.0 ) THEN
         info = -4
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -6
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
         info = -8
      END IF
*
      IF( info.EQ.0 ) THEN
         nb = ilaenv( 1, 'ZHETRD', uplo, n, -1, -1, -1 )
         lwkopt = max( 1, ( nb + 1 )*n )
         work( 1 ) = lwkopt
*
         IF( lwork.LT.max( 1, 2*n - 1 ) .AND. .NOT.lquery ) THEN
            info = -11
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZHEGV ', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
*     Form a Cholesky factorization of B.
*
      CALL zpotrf( uplo, n, b, ldb, info )
      IF( info.NE.0 ) THEN
         info = n + info
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
      CALL zhegst( itype, uplo, n, a, lda, b, ldb, info )
      CALL zheev( jobz, uplo, n, a, lda, w, work, lwork, rwork, info )
*
      IF( wantz ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
         neig = n
         IF( info.GT.0 )
     $      neig = info - 1
         IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)**H *y or inv(U)*y
*
            IF( upper ) THEN
               trans = 'N'
            ELSE
               trans = 'C'
            END IF
*
            CALL ztrsm( 'Left', uplo, trans, 'Non-unit', n, neig, one,
     $                  b, ldb, a, lda )
*
         ELSE IF( itype.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U**H *y
*
            IF( upper ) THEN
               trans = 'C'
            ELSE
               trans = 'N'
            END IF
*
            CALL ztrmm( 'Left', uplo, trans, 'Non-unit', n, neig, one,
     $                  b, ldb, a, lda )
         END IF
      END IF
*
      work( 1 ) = lwkopt
*
      RETURN
*
*     End of ZHEGV
*

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