#include "ESMF_LapackBlas.inc" !> \brief \b DLAED1 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 tridiagonal. ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DLAED1 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed1.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed1.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed1.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, ! INFO ) ! ! .. Scalar Arguments .. ! INTEGER CUTPNT, INFO, LDQ, N ! DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. ! INTEGER INDXQ( * ), IWORK( * ) ! DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DLAED1 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. DLAED7 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 occurrence the dimension of the !> secular equation problem is reduced by one. This stage is !> performed by the routine DLAED2. !> !> 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 DLAED3). !> 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] N !> \verbatim !> N is INTEGER !> The dimension of the symmetric tridiagonal matrix. N >= 0. !> \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[in,out] INDXQ !> \verbatim !> INDXQ is 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. !> \endverbatim !> !> \param[in] RHO !> \verbatim !> RHO is DOUBLE PRECISION !> The subdiagonal entry used to create the rank-1 modification. !> \endverbatim !> !> \param[in] CUTPNT !> \verbatim !> CUTPNT is INTEGER !> The location of the last eigenvalue in the leading sub-matrix. !> min(1,N) <= CUTPNT <= N/2. !> \endverbatim !> !> \param[out] WORK !> \verbatim !> WORK is DOUBLE PRECISION array, dimension (4*N + N**2) !> \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 \n !> Modified by Francoise Tisseur, University of Tennessee !> ! ===================================================================== SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, 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 CUTPNT, INFO, LDQ, N DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. INTEGER INDXQ( * ), IWORK( * ) DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * ) ! .. ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER COLTYP, I, IDLMDA, INDX, INDXC, INDXP, IQ2, IS, & IW, IZ, K, N1, N2, ZPP1 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLAED2, DLAED3, DLAMRG, 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( 'DLAED1', -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 DLAED2 and DLAED3. ! 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 DCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 ) ZPP1 = CUTPNT + 1 CALL DCOPY( N-CUTPNT, Q( ZPP1, ZPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 ) ! ! Deflate eigenvalues. ! CALL DLAED2( 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 DLAED3( 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 DLAMRG( N1, N2, D, 1, -1, INDXQ ) ELSE DO 10 I = 1, N INDXQ( I ) = I 10 CONTINUE END IF ! 20 CONTINUE RETURN ! ! End of DLAED1 ! END