#include "ESMF_LapackBlas.inc" !> \brief \b DLAED7 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 !> Download DLAED7 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed7.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed7.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed7.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DLAED7( 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 ! DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. ! INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), ! $ IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) ! DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ), ! $ QSTORE( * ), WORK( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DLAED7 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. DLAED1 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 DLAED8. !> !> The second stage consists of calculating the updated !> eigenvalues. This is done by finding the roots of the secular !> equation via the routine DLAED4 (as called by DLAED9). !> 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION !> 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DLAED7( 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 DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), & IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ), & QSTORE( * ), WORK( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 ) ! .. ! .. Local Scalars .. INTEGER COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP, & IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DLAED8, DLAED9, DLAEDA, DLAMRG, 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( 'DLAED7', -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 DLAED8 and DLAED9. ! 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 DLAEDA( 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 DLAED8( 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 DLAED9( 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 DGEMM( '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 DLAMRG( 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 DLAED7 ! END