*> \brief \b SLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaed7.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
* LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
* PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
* $ QSIZ, TLVLS
* REAL RHO
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
* $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
* REAL D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
* $ QSTORE( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLAED7 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 optionally eigenvectors of a dense symmetric matrix
*> that has been reduced to tridiagonal form. SLAED1 handles
*> the case in which all eigenvalues and eigenvectors of a symmetric
*> tridiagonal matrix are desired.
*>
*> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
*>
*> where Z = Q**Tu, 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 occurrence the dimension of the
*> secular equation problem is reduced by one. This stage is
*> performed by the routine SLAED8.
*>
*> 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 SLAED9).
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> = 0: Compute eigenvalues only.
*> = 1: Compute eigenvectors of original dense symmetric matrix
*> also. On entry, Q contains the orthogonal matrix used
*> to reduce the original matrix to tridiagonal form.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the symmetric tridiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*> QSIZ is INTEGER
*> The dimension of the orthogonal matrix used to reduce
*> the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in] TLVLS
*> \verbatim
*> TLVLS is INTEGER
*> The total number of merging levels in the overall divide and
*> conquer tree.
*> \endverbatim
*>
*> \param[in] CURLVL
*> \verbatim
*> CURLVL is INTEGER
*> The current level in the overall merge routine,
*> 0 <= CURLVL <= TLVLS.
*> \endverbatim
*>
*> \param[in] CURPBM
*> \verbatim
*> CURPBM is INTEGER
*> The current problem in the current level in the overall
*> merge routine (counting from upper left to lower right).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is REAL array, dimension (N)
*> On entry, the eigenvalues of the rank-1-perturbed matrix.
*> On exit, the eigenvalues of the repaired matrix.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is REAL array, dimension (LDQ, N)
*> On entry, the eigenvectors of the rank-1-perturbed matrix.
*> On exit, the eigenvectors of the repaired tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[out] INDXQ
*> \verbatim
*> INDXQ is INTEGER array, dimension (N)
*> The permutation which will reintegrate the subproblem just
*> solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
*> will be in ascending order.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is REAL
*> The subdiagonal element used to create the rank-1
*> modification.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*> CUTPNT is INTEGER
*> Contains the location of the last eigenvalue in the leading
*> sub-matrix. min(1,N) <= CUTPNT <= N.
*> \endverbatim
*>
*> \param[in,out] QSTORE
*> \verbatim
*> QSTORE is REAL array, dimension (N**2+1)
*> Stores eigenvectors of submatrices encountered during
*> divide and conquer, packed together. QPTR points to
*> beginning of the submatrices.
*> \endverbatim
*>
*> \param[in,out] QPTR
*> \verbatim
*> QPTR is INTEGER array, dimension (N+2)
*> List of indices pointing to beginning of submatrices stored
*> in QSTORE. The submatrices are numbered starting at the
*> bottom left of the divide and conquer tree, from left to
*> right and bottom to top.
*> \endverbatim
*>
*> \param[in] PRMPTR
*> \verbatim
*> PRMPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in PERM a
*> level's permutation is stored. PRMPTR(i+1) - PRMPTR(i)
*> indicates the size of the permutation and also the size of
*> the full, non-deflated problem.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*> PERM is INTEGER array, dimension (N lg N)
*> Contains the permutations (from deflation and sorting) to be
*> applied to each eigenblock.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER array, dimension (N lg N)
*> Contains a list of pointers which indicate where in GIVCOL a
*> level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i)
*> indicates the number of Givens rotations.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension (2, N lg N)
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*> GIVNUM is REAL array, dimension (2, N lg N)
*> Each number indicates the S value to be used in the
*> corresponding Givens rotation.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (3*N+2*QSIZ*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is 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
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE SLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
$ LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
$ PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
$ INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
$ QSIZ, TLVLS
REAL RHO
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
$ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
REAL D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
$ QSTORE( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E0, ZERO = 0.0E0 )
* ..
* .. Local Scalars ..
INTEGER COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP,
$ IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR
* ..
* .. External Subroutines ..
EXTERNAL SGEMM, SLAED8, SLAED9, SLAEDA, SLAMRG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
INFO = -3
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLAED7', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* The following values are for bookkeeping purposes only. They are
* integer pointers which indicate the portion of the workspace
* used by a particular array in SLAED8 and SLAED9.
*
IF( ICOMPQ.EQ.1 ) THEN
LDQ2 = QSIZ
ELSE
LDQ2 = N
END IF
*
IZ = 1
IDLMDA = IZ + N
IW = IDLMDA + N
IQ2 = IW + N
IS = IQ2 + N*LDQ2
*
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.
*
PTR = 1 + 2**TLVLS
DO 10 I = 1, CURLVL - 1
PTR = PTR + 2**( TLVLS-I )
10 CONTINUE
CURR = PTR + CURPBM
CALL SLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
$ GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ),
$ WORK( IZ+N ), INFO )
*
* When solving the final problem, we no longer need the stored data,
* so we will overwrite the data from this level onto the previously
* used storage space.
*
IF( CURLVL.EQ.TLVLS ) THEN
QPTR( CURR ) = 1
PRMPTR( CURR ) = 1
GIVPTR( CURR ) = 1
END IF
*
* Sort and Deflate eigenvalues.
*
CALL SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
$ WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2,
$ WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ),
$ GIVCOL( 1, GIVPTR( CURR ) ),
$ GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ),
$ IWORK( INDX ), INFO )
PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N
GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR )
*
* Solve Secular Equation.
*
IF( K.NE.0 ) THEN
CALL SLAED9( K, 1, K, N, D, WORK( IS ), K, RHO, WORK( IDLMDA ),
$ WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO )
IF( INFO.NE.0 )
$ GO TO 30
IF( ICOMPQ.EQ.1 ) THEN
CALL SGEMM( 'N', 'N', QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2,
$ QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ )
END IF
QPTR( CURR+1 ) = QPTR( CURR ) + K**2
*
* Prepare the INDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL SLAMRG( N1, N2, D, 1, -1, INDXQ )
ELSE
QPTR( CURR+1 ) = QPTR( CURR )
DO 20 I = 1, N
INDXQ( I ) = I
20 CONTINUE
END IF
*
30 CONTINUE
RETURN
*
* End of SLAED7
*
END