SYNOPSIS
 SUBROUTINE DSYGVX(
 ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO )
 CHARACTER JOBZ, RANGE, UPLO
 INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
 DOUBLE PRECISION ABSTOL, VL, VU
 INTEGER IFAIL( * ), IWORK( * )
 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ), Z( LDZ, * )
PURPOSE
DSYGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetricdefinite 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. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.ARGUMENTS
 ITYPE (input) 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 (input) CHARACTER*1

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.  RANGE (input) CHARACTER*1

= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the halfopen interval (VL,VU] will be found. = 'I': the ILth through IUth eigenvalues will be found.  UPLO (input) CHARACTER*1

= 'U': Upper triangle of A and B are stored;
= 'L': Lower triangle of A and B are stored.  N (input) INTEGER
 The order of the matrix pencil (A,B). N >= 0.
 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
 On entry, the symmetric matrix A. If UPLO = 'U', the leading NbyN upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading NbyN 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.
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,N).
 B (input/output) DOUBLE PRECISION array, dimension (LDA, N)
 On entry, the symmetric matrix B. If UPLO = 'U', the leading NbyN upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading NbyN 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 (input) INTEGER
 The leading dimension of the array B. LDB >= max(1,N).
 VL (input) DOUBLE PRECISION
 VU (input) DOUBLE PRECISION If RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
 IL (input) INTEGER
 IU (input) INTEGER If RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues 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'.
 ABSTOL (input) DOUBLE PRECISION
 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 1norm of the tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
 M (output) INTEGER
 The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IUIL+1.
 W (output) DOUBLE PRECISION array, dimension (N)
 On normal exit, the first M elements contain the selected eigenvalues in ascending order.
 Z (output) DOUBLE PRECISION 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 ith 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.
 LDZ (input) INTEGER
 The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
 WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
 LWORK (input) INTEGER
 The length of the array WORK. LWORK >= max(1,8*N). For optimal efficiency, LWORK >= (NB+3)*N, where NB is the blocksize for DSYTRD 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.
 IWORK (workspace) INTEGER array, dimension (5*N)
 IFAIL (output) 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.
 INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
> 0: DPOTRF or DSYEVX returned an error code:
<= N: if INFO = i, DSYEVX 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.
FURTHER DETAILS
Based on contributions byMark Fahey, Department of Mathematics, Univ. of Kentucky, USA