#include "ESMF_LapackBlas.inc" !> \brief \b DLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal. ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DLAED2 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed2.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed2.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed2.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W, ! Q2, INDX, INDXC, INDXP, COLTYP, INFO ) ! ! .. Scalar Arguments .. ! INTEGER INFO, K, LDQ, N, N1 ! DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. ! INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), ! $ INDXQ( * ) ! DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), ! $ W( * ), Z( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DLAED2 merges the two sets of eigenvalues together into a single !> sorted set. Then it tries to deflate the size of the problem. !> There are two ways in which deflation can occur: when two or more !> eigenvalues are close together or if there is a tiny entry in the !> Z vector. For each such occurrence the order of the related secular !> equation problem is reduced by one. !> \endverbatim ! ! Arguments: ! ========== ! !> \param[out] K !> \verbatim !> K is INTEGER !> The number of non-deflated eigenvalues, and the order of the !> related secular equation. 0 <= K <=N. !> \endverbatim !> !> \param[in] N !> \verbatim !> N is INTEGER !> The dimension of the symmetric tridiagonal matrix. N >= 0. !> \endverbatim !> !> \param[in] N1 !> \verbatim !> N1 is INTEGER !> The location of the last eigenvalue in the leading sub-matrix. !> min(1,N) <= N1 <= N/2. !> \endverbatim !> !> \param[in,out] D !> \verbatim !> D is DOUBLE PRECISION array, dimension (N) !> On entry, D contains the eigenvalues of the two submatrices to !> be combined. !> On exit, D contains the trailing (N-K) updated eigenvalues !> (those which were deflated) sorted into increasing order. !> \endverbatim !> !> \param[in,out] Q !> \verbatim !> Q is DOUBLE PRECISION array, dimension (LDQ, N) !> On entry, Q contains the eigenvectors of two submatrices in !> the two square blocks with corners at (1,1), (N1,N1) !> and (N1+1, N1+1), (N,N). !> On exit, Q contains the trailing (N-K) updated eigenvectors !> (those which were deflated) in its last N-K columns. !> \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) !> The permutation which separately sorts the two sub-problems !> in D into ascending order. Note that elements in the second !> half of this permutation must first have N1 added to their !> values. Destroyed on exit. !> \endverbatim !> !> \param[in,out] RHO !> \verbatim !> RHO is DOUBLE PRECISION !> On entry, the off-diagonal element associated with the rank-1 !> cut which originally split the two submatrices which are now !> being recombined. !> On exit, RHO has been modified to the value required by !> DLAED3. !> \endverbatim !> !> \param[in] Z !> \verbatim !> Z is DOUBLE PRECISION array, dimension (N) !> On entry, Z contains the updating vector (the last !> row of the first sub-eigenvector matrix and the first row of !> the second sub-eigenvector matrix). !> On exit, the contents of Z have been destroyed by the updating !> process. !> \endverbatim !> !> \param[out] DLAMDA !> \verbatim !> DLAMDA is DOUBLE PRECISION array, dimension (N) !> A copy of the first K eigenvalues which will be used by !> DLAED3 to form the secular equation. !> \endverbatim !> !> \param[out] W !> \verbatim !> W is DOUBLE PRECISION array, dimension (N) !> The first k values of the final deflation-altered z-vector !> which will be passed to DLAED3. !> \endverbatim !> !> \param[out] Q2 !> \verbatim !> Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2) !> A copy of the first K eigenvectors which will be used by !> DLAED3 in a matrix multiply (DGEMM) to solve for the new !> eigenvectors. !> \endverbatim !> !> \param[out] INDX !> \verbatim !> INDX is INTEGER array, dimension (N) !> The permutation used to sort the contents of DLAMDA into !> ascending order. !> \endverbatim !> !> \param[out] INDXC !> \verbatim !> INDXC is INTEGER array, dimension (N) !> The permutation used to arrange the columns of the deflated !> Q matrix into three groups: the first group contains non-zero !> elements only at and above N1, the second contains !> non-zero elements only below N1, and the third is dense. !> \endverbatim !> !> \param[out] INDXP !> \verbatim !> INDXP is INTEGER array, dimension (N) !> The permutation used to place deflated values of D at the end !> of the array. INDXP(1:K) points to the nondeflated D-values !> and INDXP(K+1:N) points to the deflated eigenvalues. !> \endverbatim !> !> \param[out] COLTYP !> \verbatim !> COLTYP is INTEGER array, dimension (N) !> During execution, a label which will indicate which of the !> following types a column in the Q2 matrix is: !> 1 : non-zero in the upper half only; !> 2 : dense; !> 3 : non-zero in the lower half only; !> 4 : deflated. !> On exit, COLTYP(i) is the number of columns of type i, !> for i=1 to 4 only. !> \endverbatim !> !> \param[out] INFO !> \verbatim !> INFO is INTEGER !> = 0: successful exit. !> < 0: if INFO = -i, the i-th argument had an illegal value. !> \endverbatim ! ! Authors: ! ======== ! !> \author Univ. of Tennessee !> \author Univ. of California Berkeley !> \author Univ. of Colorado Denver !> \author NAG Ltd. ! !> \date December 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 DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W, & Q2, INDX, INDXC, INDXP, COLTYP, 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..-- ! December 2016 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDQ, N, N1 DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), & INDXQ( * ) DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), & W( * ), Z( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0, & TWO = 2.0D0, EIGHT = 8.0D0 ) ! .. ! .. Local Arrays .. INTEGER CTOT( 4 ), PSM( 4 ) ! .. ! .. Local Scalars .. INTEGER CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1, & N2, NJ, PJ DOUBLE PRECISION C, EPS, S, T, TAU, TOL ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL IDAMAX, DLAMCH, DLAPY2 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLACPY, DLAMRG, DROT, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! 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 XERBLA( 'DLAED2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! N2 = N - N1 N1P1 = N1 + 1 ! IF( RHO.LT.ZERO ) THEN CALL DSCAL( N2, MONE, Z( N1P1 ), 1 ) END IF ! ! Normalize z so that norm(z) = 1. Since z is the concatenation of ! two normalized vectors, norm2(z) = sqrt(2). ! T = ONE / SQRT( TWO ) CALL DSCAL( N, T, Z, 1 ) ! ! RHO = ABS( norm(z)**2 * RHO ) ! RHO = ABS( TWO*RHO ) ! ! Sort the eigenvalues into increasing order ! DO 10 I = N1P1, N INDXQ( I ) = INDXQ( I ) + N1 10 CONTINUE ! ! re-integrate the deflated parts from the last pass ! DO 20 I = 1, N DLAMDA( I ) = D( INDXQ( I ) ) 20 CONTINUE CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDXC ) DO 30 I = 1, N INDX( I ) = INDXQ( INDXC( I ) ) 30 CONTINUE ! ! Calculate the allowable deflation tolerance ! IMAX = IDAMAX( N, Z, 1 ) JMAX = IDAMAX( N, D, 1 ) EPS = DLAMCH( 'Epsilon' ) TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) ) ! ! If the rank-1 modifier is small enough, no more needs to be done ! except to reorganize Q so that its columns correspond with the ! elements in D. ! IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN K = 0 IQ2 = 1 DO 40 J = 1, N I = INDX( J ) CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 ) DLAMDA( J ) = D( I ) IQ2 = IQ2 + N 40 CONTINUE CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ ) CALL DCOPY( N, DLAMDA, 1, D, 1 ) GO TO 190 END IF ! ! If there are multiple eigenvalues then the problem deflates. Here ! the number of equal eigenvalues are found. As each equal ! eigenvalue is found, an elementary reflector is computed to rotate ! the corresponding eigensubspace so that the corresponding ! components of Z are zero in this new basis. ! DO 50 I = 1, N1 COLTYP( I ) = 1 50 CONTINUE DO 60 I = N1P1, N COLTYP( I ) = 3 60 CONTINUE ! ! K = 0 K2 = N + 1 DO 70 J = 1, N NJ = INDX( J ) IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 COLTYP( NJ ) = 4 INDXP( K2 ) = NJ IF( J.EQ.N ) & GO TO 100 ELSE PJ = NJ GO TO 80 END IF 70 CONTINUE 80 CONTINUE J = J + 1 NJ = INDX( J ) IF( J.GT.N ) & GO TO 100 IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 COLTYP( NJ ) = 4 INDXP( K2 ) = NJ ELSE ! ! Check if eigenvalues are close enough to allow deflation. ! S = Z( PJ ) C = Z( NJ ) ! ! Find sqrt(a**2+b**2) without overflow or ! destructive underflow. ! TAU = DLAPY2( C, S ) T = D( NJ ) - D( PJ ) C = C / TAU S = -S / TAU IF( ABS( T*C*S ).LE.TOL ) THEN ! ! Deflation is possible. ! Z( NJ ) = TAU Z( PJ ) = ZERO IF( COLTYP( NJ ).NE.COLTYP( PJ ) ) & COLTYP( NJ ) = 2 COLTYP( PJ ) = 4 CALL DROT( N, Q( 1, PJ ), 1, Q( 1, NJ ), 1, C, S ) T = D( PJ )*C**2 + D( NJ )*S**2 D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2 D( PJ ) = T K2 = K2 - 1 I = 1 90 CONTINUE IF( K2+I.LE.N ) THEN IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN INDXP( K2+I-1 ) = INDXP( K2+I ) INDXP( K2+I ) = PJ I = I + 1 GO TO 90 ELSE INDXP( K2+I-1 ) = PJ END IF ELSE INDXP( K2+I-1 ) = PJ END IF PJ = NJ ELSE K = K + 1 DLAMDA( K ) = D( PJ ) W( K ) = Z( PJ ) INDXP( K ) = PJ PJ = NJ END IF END IF GO TO 80 100 CONTINUE ! ! Record the last eigenvalue. ! K = K + 1 DLAMDA( K ) = D( PJ ) W( K ) = Z( PJ ) INDXP( K ) = PJ ! ! Count up the total number of the various types of columns, then ! form a permutation which positions the four column types into ! four uniform groups (although one or more of these groups may be ! empty). ! DO 110 J = 1, 4 CTOT( J ) = 0 110 CONTINUE DO 120 J = 1, N CT = COLTYP( J ) CTOT( CT ) = CTOT( CT ) + 1 120 CONTINUE ! ! PSM(*) = Position in SubMatrix (of types 1 through 4) ! PSM( 1 ) = 1 PSM( 2 ) = 1 + CTOT( 1 ) PSM( 3 ) = PSM( 2 ) + CTOT( 2 ) PSM( 4 ) = PSM( 3 ) + CTOT( 3 ) K = N - CTOT( 4 ) ! ! Fill out the INDXC array so that the permutation which it induces ! will place all type-1 columns first, all type-2 columns next, ! then all type-3's, and finally all type-4's. ! DO 130 J = 1, N JS = INDXP( J ) CT = COLTYP( JS ) INDX( PSM( CT ) ) = JS INDXC( PSM( CT ) ) = J PSM( CT ) = PSM( CT ) + 1 130 CONTINUE ! ! Sort the eigenvalues and corresponding eigenvectors into DLAMDA ! and Q2 respectively. The eigenvalues/vectors which were not ! deflated go into the first K slots of DLAMDA and Q2 respectively, ! while those which were deflated go into the last N - K slots. ! I = 1 IQ1 = 1 IQ2 = 1 + ( CTOT( 1 )+CTOT( 2 ) )*N1 DO 140 J = 1, CTOT( 1 ) JS = INDX( I ) CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 ) Z( I ) = D( JS ) I = I + 1 IQ1 = IQ1 + N1 140 CONTINUE ! DO 150 J = 1, CTOT( 2 ) JS = INDX( I ) CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 ) CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 ) Z( I ) = D( JS ) I = I + 1 IQ1 = IQ1 + N1 IQ2 = IQ2 + N2 150 CONTINUE ! DO 160 J = 1, CTOT( 3 ) JS = INDX( I ) CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 ) Z( I ) = D( JS ) I = I + 1 IQ2 = IQ2 + N2 160 CONTINUE ! IQ1 = IQ2 DO 170 J = 1, CTOT( 4 ) JS = INDX( I ) CALL DCOPY( N, Q( 1, JS ), 1, Q2( IQ2 ), 1 ) IQ2 = IQ2 + N Z( I ) = D( JS ) I = I + 1 170 CONTINUE ! ! The deflated eigenvalues and their corresponding vectors go back ! into the last N - K slots of D and Q respectively. ! IF( K.LT.N ) THEN CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N, & Q( 1, K+1 ), LDQ ) CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 ) END IF ! ! Copy CTOT into COLTYP for referencing in DLAED3. ! DO 180 J = 1, 4 COLTYP( J ) = CTOT( J ) 180 CONTINUE ! 190 CONTINUE RETURN ! ! End of DLAED2 ! END