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SUBROUTINE <a name="DLAED2.1"></a><a href="dlaed2.f.html#DLAED2.1">DLAED2</a>( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
$ Q2, INDX, INDXC, INDXP, COLTYP, 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 INFO, K, LDQ, N, N1
DOUBLE PRECISION RHO
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
$ INDXQ( * )
DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
$ W( * ), Z( * )
<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="DLAED2.22"></a><a href="dlaed2.f.html#DLAED2.1">DLAED2</a> merges the two sets of eigenvalues together into a single
</span><span class="comment">*</span><span class="comment"> sorted set. Then it tries to deflate the size of the problem.
</span><span class="comment">*</span><span class="comment"> There are two ways in which deflation can occur: when two or more
</span><span class="comment">*</span><span class="comment"> eigenvalues are close together or if there is a tiny entry in the
</span><span class="comment">*</span><span class="comment"> Z vector. For each such occurrence the order of the related secular
</span><span class="comment">*</span><span class="comment"> equation problem is reduced by one.
</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"> K (output) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of non-deflated eigenvalues, and the order of the
</span><span class="comment">*</span><span class="comment"> related secular equation. 0 <= K <=N.
</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"> N1 (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) <= N1 <= N/2.
</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, D contains the eigenvalues of the two submatrices to
</span><span class="comment">*</span><span class="comment"> be combined.
</span><span class="comment">*</span><span class="comment"> On exit, D contains the trailing (N-K) updated eigenvalues
</span><span class="comment">*</span><span class="comment"> (those which were deflated) sorted into increasing order.
</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, Q contains the eigenvectors of two submatrices in
</span><span class="comment">*</span><span class="comment"> the two square blocks with corners at (1,1), (N1,N1)
</span><span class="comment">*</span><span class="comment"> and (N1+1, N1+1), (N,N).
</span><span class="comment">*</span><span class="comment"> On exit, Q contains the trailing (N-K) updated eigenvectors
</span><span class="comment">*</span><span class="comment"> (those which were deflated) in its last N-K columns.
</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"> The permutation which separately sorts the two sub-problems
</span><span class="comment">*</span><span class="comment"> in D into ascending order. Note that elements in the second
</span><span class="comment">*</span><span class="comment"> half of this permutation must first have N1 added to their
</span><span class="comment">*</span><span class="comment"> values. Destroyed on exit.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RHO (input/output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> On entry, the off-diagonal element associated with the rank-1
</span><span class="comment">*</span><span class="comment"> cut which originally split the two submatrices which are now
</span><span class="comment">*</span><span class="comment"> being recombined.
</span><span class="comment">*</span><span class="comment"> On exit, RHO has been modified to the value required by
</span><span class="comment">*</span><span class="comment"> <a name="DLAED3.70"></a><a href="dlaed3.f.html#DLAED3.1">DLAED3</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z (input) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On entry, Z contains the updating vector (the last
</span><span class="comment">*</span><span class="comment"> row of the first sub-eigenvector matrix and the first row of
</span><span class="comment">*</span><span class="comment"> the second sub-eigenvector matrix).
</span><span class="comment">*</span><span class="comment"> On exit, the contents of Z have been destroyed by the updating
</span><span class="comment">*</span><span class="comment"> process.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> DLAMDA (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> A copy of the first K eigenvalues which will be used by
</span><span class="comment">*</span><span class="comment"> <a name="DLAED3.81"></a><a href="dlaed3.f.html#DLAED3.1">DLAED3</a> to form the secular equation.
</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"> The first k values of the final deflation-altered z-vector
</span><span class="comment">*</span><span class="comment"> which will be passed to <a name="DLAED3.85"></a><a href="dlaed3.f.html#DLAED3.1">DLAED3</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q2 (output) DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
</span><span class="comment">*</span><span class="comment"> A copy of the first K eigenvectors which will be used by
</span><span class="comment">*</span><span class="comment"> <a name="DLAED3.89"></a><a href="dlaed3.f.html#DLAED3.1">DLAED3</a> in a matrix multiply (DGEMM) to solve for the new
</span><span class="comment">*</span><span class="comment"> eigenvectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INDX (workspace) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> The permutation used to sort the contents of DLAMDA into
</span><span class="comment">*</span><span class="comment"> ascending order.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INDXC (output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> The permutation used to arrange the columns of the deflated
</span><span class="comment">*</span><span class="comment"> Q matrix into three groups: the first group contains non-zero
</span><span class="comment">*</span><span class="comment"> elements only at and above N1, the second contains
</span><span class="comment">*</span><span class="comment"> non-zero elements only below N1, and the third is dense.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INDXP (workspace) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> The permutation used to place deflated values of D at the end
</span><span class="comment">*</span><span class="comment"> of the array. INDXP(1:K) points to the nondeflated D-values
</span><span class="comment">*</span><span class="comment"> and INDXP(K+1:N) points to the deflated eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> COLTYP (workspace/output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> During execution, a label which will indicate which of the
</span><span class="comment">*</span><span class="comment"> following types a column in the Q2 matrix is:
</span><span class="comment">*</span><span class="comment"> 1 : non-zero in the upper half only;
</span><span class="comment">*</span><span class="comment"> 2 : dense;
</span><span class="comment">*</span><span class="comment"> 3 : non-zero in the lower half only;
</span><span class="comment">*</span><span class="comment"> 4 : deflated.
</span><span class="comment">*</span><span class="comment"> On exit, COLTYP(i) is the number of columns of type i,
</span><span class="comment">*</span><span class="comment"> for i=1 to 4 only.
</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">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Based on contributions by
</span><span class="comment">*</span><span class="comment"> Jeff Rutter, Computer Science Division, University of California
</span><span class="comment">*</span><span class="comment"> at Berkeley, USA
</span><span class="comment">*</span><span class="comment"> Modified by Francoise Tisseur, University of Tennessee.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, EIGHT = 8.0D0 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Arrays ..
</span> INTEGER CTOT( 4 ), PSM( 4 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> INTEGER CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
$ N2, NJ, PJ
DOUBLE PRECISION C, EPS, S, T, TAU, TOL
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Functions ..
</span> INTEGER IDAMAX
DOUBLE PRECISION <a name="DLAMCH.146"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, <a name="DLAPY2.146"></a><a href="dlapy2.f.html#DLAPY2.1">DLAPY2</a>
EXTERNAL IDAMAX, <a name="DLAMCH.147"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, <a name="DLAPY2.147"></a><a href="dlapy2.f.html#DLAPY2.1">DLAPY2</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Subroutines ..
</span> EXTERNAL DCOPY, <a name="DLACPY.150"></a><a href="dlacpy.f.html#DLACPY.1">DLACPY</a>, <a name="DLAMRG.150"></a><a href="dlamrg.f.html#DLAMRG.1">DLAMRG</a>, DROT, DSCAL, <a name="XERBLA.150"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
</span> INTRINSIC ABS, MAX, MIN, SQRT
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Test the input parameters.
</span><span class="comment">*</span><span class="comment">
</span> INFO = 0
<span class="comment">*</span><span class="comment">
</span> IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( MIN( 1, ( N / 2 ) ).GT.N1 .OR. ( N / 2 ).LT.N1 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL <a name="XERBLA.169"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>( <span class="string">'<a name="DLAED2.169"></a><a href="dlaed2.f.html#DLAED2.1">DLAED2</a>'</span>, -INFO )
RETURN
END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Quick return if possible
</span><span class="comment">*</span><span class="comment">
</span> IF( N.EQ.0 )
$ RETURN
<span class="comment">*</span><span class="comment">
</span> N2 = N - N1
N1P1 = N1 + 1
<span class="comment">*</span><span class="comment">
</span> IF( RHO.LT.ZERO ) THEN
CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Normalize z so that norm(z) = 1. Since z is the concatenation of
</span><span class="comment">*</span><span class="comment"> two normalized vectors, norm2(z) = sqrt(2).
</span><span class="comment">*</span><span class="comment">
</span> T = ONE / SQRT( TWO )
CALL DSCAL( N, T, Z, 1 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RHO = ABS( norm(z)**2 * RHO )
</span><span class="comment">*</span><span class="comment">
</span> RHO = ABS( TWO*RHO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Sort the eigenvalues into increasing order
</span><span class="comment">*</span><span class="comment">
</span> DO 10 I = N1P1, N
INDXQ( I ) = INDXQ( I ) + N1
10 CONTINUE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> re-integrate the deflated parts from the last pass
</span><span class="comment">*</span><span class="comment">
</span> DO 20 I = 1, N
DLAMDA( I ) = D( INDXQ( I ) )
20 CONTINUE
CALL <a name="DLAMRG.206"></a><a href="dlamrg.f.html#DLAMRG.1">DLAMRG</a>( N1, N2, DLAMDA, 1, 1, INDXC )
DO 30 I = 1, N
INDX( I ) = INDXQ( INDXC( I ) )
30 CONTINUE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Calculate the allowable deflation tolerance
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