DSYGVD computes all the eigenvalues, and optionally, the eigenvectors


 of a real generalized symmetric-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 symmetric and B is also positive definite.


 If eigenvectors are desired, it uses a divide and conquer algorithm.


 The divide and conquer algorithm makes very mild assumptions about


 floating point arithmetic. It will work on machines with a guard


 digit in add/subtract, or on those binary machines without guard


 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or


 Cray-2. It could conceivably fail on hexadecimal or decimal machines


 without guard digits, but we know of none.

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

JOBZ

          JOBZ is CHARACTER*1


          = 'N':  Compute eigenvalues only;


          = 'V':  Compute eigenvalues and eigenvectors.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangles of A and B are stored;


          = 'L':  Lower triangles of A and B are stored.

N

          N is INTEGER


          The order of the matrices A and B.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA, N)


          On entry, the symmetric 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**T*B*Z = I;


          if ITYPE = 3, Z**T*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.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is DOUBLE PRECISION array, dimension (LDB, N)


          On entry, the symmetric 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**T*U or B = L*L**T.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

W

          W is DOUBLE PRECISION array, dimension (N)


          If INFO = 0, the eigenvalues in ascending order.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          If N <= 1,               LWORK >= 1.


          If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.


          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal sizes of the WORK and IWORK


          arrays, returns these values as the first entries of the WORK


          and IWORK arrays, and no error message related to LWORK or


          LIWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))


          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER


          The dimension of the array IWORK.


          If N <= 1,                LIWORK >= 1.


          If JOBZ  = 'N' and N > 1, LIWORK >= 1.


          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.


          If LIWORK = -1, then a workspace query is assumed; the


          routine only calculates the optimal sizes of the WORK and


          IWORK arrays, returns these values as the first entries of


          the WORK and IWORK arrays, and no error message related to


          LWORK or LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value


          > 0:  DPOTRF or DSYEVD returned an error code:


             <= N:  if INFO = i and JOBZ = 'N', then the algorithm


                    failed to converge; i off-diagonal elements of an


                    intermediate tridiagonal form did not converge to


                    zero;


                    if INFO = i and JOBZ = 'V', then the algorithm


                    failed to compute an eigenvalue while working on


                    the submatrix lying in rows and columns INFO/(N+1)


                    through mod(INFO,N+1);


             > 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

  Modified so that no backsubstitution is performed if DSYEVD fails to


  converge (NEIG in old code could be greater than N causing out of


  bounds reference to A - reported by Ralf Meyer).  Also corrected the


  description of INFO and the test on ITYPE. Sven, 16 Feb 05.

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA