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SUBROUTINE <a name="SLAED1.1"></a><a href="slaed1.f.html#SLAED1.1">SLAED1</a>( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, 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 CUTPNT, INFO, LDQ, N
REAL RHO
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER INDXQ( * ), IWORK( * )
REAL D( * ), Q( LDQ, * ), 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="SLAED1.20"></a><a href="slaed1.f.html#SLAED1.1">SLAED1</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 eigenvectors of a tridiagonal matrix. <a name="SLAED7.23"></a><a href="slaed7.f.html#SLAED7.1">SLAED7</a> handles
</span><span class="comment">*</span><span class="comment"> the case in which eigenvalues only or eigenvalues and eigenvectors
</span><span class="comment">*</span><span class="comment"> of a full symmetric matrix (which was reduced to tridiagonal form)
</span><span class="comment">*</span><span class="comment"> 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="SLAED2.40"></a><a href="slaed2.f.html#SLAED2.1">SLAED2</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="SLAED4.44"></a><a href="slaed4.f.html#SLAED4.1">SLAED4</a> (as called by <a name="SLAED3.44"></a><a href="slaed3.f.html#SLAED3.1">SLAED3</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"> 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"> D (input/output) REAL 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) REAL 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 (input/output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On entry, the permutation which separately sorts the two
</span><span class="comment">*</span><span class="comment"> subproblems in D into ascending order.
</span><span class="comment">*</span><span class="comment"> On exit, the permutation which will reintegrate the
</span><span class="comment">*</span><span class="comment"> subproblems back into sorted order,
</span><span class="comment">*</span><span class="comment"> i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RHO (input) REAL
</span><span class="comment">*</span><span class="comment"> The subdiagonal entry used to create the rank-1 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"> The location of the last eigenvalue in the leading sub-matrix.
</span><span class="comment">*</span><span class="comment"> min(1,N) <= CUTPNT <= N/2.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WORK (workspace) REAL array, dimension (4*N + N**2)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IWORK (workspace) INTEGER array, dimension (4*N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INFO (output) INTEGER
</span><span class="comment">*</span><span class="comment"> = 0: successful exit.
</span><span class="comment">*</span><span class="comment"> < 0: if INFO = -i, the i-th argument had an illegal value.
</span><span class="comment">*</span><span class="comment"> > 0: if INFO = 1, an eigenvalue did not converge
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