#include "ESMF_LapackBlas.inc" !> \brief \b DLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense. ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DLAED8 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed8.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed8.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed8.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, ! CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, ! GIVCOL, GIVNUM, INDXP, INDX, INFO ) ! ! .. Scalar Arguments .. ! INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, ! $ QSIZ ! DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. ! INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), ! $ INDXQ( * ), PERM( * ) ! DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), ! $ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DLAED8 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 element in the !> Z vector. For each such occurrence the order of the related secular !> equation problem is reduced by one. !> \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[out] K !> \verbatim !> K is INTEGER !> The number of non-deflated eigenvalues, and the order of the !> related secular equation. !> \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,out] D !> \verbatim !> D is DOUBLE PRECISION array, dimension (N) !> On entry, the eigenvalues of the two submatrices to be !> combined. On exit, 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) !> If ICOMPQ = 0, Q is not referenced. Otherwise, !> on entry, Q contains the eigenvectors of the partially solved !> system which has been previously updated in matrix !> multiplies with other partially solved eigensystems. !> 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] 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 CUTPNT added to !> their values in order to be accurate. !> \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] CUTPNT !> \verbatim !> CUTPNT is INTEGER !> The location of the last eigenvalue in the leading !> sub-matrix. min(1,N) <= CUTPNT <= N. !> \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 are 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] Q2 !> \verbatim !> Q2 is DOUBLE PRECISION array, dimension (LDQ2,N) !> If ICOMPQ = 0, Q2 is not referenced. Otherwise, !> a copy of the first K eigenvectors which will be used by !> DLAED7 in a matrix multiply (DGEMM) to update the new !> eigenvectors. !> \endverbatim !> !> \param[in] LDQ2 !> \verbatim !> LDQ2 is INTEGER !> The leading dimension of the array Q2. LDQ2 >= max(1,N). !> \endverbatim !> !> \param[out] W !> \verbatim !> W is DOUBLE PRECISION array, dimension (N) !> The first k values of the final deflation-altered z-vector and !> will be passed to DLAED3. !> \endverbatim !> !> \param[out] PERM !> \verbatim !> PERM is INTEGER array, dimension (N) !> The permutations (from deflation and sorting) to be applied !> to each eigenblock. !> \endverbatim !> !> \param[out] GIVPTR !> \verbatim !> GIVPTR is INTEGER !> The number of Givens rotations which took place in this !> subproblem. !> \endverbatim !> !> \param[out] GIVCOL !> \verbatim !> GIVCOL is INTEGER array, dimension (2, N) !> Each pair of numbers indicates a pair of columns to take place !> in a Givens rotation. !> \endverbatim !> !> \param[out] GIVNUM !> \verbatim !> GIVNUM is DOUBLE PRECISION array, dimension (2, N) !> Each number indicates the S value to be used in the !> corresponding Givens rotation. !> \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] INDX !> \verbatim !> INDX is INTEGER array, dimension (N) !> The permutation used to sort the contents of D into ascending !> order. !> \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 ! ! ===================================================================== SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, & CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, & GIVCOL, GIVNUM, INDXP, INDX, 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 CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, & QSIZ DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), & INDXQ( * ), PERM( * ) DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), & Q( LDQ, * ), Q2( LDQ2, * ), 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 Scalars .. ! INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2 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( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN INFO = -4 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN INFO = -10 ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN INFO = -14 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAED8', -INFO ) RETURN END IF ! ! Need to initialize GIVPTR to O here in case of quick exit ! to prevent an unspecified code behavior (usually sigfault) ! when IWORK array on entry to *stedc is not zeroed ! (or at least some IWORK entries which used in *laed7 for GIVPTR). ! GIVPTR = 0 ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! N1 = CUTPNT 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 ! T = ONE / SQRT( TWO ) DO 10 J = 1, N INDX( J ) = J 10 CONTINUE CALL DSCAL( N, T, Z, 1 ) RHO = ABS( TWO*RHO ) ! ! Sort the eigenvalues into increasing order ! DO 20 I = CUTPNT + 1, N INDXQ( I ) = INDXQ( I ) + CUTPNT 20 CONTINUE DO 30 I = 1, N DLAMDA( I ) = D( INDXQ( I ) ) W( I ) = Z( INDXQ( I ) ) 30 CONTINUE I = 1 J = CUTPNT + 1 CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDX ) DO 40 I = 1, N D( I ) = DLAMDA( INDX( I ) ) Z( I ) = W( INDX( I ) ) 40 CONTINUE ! ! Calculate the allowable deflation tolerence ! IMAX = IDAMAX( N, Z, 1 ) JMAX = IDAMAX( N, D, 1 ) EPS = DLAMCH( 'Epsilon' ) TOL = EIGHT*EPS*ABS( D( JMAX ) ) ! ! 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 IF( ICOMPQ.EQ.0 ) THEN DO 50 J = 1, N PERM( J ) = INDXQ( INDX( J ) ) 50 CONTINUE ELSE DO 60 J = 1, N PERM( J ) = INDXQ( INDX( J ) ) CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 ) 60 CONTINUE CALL DLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ), & LDQ ) END IF RETURN 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. ! K = 0 K2 = N + 1 DO 70 J = 1, N IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 INDXP( K2 ) = J IF( J.EQ.N ) & GO TO 110 ELSE JLAM = J GO TO 80 END IF 70 CONTINUE 80 CONTINUE J = J + 1 IF( J.GT.N ) & GO TO 100 IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 INDXP( K2 ) = J ELSE ! ! Check if eigenvalues are close enough to allow deflation. ! S = Z( JLAM ) C = Z( J ) ! ! Find sqrt(a**2+b**2) without overflow or ! destructive underflow. ! TAU = DLAPY2( C, S ) T = D( J ) - D( JLAM ) C = C / TAU S = -S / TAU IF( ABS( T*C*S ).LE.TOL ) THEN ! ! Deflation is possible. ! Z( J ) = TAU Z( JLAM ) = ZERO ! ! Record the appropriate Givens rotation ! GIVPTR = GIVPTR + 1 GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) ) GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) ) GIVNUM( 1, GIVPTR ) = C GIVNUM( 2, GIVPTR ) = S IF( ICOMPQ.EQ.1 ) THEN CALL DROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1, & Q( 1, INDXQ( INDX( J ) ) ), 1, C, S ) END IF T = D( JLAM )*C*C + D( J )*S*S D( J ) = D( JLAM )*S*S + D( J )*C*C D( JLAM ) = T K2 = K2 - 1 I = 1 90 CONTINUE IF( K2+I.LE.N ) THEN IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN INDXP( K2+I-1 ) = INDXP( K2+I ) INDXP( K2+I ) = JLAM I = I + 1 GO TO 90 ELSE INDXP( K2+I-1 ) = JLAM END IF ELSE INDXP( K2+I-1 ) = JLAM END IF JLAM = J ELSE K = K + 1 W( K ) = Z( JLAM ) DLAMDA( K ) = D( JLAM ) INDXP( K ) = JLAM JLAM = J END IF END IF GO TO 80 100 CONTINUE ! ! Record the last eigenvalue. ! K = K + 1 W( K ) = Z( JLAM ) DLAMDA( K ) = D( JLAM ) INDXP( K ) = JLAM ! 110 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. ! IF( ICOMPQ.EQ.0 ) THEN DO 120 J = 1, N JP = INDXP( J ) DLAMDA( J ) = D( JP ) PERM( J ) = INDXQ( INDX( JP ) ) 120 CONTINUE ELSE DO 130 J = 1, N JP = INDXP( J ) DLAMDA( J ) = D( JP ) PERM( J ) = INDXQ( INDX( JP ) ) CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 ) 130 CONTINUE END IF ! ! 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 IF( ICOMPQ.EQ.0 ) THEN CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 ) ELSE CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 ) CALL DLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2, & Q( 1, K+1 ), LDQ ) END IF END IF ! RETURN ! ! End of DLAED8 ! END