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SUBROUTINE <a name="DLAED7.1"></a><a href="dlaed7.f.html#DLAED7.1">DLAED7</a>( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
$ LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
$ PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
$ INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK 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> INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
$ QSIZ, TLVLS
DOUBLE PRECISION RHO
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
$ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
$ QSTORE( * ), WORK( * )
<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="DLAED7.25"></a><a href="dlaed7.f.html#DLAED7.1">DLAED7</a> computes the updated eigensystem of a diagonal
</span><span class="comment">*</span><span class="comment"> matrix after modification by a rank-one symmetric matrix. This
</span><span class="comment">*</span><span class="comment"> routine is used only for the eigenproblem which requires all
</span><span class="comment">*</span><span class="comment"> eigenvalues and optionally eigenvectors of a dense symmetric matrix
</span><span class="comment">*</span><span class="comment"> that has been reduced to tridiagonal form. <a name="DLAED1.29"></a><a href="dlaed1.f.html#DLAED1.1">DLAED1</a> handles
</span><span class="comment">*</span><span class="comment"> the case in which all eigenvalues and eigenvectors of a symmetric
</span><span class="comment">*</span><span class="comment"> tridiagonal matrix are desired.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where Z = Q'u, u is a vector of length N with ones in the
</span><span class="comment">*</span><span class="comment"> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The eigenvectors of the original matrix are stored in Q, and the
</span><span class="comment">*</span><span class="comment"> eigenvalues are in D. The algorithm consists of three stages:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The first stage consists of deflating the size of the problem
</span><span class="comment">*</span><span class="comment"> when there are multiple eigenvalues or if there is a zero in
</span><span class="comment">*</span><span class="comment"> the Z vector. For each such occurence the dimension of the
</span><span class="comment">*</span><span class="comment"> secular equation problem is reduced by one. This stage is
</span><span class="comment">*</span><span class="comment"> performed by the routine <a name="DLAED8.45"></a><a href="dlaed8.f.html#DLAED8.1">DLAED8</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The second stage consists of calculating the updated
</span><span class="comment">*</span><span class="comment"> eigenvalues. This is done by finding the roots of the secular
</span><span class="comment">*</span><span class="comment"> equation via the routine <a name="DLAED4.49"></a><a href="dlaed4.f.html#DLAED4.1">DLAED4</a> (as called by <a name="DLAED9.49"></a><a href="dlaed9.f.html#DLAED9.1">DLAED9</a>).
</span><span class="comment">*</span><span class="comment"> This routine also calculates the eigenvectors of the current
</span><span class="comment">*</span><span class="comment"> problem.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The final stage consists of computing the updated eigenvectors
</span><span class="comment">*</span><span class="comment"> directly using the updated eigenvalues. The eigenvectors for
</span><span class="comment">*</span><span class="comment"> the current problem are multiplied with the eigenvectors from
</span><span class="comment">*</span><span class="comment"> the overall problem.
</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"> ICOMPQ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> = 0: Compute eigenvalues only.
</span><span class="comment">*</span><span class="comment"> = 1: Compute eigenvectors of original dense symmetric matrix
</span><span class="comment">*</span><span class="comment"> also. On entry, Q contains the orthogonal matrix used
</span><span class="comment">*</span><span class="comment"> to reduce the original matrix to tridiagonal form.
</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 dimension of the symmetric tridiagonal matrix. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> QSIZ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The dimension of the orthogonal matrix used to reduce
</span><span class="comment">*</span><span class="comment"> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TLVLS (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The total number of merging levels in the overall divide and
</span><span class="comment">*</span><span class="comment"> conquer tree.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> CURLVL (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The current level in the overall merge routine,
</span><span class="comment">*</span><span class="comment"> 0 <= CURLVL <= TLVLS.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> CURPBM (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The current problem in the current level in the overall
</span><span class="comment">*</span><span class="comment"> merge routine (counting from upper left to lower right).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> D (input/output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On entry, the eigenvalues of the rank-1-perturbed matrix.
</span><span class="comment">*</span><span class="comment"> On exit, the eigenvalues of the repaired matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
</span><span class="comment">*</span><span class="comment"> On entry, the eigenvectors of the rank-1-perturbed matrix.
</span><span class="comment">*</span><span class="comment"> On exit, the eigenvectors of the repaired tridiagonal matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDQ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array Q. LDQ >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INDXQ (output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> The permutation which will reintegrate the subproblem just
</span><span class="comment">*</span><span class="comment"> solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
</span><span class="comment">*</span><span class="comment"> will be in ascending order.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RHO (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> The subdiagonal element used to create the rank-1
</span><span class="comment">*</span><span class="comment"> modification.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> CUTPNT (input) INTEGER
</span><span class="comment">*</span><span class="comment"> Contains the location of the last eigenvalue in the leading
</span><span class="comment">*</span><span class="comment"> sub-matrix. min(1,N) <= CUTPNT <= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> QSTORE (input/output) DOUBLE PRECISION array, dimension (N**2+1)
</span><span class="comment">*</span><span class="comment"> Stores eigenvectors of submatrices encountered during
</span><span class="comment">*</span><span class="comment"> divide and conquer, packed together. QPTR points to
</span><span class="comment">*</span><span class="comment"> beginning of the submatrices.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> QPTR (input/output) INTEGER array, dimension (N+2)
</span><span class="comment">*</span><span class="comment"> List of indices pointing to beginning of submatrices stored
</span><span class="comment">*</span><span class="comment"> in QSTORE. The submatrices are numbered starting at the
</span><span class="comment">*</span><span class="comment"> bottom left of the divide and conquer tree, from left to
</span><span class="comment">*</span><span class="comment"> right and bottom to top.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> PRMPTR (input) INTEGER array, dimension (N lg N)
</span><span class="comment">*</span><span class="comment"> Contains a list of pointers which indicate where in PERM a
</span><span class="comment">*</span><span class="comment"> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
</span><span class="comment">*</span><span class="comment"> indicates the size of the permutation and also the size of
</span><span class="comment">*</span><span class="comment"> the full, non-deflated problem.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> PERM (input) INTEGER array, dimension (N lg N)
</span><span class="comment">*</span><span class="comment"> Contains the permutations (from deflation and sorting) to be
</span><span class="comment">*</span><span class="comment"> applied to each eigenblock.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> GIVPTR (input) INTEGER array, dimension (N lg N)
</span><span class="comment">*</span><span class="comment"> Contains a list of pointers which indicate where in GIVCOL a
</span><span class="comment">*</span><span class="comment"> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
</span><span class="comment">*</span><span class="comment"> indicates the number of Givens rotations.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> GIVCOL (input) INTEGER array, dimension (2, N lg N)
</span><span class="comment">*</span><span class="comment"> Each pair of numbers indicates a pair of columns to take place
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