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SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
$ INFO )
*
* -- LAPACK routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
INTEGER CUTPNT, INFO, LDQ, N
REAL RHO
* ..
* .. Array Arguments ..
INTEGER INDXQ( * ), IWORK( * )
REAL D( * ), Q( LDQ, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SLAED1 computes the updated eigensystem of a diagonal
* matrix after modification by a rank-one symmetric matrix. This
* routine is used only for the eigenproblem which requires all
* eigenvalues and eigenvectors of a tridiagonal matrix. SLAED7 handles
* the case in which eigenvalues only or eigenvalues and eigenvectors
* of a full symmetric matrix (which was reduced to tridiagonal form)
* are desired.
*
* T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
*
* where Z = Q**T*u, u is a vector of length N with ones in the
* CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
*
* The eigenvectors of the original matrix are stored in Q, and the
* eigenvalues are in D. The algorithm consists of three stages:
*
* The first stage consists of deflating the size of the problem
* when there are multiple eigenvalues or if there is a zero in
* the Z vector. For each such occurence the dimension of the
* secular equation problem is reduced by one. This stage is
* performed by the routine SLAED2.
*
* The second stage consists of calculating the updated
* eigenvalues. This is done by finding the roots of the secular
* equation via the routine SLAED4 (as called by SLAED3).
* This routine also calculates the eigenvectors of the current
* problem.
*
* The final stage consists of computing the updated eigenvectors
* directly using the updated eigenvalues. The eigenvectors for
* the current problem are multiplied with the eigenvectors from
* the overall problem.
*
* Arguments
* =========
*
* N (input) INTEGER
* The dimension of the symmetric tridiagonal matrix. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the eigenvalues of the rank-1-perturbed matrix.
* On exit, the eigenvalues of the repaired matrix.
*
* Q (input/output) REAL array, dimension (LDQ,N)
* On entry, the eigenvectors of the rank-1-perturbed matrix.
* On exit, the eigenvectors of the repaired tridiagonal matrix.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= max(1,N).
*
* INDXQ (input/output) INTEGER array, dimension (N)
* On entry, the permutation which separately sorts the two
* subproblems in D into ascending order.
* On exit, the permutation which will reintegrate the
* subproblems back into sorted order,
* i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
*
* RHO (input) REAL
* The subdiagonal entry used to create the rank-1 modification.
*
* CUTPNT (input) INTEGER
* The location of the last eigenvalue in the leading sub-matrix.
* min(1,N) <= CUTPNT <= N/2.
*
* WORK (workspace) REAL array, dimension (4*N + N**2)
*
* IWORK (workspace) INTEGER array, dimension (4*N)
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = 1, an eigenvalue did not converge
*
* Further Details
* ===============
*
* Based on contributions by
* Jeff Rutter, Computer Science Division, University of California
* at Berkeley, USA
* Modified by Francoise Tisseur, University of Tennessee.
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER COLTYP, CPP1, I, IDLMDA, INDX, INDXC, INDXP,
$ IQ2, IS, IW, IZ, K, N1, N2
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLAED2, SLAED3, SLAMRG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLAED1', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* The following values are integer pointers which indicate
* the portion of the workspace
* used by a particular array in SLAED2 and SLAED3.
*
IZ = 1
IDLMDA = IZ + N
IW = IDLMDA + N
IQ2 = IW + N
*
INDX = 1
INDXC = INDX + N
COLTYP = INDXC + N
INDXP = COLTYP + N
*
*
* Form the z-vector which consists of the last row of Q_1 and the
* first row of Q_2.
*
CALL SCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
CPP1 = CUTPNT + 1
CALL SCOPY( N-CUTPNT, Q( CPP1, CPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 )
*
* Deflate eigenvalues.
*
CALL SLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
$ WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
$ IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
$ IWORK( COLTYP ), INFO )
*
IF( INFO.NE.0 )
$ GO TO 20
*
* Solve Secular Equation.
*
IF( K.NE.0 ) THEN
IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
$ ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
CALL SLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
$ WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
$ WORK( IW ), WORK( IS ), INFO )
IF( INFO.NE.0 )
$ GO TO 20
*
* Prepare the INDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL SLAMRG( N1, N2, D, 1, -1, INDXQ )
ELSE
DO 10 I = 1, N
INDXQ( I ) = I
10 CONTINUE
END IF
*
20 CONTINUE
RETURN
*
* End of SLAED1
*
END
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