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SUBROUTINE <a name="ZHPGVX.1"></a><a href="zhpgvx.f.html#ZHPGVX.1">ZHPGVX</a>( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
$ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
$ 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, LDZ, 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 RWORK( * ), W( * )
COMPLEX*16 AP( * ), BP( * ), 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="ZHPGVX.23"></a><a href="zhpgvx.f.html#ZHPGVX.1">ZHPGVX</a> computes selected eigenvalues and, optionally, eigenvectors
</span><span class="comment">*</span><span class="comment"> of a complex generalized Hermitian-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 and
</span><span class="comment">*</span><span class="comment"> B are assumed to be Hermitian, stored in packed format, and B is also
</span><span class="comment">*</span><span class="comment"> positive definite. Eigenvalues and eigenvectors can be selected by
</span><span class="comment">*</span><span class="comment"> specifying either a range of values or a range of indices for the
</span><span class="comment">*</span><span class="comment"> 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 triangles of A and B are stored;
</span><span class="comment">*</span><span class="comment"> = 'L': Lower triangles 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 matrices A and B. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
</span><span class="comment">*</span><span class="comment"> On entry, the upper or lower triangle of the Hermitian matrix
</span><span class="comment">*</span><span class="comment"> A, packed columnwise in a linear array. The j-th column of A
</span><span class="comment">*</span><span class="comment"> is stored in the array AP as follows:
</span><span class="comment">*</span><span class="comment"> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
</span><span class="comment">*</span><span class="comment"> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> On exit, the contents of AP are destroyed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> BP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
</span><span class="comment">*</span><span class="comment"> On entry, the upper or lower triangle of the Hermitian matrix
</span><span class="comment">*</span><span class="comment"> B, packed columnwise in a linear array. The j-th column of B
</span><span class="comment">*</span><span class="comment"> is stored in the array BP as follows:
</span><span class="comment">*</span><span class="comment"> if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
</span><span class="comment">*</span><span class="comment"> if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> On exit, the triangular factor U or L from the Cholesky
</span><span class="comment">*</span><span class="comment"> factorization B = U**H*U or B = L*L**H, in the same storage
</span><span class="comment">*</span><span class="comment"> format as B.
</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 AP 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.104"></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.107"></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) COMPLEX*16 array, dimension (LDZ, N)
</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**H*B*Z = I;
</span><span class="comment">*</span><span class="comment"> if ITYPE = 3, Z**H*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) COMPLEX*16 array, dimension (2*N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
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