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SUBROUTINE <a name="DSYGVX.1"></a><a href="dsygvx.f.html#DSYGVX.1">DSYGVX</a>( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
$ VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
$ LWORK, IWORK, IFAIL, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK driver routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
DOUBLE PRECISION ABSTOL, VL, VU
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
$ Z( LDZ, * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DSYGVX.23"></a><a href="dsygvx.f.html#DSYGVX.1">DSYGVX</a> computes selected eigenvalues, and optionally, eigenvectors
</span><span class="comment">*</span><span class="comment"> of a real generalized symmetric-definite eigenproblem, of the form
</span><span class="comment">*</span><span class="comment"> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A
</span><span class="comment">*</span><span class="comment"> and B are assumed to be symmetric and B is also positive definite.
</span><span class="comment">*</span><span class="comment"> Eigenvalues and eigenvectors can be selected by specifying either a
</span><span class="comment">*</span><span class="comment"> range of values or a range of indices for the desired eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ITYPE (input) INTEGER
</span><span class="comment">*</span><span class="comment"> Specifies the problem type to be solved:
</span><span class="comment">*</span><span class="comment"> = 1: A*x = (lambda)*B*x
</span><span class="comment">*</span><span class="comment"> = 2: A*B*x = (lambda)*x
</span><span class="comment">*</span><span class="comment"> = 3: B*A*x = (lambda)*x
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> JOBZ (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'N': Compute eigenvalues only;
</span><span class="comment">*</span><span class="comment"> = 'V': Compute eigenvalues and eigenvectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RANGE (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'A': all eigenvalues will be found.
</span><span class="comment">*</span><span class="comment"> = 'V': all eigenvalues in the half-open interval (VL,VU]
</span><span class="comment">*</span><span class="comment"> will be found.
</span><span class="comment">*</span><span class="comment"> = 'I': the IL-th through IU-th eigenvalues will be found.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> UPLO (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'U': Upper triangle of A and B are stored;
</span><span class="comment">*</span><span class="comment"> = 'L': Lower triangle of A and B are stored.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrix pencil (A,B). N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
</span><span class="comment">*</span><span class="comment"> On entry, the symmetric matrix A. If UPLO = 'U', the
</span><span class="comment">*</span><span class="comment"> leading N-by-N upper triangular part of A contains the
</span><span class="comment">*</span><span class="comment"> upper triangular part of the matrix A. If UPLO = 'L',
</span><span class="comment">*</span><span class="comment"> the leading N-by-N lower triangular part of A contains
</span><span class="comment">*</span><span class="comment"> the lower triangular part of the matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> On exit, the lower triangle (if UPLO='L') or the upper
</span><span class="comment">*</span><span class="comment"> triangle (if UPLO='U') of A, including the diagonal, is
</span><span class="comment">*</span><span class="comment"> destroyed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array A. LDA >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B (input/output) DOUBLE PRECISION array, dimension (LDA, N)
</span><span class="comment">*</span><span class="comment"> On entry, the symmetric matrix B. If UPLO = 'U', the
</span><span class="comment">*</span><span class="comment"> leading N-by-N upper triangular part of B contains the
</span><span class="comment">*</span><span class="comment"> upper triangular part of the matrix B. If UPLO = 'L',
</span><span class="comment">*</span><span class="comment"> the leading N-by-N lower triangular part of B contains
</span><span class="comment">*</span><span class="comment"> the lower triangular part of the matrix B.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> On exit, if INFO <= N, the part of B containing the matrix is
</span><span class="comment">*</span><span class="comment"> overwritten by the triangular factor U or L from the Cholesky
</span><span class="comment">*</span><span class="comment"> factorization B = U**T*U or B = L*L**T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array B. LDB >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VL (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> VU (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> If RANGE='V', the lower and upper bounds of the interval to
</span><span class="comment">*</span><span class="comment"> be searched for eigenvalues. VL < VU.
</span><span class="comment">*</span><span class="comment"> Not referenced if RANGE = 'A' or 'I'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IL (input) INTEGER
</span><span class="comment">*</span><span class="comment"> IU (input) INTEGER
</span><span class="comment">*</span><span class="comment"> If RANGE='I', the indices (in ascending order) of the
</span><span class="comment">*</span><span class="comment"> smallest and largest eigenvalues to be returned.
</span><span class="comment">*</span><span class="comment"> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
</span><span class="comment">*</span><span class="comment"> Not referenced if RANGE = 'A' or 'V'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ABSTOL (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> The absolute error tolerance for the eigenvalues.
</span><span class="comment">*</span><span class="comment"> An approximate eigenvalue is accepted as converged
</span><span class="comment">*</span><span class="comment"> when it is determined to lie in an interval [a,b]
</span><span class="comment">*</span><span class="comment"> of width less than or equal to
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ABSTOL + EPS * max( |a|,|b| ) ,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where EPS is the machine precision. If ABSTOL is less than
</span><span class="comment">*</span><span class="comment"> or equal to zero, then EPS*|T| will be used in its place,
</span><span class="comment">*</span><span class="comment"> where |T| is the 1-norm of the tridiagonal matrix obtained
</span><span class="comment">*</span><span class="comment"> by reducing A to tridiagonal form.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Eigenvalues will be computed most accurately when ABSTOL is
</span><span class="comment">*</span><span class="comment"> set to twice the underflow threshold 2*<a name="DLAMCH.111"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>('S'), not zero.
</span><span class="comment">*</span><span class="comment"> If this routine returns with INFO>0, indicating that some
</span><span class="comment">*</span><span class="comment"> eigenvectors did not converge, try setting ABSTOL to
</span><span class="comment">*</span><span class="comment"> 2*<a name="DLAMCH.114"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>('S').
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> M (output) INTEGER
</span><span class="comment">*</span><span class="comment"> The total number of eigenvalues found. 0 <= M <= N.
</span><span class="comment">*</span><span class="comment"> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> W (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On normal exit, the first M elements contain the selected
</span><span class="comment">*</span><span class="comment"> eigenvalues in ascending order.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
</span><span class="comment">*</span><span class="comment"> If JOBZ = 'N', then Z is not referenced.
</span><span class="comment">*</span><span class="comment"> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
</span><span class="comment">*</span><span class="comment"> contain the orthonormal eigenvectors of the matrix A
</span><span class="comment">*</span><span class="comment"> corresponding to the selected eigenvalues, with the i-th
</span><span class="comment">*</span><span class="comment"> column of Z holding the eigenvector associated with W(i).
</span><span class="comment">*</span><span class="comment"> The eigenvectors are normalized as follows:
</span><span class="comment">*</span><span class="comment"> if ITYPE = 1 or 2, Z**T*B*Z = I;
</span><span class="comment">*</span><span class="comment"> if ITYPE = 3, Z**T*inv(B)*Z = I.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If an eigenvector fails to converge, then that column of Z
</span><span class="comment">*</span><span class="comment"> contains the latest approximation to the eigenvector, and the
</span><span class="comment">*</span><span class="comment"> index of the eigenvector is returned in IFAIL.
</span><span class="comment">*</span><span class="comment"> Note: the user must ensure that at least max(1,M) columns are
</span><span class="comment">*</span><span class="comment"> supplied in the array Z; if RANGE = 'V', the exact value of M
</span><span class="comment">*</span><span class="comment"> is not known in advance and an upper bound must be used.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDZ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array Z. LDZ >= 1, and if
</span><span class="comment">*</span><span class="comment"> JOBZ = 'V', LDZ >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
</span><span class="comment">*</span><span class="comment"> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LWORK (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The length of the array WORK. LWORK >= max(1,8*N).
</span><span class="comment">*</span><span class="comment"> For optimal efficiency, LWORK >= (NB+3)*N,
</span><span class="comment">*</span><span class="comment"> where NB is the blocksize for <a name="DSYTRD.151"></a><a href="dsytrd.f.html#DSYTRD.1">DSYTRD</a> returned by <a name="ILAENV.151"></a><a href="hfy-index.html#ILAENV">ILAENV</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If LWORK = -1, then a workspace query is assumed; the routine
</span><span class="comment">*</span><span class="comment"> only calculates the optimal size of the WORK array, returns
</span><span class="comment">*</span><span class="comment"> this value as the first entry of the WORK array, and no error
</span><span class="comment">*</span><span class="comment"> message related to LWORK is issued by <a name="XERBLA.156"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IWORK (workspace) INTEGER array, dimension (5*N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IFAIL (output) INTEGER array, dimension (N)
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