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SUBROUTINE <a name="ZHEEVR.1"></a><a href="zheevr.f.html#ZHEEVR.1">ZHEEVR</a>( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
$ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
$ RWORK, LRWORK, IWORK, LIWORK, 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, IU, LDA, LDZ, LIWORK, LRWORK, 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 ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION RWORK( * ), W( * )
COMPLEX*16 A( LDA, * ), 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="ZHEEVR.24"></a><a href="zheevr.f.html#ZHEEVR.1">ZHEEVR</a> computes selected eigenvalues and, optionally, eigenvectors
</span><span class="comment">*</span><span class="comment"> of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
</span><span class="comment">*</span><span class="comment"> be selected by specifying either a range of values or a range of
</span><span class="comment">*</span><span class="comment"> indices for the desired eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="ZHEEVR.29"></a><a href="zheevr.f.html#ZHEEVR.1">ZHEEVR</a> first reduces the matrix A to tridiagonal form T with a call
</span><span class="comment">*</span><span class="comment"> to <a name="ZHETRD.30"></a><a href="zhetrd.f.html#ZHETRD.1">ZHETRD</a>. Then, whenever possible, <a name="ZHEEVR.30"></a><a href="zheevr.f.html#ZHEEVR.1">ZHEEVR</a> calls <a name="ZSTEMR.30"></a><a href="zstemr.f.html#ZSTEMR.1">ZSTEMR</a> to compute
</span><span class="comment">*</span><span class="comment"> eigenspectrum using Relatively Robust Representations. <a name="ZSTEMR.31"></a><a href="zstemr.f.html#ZSTEMR.1">ZSTEMR</a>
</span><span class="comment">*</span><span class="comment"> computes eigenvalues by the dqds algorithm, while orthogonal
</span><span class="comment">*</span><span class="comment"> eigenvectors are computed from various "good" L D L^T representations
</span><span class="comment">*</span><span class="comment"> (also known as Relatively Robust Representations). Gram-Schmidt
</span><span class="comment">*</span><span class="comment"> orthogonalization is avoided as far as possible. More specifically,
</span><span class="comment">*</span><span class="comment"> the various steps of the algorithm are as follows.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> For each unreduced block (submatrix) of T,
</span><span class="comment">*</span><span class="comment"> (a) Compute T - sigma I = L D L^T, so that L and D
</span><span class="comment">*</span><span class="comment"> define all the wanted eigenvalues to high relative accuracy.
</span><span class="comment">*</span><span class="comment"> This means that small relative changes in the entries of D and L
</span><span class="comment">*</span><span class="comment"> cause only small relative changes in the eigenvalues and
</span><span class="comment">*</span><span class="comment"> eigenvectors. The standard (unfactored) representation of the
</span><span class="comment">*</span><span class="comment"> tridiagonal matrix T does not have this property in general.
</span><span class="comment">*</span><span class="comment"> (b) Compute the eigenvalues to suitable accuracy.
</span><span class="comment">*</span><span class="comment"> If the eigenvectors are desired, the algorithm attains full
</span><span class="comment">*</span><span class="comment"> accuracy of the computed eigenvalues only right before
</span><span class="comment">*</span><span class="comment"> the corresponding vectors have to be computed, see steps c) and d).
</span><span class="comment">*</span><span class="comment"> (c) For each cluster of close eigenvalues, select a new
</span><span class="comment">*</span><span class="comment"> shift close to the cluster, find a new factorization, and refine
</span><span class="comment">*</span><span class="comment"> the shifted eigenvalues to suitable accuracy.
</span><span class="comment">*</span><span class="comment"> (d) For each eigenvalue with a large enough relative separation compute
</span><span class="comment">*</span><span class="comment"> the corresponding eigenvector by forming a rank revealing twisted
</span><span class="comment">*</span><span class="comment"> factorization. Go back to (c) for any clusters that remain.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The desired accuracy of the output can be specified by the input
</span><span class="comment">*</span><span class="comment"> parameter ABSTOL.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> For more details, see <a name="DSTEMR.59"></a><a href="dstemr.f.html#DSTEMR.1">DSTEMR</a>'s documentation and:
</span><span class="comment">*</span><span class="comment"> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
</span><span class="comment">*</span><span class="comment"> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
</span><span class="comment">*</span><span class="comment"> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
</span><span class="comment">*</span><span class="comment"> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
</span><span class="comment">*</span><span class="comment"> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
</span><span class="comment">*</span><span class="comment"> 2004. Also LAPACK Working Note 154.
</span><span class="comment">*</span><span class="comment"> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
</span><span class="comment">*</span><span class="comment"> tridiagonal eigenvalue/eigenvector problem",
</span><span class="comment">*</span><span class="comment"> Computer Science Division Technical Report No. UCB/CSD-97-971,
</span><span class="comment">*</span><span class="comment"> UC Berkeley, May 1997.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Note 1 : <a name="ZHEEVR.72"></a><a href="zheevr.f.html#ZHEEVR.1">ZHEEVR</a> calls <a name="ZSTEMR.72"></a><a href="zstemr.f.html#ZSTEMR.1">ZSTEMR</a> when the full spectrum is requested
</span><span class="comment">*</span><span class="comment"> on machines which conform to the ieee-754 floating point standard.
</span><span class="comment">*</span><span class="comment"> <a name="ZHEEVR.74"></a><a href="zheevr.f.html#ZHEEVR.1">ZHEEVR</a> calls <a name="DSTEBZ.74"></a><a href="dstebz.f.html#DSTEBZ.1">DSTEBZ</a> and <a name="ZSTEIN.74"></a><a href="zstein.f.html#ZSTEIN.1">ZSTEIN</a> on non-ieee machines and
</span><span class="comment">*</span><span class="comment"> when partial spectrum requests are made.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Normal execution of <a name="ZSTEMR.77"></a><a href="zstemr.f.html#ZSTEMR.1">ZSTEMR</a> may create NaNs and infinities and
</span><span class="comment">*</span><span class="comment"> hence may abort due to a floating point exception in environments
</span><span class="comment">*</span><span class="comment"> which do not handle NaNs and infinities in the ieee standard default
</span><span class="comment">*</span><span class="comment"> manner.
</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"> 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">********* For RANGE = 'V' or 'I' and IU - IL < N - 1, <a name="DSTEBZ.94"></a><a href="dstebz.f.html#DSTEBZ.1">DSTEBZ</a> and
</span><span class="comment">*</span><span class="comment">********* <a name="ZSTEIN.95"></a><a href="zstein.f.html#ZSTEIN.1">ZSTEIN</a> are called
</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 is stored;
</span><span class="comment">*</span><span class="comment"> = 'L': Lower triangle of A is 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 A. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) COMPLEX*16 array, dimension (LDA, N)
</span><span class="comment">*</span><span class="comment"> On entry, the Hermitian 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"> 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"> 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">
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